# Solved papers for JEE Main & Advanced Physics Rotational Motion JEE PYQ-Rotational Motion

### done JEE PYQ-Rotational Motion Total Questions - 122

• question_answer1) Two identical particles move towards each other with velocity 2v and v respectively. The velocity of centre of mass is                                                                                                                                        [AIEEE 2002]

A)
v

B)
$v/3$

C)
$v/2$

D)
zero

• question_answer2) Initial angular velocity of a circular disc of mass M is C${{\omega }_{\,1}}$. Then, two small spheres of mass m are attached gently to two diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?                                                                                                                                             [AIEEE 2002]

A)
$\left( \frac{M+m}{M} \right){{\omega }_{1}}$

B)
$\left( \frac{M+m}{m} \right){{\omega }_{1}}$

C)
$\left( \frac{M}{M+4m} \right){{\omega }_{1}}$

D)
$\left( \frac{M}{M+2m} \right){{\omega }_{1}}$

• question_answer3) A solid sphere, a hollow sphere and a ring are released from top of an inclined plane (frictionless) so that they slide down the plane. Then, maximum acceleration down the plane is for (no rolling)                             [AIEEE 2002]

A)
solid sphere

B)
hollow sphere

C)
ring

D)
All same

• question_answer4) Moment of inertia of a circular wire of mass M and radius R about its diameter is                     [AIEEE 2002]

A)
$M{{R}^{2}}/2$

B)
$M{{R}^{2}}$

C)
$2M{{R}^{2}}$

D)
$M{{R}^{2}}/4$

• question_answer5) A particle of mass m moves along line PC with velocity v as shown. What is the angular momentum of the particle about O? [AIEEE 2002]

A)
mvL

B)
mvl

C)
mvr

D)
0

• question_answer6) A circular disc X of radius R is made from an iron plate of thickness t and another disc Y of radius 4R is made from an iron plate of thickness t/4. Then, the relation between the moment of inertia ${{I}_{X}}$ and ${{I}_{Y}}$ is [AIEEE 2003]

A)
${{l}_{Y}}=32\,{{l}_{X}}$

B)
${{l}_{Y}}=16\,{{l}_{X}}$

C)
${{l}_{Y}}=\,{{l}_{X}}$

D)
${{l}_{Y}}=\,64{{l}_{X}}$

• question_answer7) A particle performing uniform circular motion has angular momentum L. If its angular frequency is doubled and its kinetic energy halved, then the new angular momentum is                                                [AIEEE 2003]

A)
$\frac{L}{4}$

B)
2L

C)
4L

D)
$\frac{L}{2}$

• question_answer8) Let F be the force acting on a particle having position vector r and t be the torque of this force about the origin. Then,                [AIEEE 2003]

A)
$r\,.\,\tau =0$ and $F\,.\,\,\tau \ne 0$

B)
$r\,.\,\,\tau \ne 0$ and $F\,.\,\,\tau =0$

C)
$r\,.\,\,\tau \ne 0$ and $F\,.\,\,\tau \ne 0$

D)
$r\,.\,\,\tau =0$ and $F\,.\,\,\tau =0$

• question_answer9) A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?                                                                 [AIEEE 2004]

A)
Moment of inertia

B)
Angular momentum

C)
Angular velocity

D)
Rotational kinetic energy

• question_answer10) One solid sphere A and another hollow sphere B are of same mass and same outer radii. Their moment of inertia about their diameters are respectively${{I}_{A}}$and${{I}_{B}}$such that where${{d}_{A}}$and${{d}_{B}}$are their densities.                                                                                                  [AIEEE 2004]

A)
${{I}_{A}}={{I}_{B}}$

B)
${{I}_{A}}>{{I}_{B}}$

C)
${{I}_{A}}<{{I}_{B}}$

D)
$\frac{{{I}_{A}}}{{{I}_{B}}}<\frac{{{d}_{A}}}{{{d}_{B}}}$

• question_answer11) An annular ring with inner and outer radii${{R}_{1}}$and${{R}_{2}}$is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring,$\frac{{{F}_{1}}}{{{F}_{2}}}$is                                                                           [AIEEE 2005]

A)
$\frac{{{R}_{2}}}{{{R}_{1}}}$

B)
${{\left( \frac{{{R}_{1}}}{{{R}_{2}}} \right)}^{2}}$

C)
1

D)
$\frac{{{R}_{1}}}{{{R}_{2}}}$

• question_answer12) The moment of inertia of uniform semi-circular disc of mass M and radius r about a line perpendicular to the plane of the disc through the centre is                                                                                     [AIEEE 2005]

A)
$\frac{1}{4}M{{r}^{2}}$

B)
$\frac{2}{5}M{{r}^{2}}$

C)
$M{{r}^{2}}$

D)
$\frac{1}{2}M{{r}^{2}}$

• question_answer13) AT shaped object with dimensions shown in the figure, is lying on a smooth floor. A force F is applied at the point P parallel to AB, such that the object has only the translational motion without rotation. Find the location of P with respect to C.                                                                                                                                                   [AIEEE 2005]

A)
$\frac{2}{3}l$

B)
$\frac{3}{2}l$

C)
$\frac{4}{3}l$

D)
$l$

• question_answer14) Consider a two particle system with particles having masses${{m}_{1}}$and${{m}_{2}}$. If the first particle is pushed towards the centre of mass through a distance d, by what distance should the second particle be moved, so as to keep the centre of mass at the same position?                                                                                [AIEEE 2006]

A)
$\frac{{{m}_{2}}}{{{m}_{1}}}d$

B)
$\frac{{{m}_{1}}}{{{m}_{1}}+{{m}_{2}}}d$

C)
$\frac{{{m}_{1}}}{{{m}_{2}}}d$

D)
$d$

• question_answer15) Four point masses, each of value m, are placed at the corners of a square ABCD of side 1. The moment of inertia of this system about an axis passing through A and parallel to BD is                                  [AIEEE 2006]

A)
$2m{{l}^{2}}$

B)
$\sqrt{3}\,m{{l}^{2}}$

C)
$3\,m{{l}^{2}}$

D)
$\,m{{l}^{2}}$

• question_answer16) A force of $-F\hat{k}$ acts on O, the   origin   of   the coordinate system. The torque about the point $(1,-1)$is                                                                                                                                                [AIEEE 2006]

A)
$F(\hat{i}-\hat{j})$

B)
$-F(\hat{i}+\hat{j})$

C)
$F(\hat{i}+\hat{j})$

D)
$-F(\hat{i}-\hat{j})$

• question_answer17) A thin circular ring of mass m and radius R is rotating about its axis with a constant angular velocity$\omega$. Two objects each of mass M are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity$\omega '$is equal to                                                                                  [AIEEE 2006]

A)
$\frac{\omega (m+2M)}{m}$

B)
$\frac{\omega (m-2M)}{(m-2M)}$

C)
$\frac{\omega m}{(m+M)}$

D)
$\frac{\omega m}{(m+2M)}$

• question_answer18)  For the given uniform square lamina ABCD, whose centre is O                                    [AIEEE 2007]

A)
$f(x)=$

B)
${{\log }_{e}}x$

C)
$2{{\log }_{3}}e$

D)
$\frac{1}{2}{{\log }_{e}}3$

• question_answer19) A circular disc of radius R is removed from a bigger circular disc of radius 2R, such that the circumference of the discs coincide. The centre of mass of the new disc is$(3,\infty )$from the centre of the bigger disc. The value of a is [AIEEE 2007]

A)
$(-\infty ,-3)$

B)
$x=(2\times {{10}^{-2}})\cos \pi t$

C)
$a=2\times {{10}^{-2}}m=2cm$

D)
$t=0,$

• question_answer20) A round uniform body of radius R, mass M and moment of inertia$x=2\text{ }cm$rolls down (without slipping) an inclined plane making an angle 6 with the horizontal.                                                                   [AIEEE 2007]             Then, its acceleration is

A)
$\frac{1}{4}$

B)
$=\frac{T}{4}$

C)
$\omega =\frac{2\pi }{T}=\pi$

D)
$\Rightarrow$

• question_answer21) Angular momentum of the particle rotating with a central force is constant due to             [AIEEE 2007]

A)
constant force

B)
constant linear momentum

C)
zero torque

D)
constant torque

• question_answer22) Consider a uniform square plate of side a and mass m. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is                                                                    [AIEEE 2008]

A)
$\frac{7}{12}m{{a}^{2}}$

B)
$\frac{2}{3}m{{a}^{2}}$

C)
$\frac{5}{6}m{{a}^{2}}$

D)
$\frac{1}{12}m{{a}^{2}}$

• question_answer23) A thin rod of length L is lying along the x-axis with its ends at $x=0$ and $x=L$. Its linear density (mass/length) varies with x as $k{{\left( \frac{x}{L} \right)}^{n}}$, where n can be zero or any positive number. If the position ${{x}_{CM}}$ of the centre of mass of the rod is plotted against n, which of the following graphs best approximates the dependence of ${{x}_{CM}}$ on n?                            [AIEEE 2008]

A)

B)

C)

D)

• question_answer24) A thin uniform rod of length l and mass m is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega$. Its centre of mass rises to a maximum height of                         [AIEEE 2009]

A)
$\frac{1}{3}\frac{{{\ell }^{2}}{{\omega }^{2}}}{g}$

B)
$\frac{1}{6}\frac{\ell \omega }{g}$

C)
$\frac{1}{2}\frac{{{\ell }^{2}}{{\omega }^{2}}}{g}$

D)
$\frac{1}{6}\frac{{{\ell }^{2}}{{\omega }^{2}}}{g}$

• question_answer25)  A small particle of mass m is projected at an angle$\theta$with the x-axis with an initial velocity v0 in the$x-y$plane as shown in the figure. At a time $t<\frac{{{v}_{0}}\sin \theta }{g},$ the angular momentum of the particle is -                  [AIEEE 2010]

A)
$\frac{1}{2}mg{{v}_{0}}{{t}^{2}}\cos \theta \hat{i}$

B)
$-mg{{v}_{0}}{{t}^{2}}\cos \theta \hat{j}$

C)
$mg{{v}_{0}}t\cos \theta \hat{k}$

D)
$-\frac{1}{2}mg{{v}_{0}}{{t}^{2}}\cos \theta \hat{k}$

• question_answer26) A pulley of radius 2 m is rotated about its axis by a force $F=(20\,t-5{{t}^{2}})$ newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg ${{m}^{2}}$, the number of rotations made by the pulley before its direction of motion if reversed is                                   [AIEEE 2011]

A)
More than 9

B)
Less than 3

C)
More than 3 but less than 6

D)
More than 6 but less than 9

• question_answer27) A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect the angular speed of the disc                                                 [AIEEE 2011]

A)
First increase and then decrease

B)
Remains unchanged

C)
Continuously decreases

D)
Continuously increases

• question_answer28) A mass m hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass m and radius R. Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass m, if the string does not slip on the pulley, is                                                                                                                       [AIEEE 2011]

A)
$\frac{g}{3}$

B)
$\frac{3}{2}g$

C)
g

D)
$\frac{2}{3}g$

• question_answer29) A diatomic molecule is made of two masses ${{m}_{1}}$ and ${{m}_{2}}$ which are separated by a distance r. If we calculate its rotational energy by applying Bohrs rule of angular momentum quantization, its energy will be given by: (n is an integer)                                                                                                                   [AIEEE 2012]

A)
$\frac{{{({{m}_{1}}+{{m}_{2}})}^{2}}{{n}^{2}}{{h}^{2}}}{2m_{1}^{2}m_{2}^{2}{{r}^{2}}}$

B)
$\frac{{{n}^{2}}{{h}^{2}}}{2({{m}_{1}}+{{m}_{2}}){{r}^{2}}}$

C)
$\frac{2{{n}^{2}}{{h}^{2}}}{({{m}_{1}}+{{m}_{2}}){{r}^{2}}}$

D)
$\frac{({{m}_{1}}+{{m}_{2}}){{n}^{2}}{{h}^{2}}}{2{{m}_{1}}{{m}_{2}}{{r}^{2}}}$

• question_answer30) A circular hole of diameter R is cut from a disc of mass M and radius R, -the circumference of the cut passes through the centre of the disc. The moment of inertia of the remaining portion of the disc about an axis perpendicular to the disc and passing through its centre is  [JEE ONLINE 07-05-2012]

A)
$\left( \frac{15}{32} \right)M{{R}^{2}}$

B)
$\left( \frac{1}{8} \right)M{{R}^{2}}$

C)
$\left( \frac{3}{8} \right)M{{R}^{2}}$

D)
$\left( \frac{13}{32} \right)M{{R}^{2}}$

• question_answer31) A solid sphere having mass m and radius r rolls down an inclined plane. Then its kinetic energy is                                                                                                                                      [JEE ONLINE 07-05-2012]

A)
$\frac{5}{7}$rotational and $\frac{2}{7}$translational

B)
$\frac{2}{7}$rotational and $\frac{5}{7}$translational

C)
$\frac{2}{5}$rotational and $\frac{3}{5}$translational

D)
$\frac{1}{2}$rotational and $\frac{1}{2}$translational

• question_answer32)  A solid sphere is rolling on a surface as shown in figure, with a translational velocity $v\,m\,{{s}^{-1}}.$ If it is to climb the inclined surface continuing to roll without slipping, then minimum velocity for this to happen is [JEE ONLINE 12-05-2012]

A)
$\sqrt{2gh}$

B)
$\sqrt{\frac{7}{5}gh}$

C)
$\sqrt{\frac{7}{2}gh}$

D)
$\sqrt{\frac{10}{7}gh}$

• question_answer33)  This question has Statement land Statement 2.                                           [JEE ONLINE 12-05-2012] Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement 1: When moment of inertia/of a body rotating about an axis with angular speed $\omega$increases, its angular momentum L is unchanged but the kinetic energy K increases if there is no torque applied on it. Statement 2: $L=I\omega ,$ kinetic energy of rotation $=\frac{1}{2}I{{\omega }^{2}}$

A)
Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation of Statement 1.

B)
Statement 1 is false, Statement 2 is true.

C)
Statement 1 is true, Statement 2 is true. Statement 2 is correct explanation of the Statement 1.

D)
Statement 1 is true, Statement 2 is false.

• question_answer34) A stone of mass m, tied to the end of a string, is whirled around in a circle on a horizontal frictionless table. The length of the string is reduced gradually keeping the angular momentum of the stone about the centre of the circle constant. Then, the tension in the string is given by $T=A{{r}^{n}},$where A is a constant, r is the instantaneous radius of the circle. The value of n is equal to                                                                             [JEE ONLINE 26-05-2012]

A)
- 1

B)
- 2

C)
- 4

D)
- 3

• question_answer35) A thick-walled hollow sphere has outside radius ${{R}_{0}}.$It rolls down an incline without slipping and its speed at the bottom is ${{v}_{0}}.$ Now the incline is waxed, so that it is practically frictionless and the sphere is observed to slide down (without any rolling). Its speed at the bottom is observed to be $5{{v}_{0}}/4.$The radius of gyration of the hollow sphere about an axis through its centre is                                                   [JEE ONLINE 26-05-2012]

A)
$3{{R}_{0}}/2$

B)
$3{{R}_{0}}/4$

C)
$9{{R}_{0}}/16$

D)
$3{{R}_{0}}$

• question_answer36) A hoop of radius r and mass m rotating with an angular velocity ω0 is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?                                                                                                                                             [JEE MAIN 2013]

A)
$\frac{r{{\omega }_{0}}}{4}$

B)
$\frac{r{{\omega }_{0}}}{3}$

C)
$\frac{r{{\omega }_{0}}}{2}$

D)
$r{{\omega }_{0}}$

• question_answer37) A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 Kg. It is hinged at one end and rotates about a vertical axis practically without friction. The angular speed of the door just ager the bullet embeds into it will be:                           [JEE ONLINE 09-04-2013]

A)

B)

C)

D)

• question_answer38) A tennis ball (treated as hollow spherical shell) starting from O rolls down a hill. At point A the ball becomes air borne leaving at an angle of ${{30}^{0}}$ with the horizontal. The ball strikes the ground at B. What is the value of the distance AB? (Moment of inertia of spherical shell of mass m and radius R about its diameter $=\frac{2}{3}{{\operatorname{mR}}^{2}})$ [JEE ONLINE 22-04-2013]

A)
1.87 m

B)
2.08 m

C)
1.57 m

D)
1.77 m

• question_answer39) A particle of mass 2 kg is moving such hat at time t, its position, in meter, is given by $\overset{\to }{\mathop{\operatorname{r}}}\,(\operatorname{t})=5\hat{i}-2{{\operatorname{t}}^{2}}\hat{j}.$ The angular momentum of the particle at $\operatorname{t}=2s$ about the origin in kg ${{\operatorname{m}}^{-2}}$${{\operatorname{s}}^{-2}}$ is:                                        [JEE ONLINE 23-04-2013]

A)
$-80\hat{k}$

B)
$\left( 10\hat{i}-16\hat{j} \right)$

C)
$-40\hat{k}$

D)
$40\hat{k}$­­

• question_answer40) A boy of mass 20 kg is standing on a 80 kg free to move long cart. There is negligible friction between cart and ground. Initially, the boy is standing 25 m from a wall. If he walks 10 m on the cart toward the wall, then the final distance of the boy from the wall will be:                                                                    [JEE ONLINE 23-04-2013]

A)
15 m

B)
12.5 m

C)
15.5 m

D)
17 m

• question_answer41) A 70 Kg man leaps vertically into the air from a crouching position. To take the leap the man pushes the pushes the ground with a constant force F to raise himself. The center of gravity rises by 0.5 m before he leaps. After the leap the c.g. rises by another 1 m. The maximum power delivered by the muscles is: (Take g = 10 $\text{m}{{\text{s}}^{\text{2}}}$).                                                                                                                                    [JEE ONLINE 23-04-2013]

A)
$\text{6}\text{.26}\times \text{1}{{\text{0}}^{\text{3}}}$ Watts at the start

B)
$\text{6}\text{.26}\times \text{1}{{\text{0}}^{\text{3}}}$ Watts at take off

C)
$\text{6}\text{.26}\times \text{1}{{\text{0}}^{\text{4}}}$ Watts at the start

D)
$\text{6}\text{.26}\times \text{1}{{\text{0}}^{\text{4}}}$ Watts at take off

• question_answer42)   A ring of mass M and radius R is rotating about its axis with angular velocity $\omega$. Two identical bodies each of mass m are now gently attached at the two ends of a diameter of the ring. Because of this, the kinetic energy loss will be:                                                                                                 [JEE ONLINE 25-04-2013]

A)
$\frac{\operatorname{m}(\operatorname{M}+2m)}{\operatorname{M}}{{\omega }^{2}}{{R}^{2}}$

B)
$\frac{\operatorname{Mm}}{(\operatorname{M}+m)}{{\omega }^{2}}{{R}^{2}}$

C)
$\frac{\operatorname{Mm}}{(\operatorname{M}+2m)}{{\omega }^{2}}{{R}^{2}}$

D)
$\frac{\left(\operatorname{M}+m \right)\operatorname{M}}{(\operatorname{M}+2m)}{{\omega }^{2}}{{R}^{2}}$

• question_answer43) A bob of mass m attached to an inextensible string of length l is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $\omega$rad/s about the vertical. About the point of suspension:                                                                                                                               [JEE MAIN 2014]

A)
angular momentum changes in direction but not in magnitude.

B)
angular momentum changes both in direction and magnitude.

C)
angular momentum is conserved.

D)
angular momentum changes in magnitude but not in direction.

• question_answer44)  A mass m is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?     [JEE MAIN 2014]

A)
$\frac{5g}{6}$

B)
g

C)
$\frac{2g}{3}$

D)
$\frac{g}{2}$

• question_answer45)  A cylinder of mass $\text{Mc}$and sphere of mass $\text{Ms}$ are placed at points A and B of two inclines, respectively (See Figure). If they roll on the incline without sipping such that their accelerations are the same, then the ratio$\frac{\sin {{\theta }_{c}}}{\sin {{\theta }_{s}}}$is                                                          [JEE ONLINE 09-04-2014]

A)
$\sqrt{\frac{8}{7}}$

B)
$\sqrt{\frac{15}{14}}$

C)
$\frac{8}{7}$

D)
$\frac{15}{14}$

• question_answer46) A thin bar of length L has a mass per unit length $\lambda$, that increases linearly with distance from one end. If its total mass is M and its mass per unit length at the lighter end is ${{\lambda}_{O}},$ then the distance of the centre of mass from the lighter end is:                                                                                          [JEE ONLINE 11-04-2014]

A)
$\frac{L}{2}-\frac{{{\lambda }_{o}}{{L}^{2}}}{4M}$

B)
$\frac{L}{3}+\frac{{{\lambda }_{o}}{{L}^{2}}}{8M}$

C)
$\frac{L}{3}+\frac{{{\lambda }_{o}}{{L}^{2}}}{4M}$

D)
$\frac{2L}{3}-\frac{{{\lambda }_{o}}{{L}^{2}}}{6M}$

• question_answer47)  A particle is moving in a circular path of radius a, with a constant velocity v as shown in the figure. The centre of circle is marked by C. The angular momentum from the origin O can be written as:    [JEE ONLINE 12-04-2014]

A)
$va(1+cos2\theta )$

B)
$va(1+cos\theta )$

C)
$vacos2\theta$

D)
$va$

• question_answer48)  Consider a cylinder of mass M resting on a rough horizontal rug that is pulled out from under it with acceleration a perpendicular to the axis of the cylinder. What is Ffriction at point P? It is assumed that the cylinder does not slip. [JEE ONLINE 19-04-2014]

A)
Ma

B)
Ma

C)
$\frac{Ma}{2}$

D)
$\frac{Ma}{3}$

• question_answer49) A ball of mass 160 g is thrown up at an angle of 60° to the horizontal at a speed of $10m{{s}^{-1}}.$The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly $(g=10m{{s}^{-2}})$                                                                                        [JEE ONLINE 19-04-2014]

A)
$1.73kg{{m}^{2}}/s$

B)
$3.0kg{{m}^{2}}/s$

C)
$3.46kg{{m}^{2}}/s$\

D)
$6.0kg{{m}^{2}}/s$

• question_answer50) Distance of the centre of mass of a solid uniform cone from its vertex is ${{z}_{0}}$. If the radius of its base is R and its height is h then ${{z}_{0}}$is equal to:                                                                      [JEE MAIN 2015]

A)
$\frac{5h}{8}$

B)
$\frac{3{{h}^{2}}}{8R}$

C)
$\frac{{{h}^{2}}}{8R}$

D)
$\frac{3h}{4}$

• question_answer51) From a solid sphere of mass M and radius R a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its faces is:                        [JEE MAIN 2015]

A)
$\frac{4M{{R}^{2}}}{9\sqrt{3\pi }}$

B)
$\frac{4M{{R}^{2}}}{3\sqrt{3\pi }}$

C)
$\frac{M{{R}^{2}}}{32\sqrt{2\pi }}$

D)
$\frac{M{{R}^{2}}}{16\sqrt{2\pi }}$

• question_answer52) A uniform solid cylindrical roller of mass m is being pulled on a horizontal surface with force F parallel to the surface and applied at its centre. If the acceleration of the cylinder is a and it is rolling without slipping then the value of F is:                       [JEE ONLINE 10-04-2015]

A)
ma

B)
2 ma

C)
$\frac{5}{3}$

D)
$\frac{3}{2}ma$

• question_answer53) Consider a thin uniform square sheet made of a rigid material. If its side is 'a', mass m and moment of inertia I about one of its diagonals, then:                                                                          [JEE ONLINE 10-04-2015]

A)
$I=\frac{m{{a}^{2}}}{12}$

B)
$I>\frac{m{{a}^{2}}}{12}$

C)
$I=\frac{m{{a}^{2}}}{24}$

D)
$\frac{m{{a}^{2}}}{24}<I<\frac{m{{a}^{2}}}{12}$

• question_answer54)             A large number (n) of identical beads, each of mass m and radius r are strung on a thin smooth rigid horizontal rod of length $L(L>>r)$ and are at rest at random positions. The rod is mounted between two rigid supports (see figure). If one of the beads is now given a speed v, the average force experienced by each support after a long time is (assume all collisions are elastic) : [JEE MAIN 11-04-2015]

A)
$\frac{m{{\upsilon }^{2}}}{L-nr}$

B)
$\frac{m{{\upsilon }^{2}}}{L-2nr}$

C)
$\frac{m{{\upsilon }^{2}}}{2(L-nr)}$

D)
zero

• question_answer55) A uniform thin rod AB of length L has linear mass density $\mu (x)=a+\frac{bx}{L},$where x is measured from A. If the CM of the rod lies at a distance of $\left( \frac{7}{12}L \right)$from A, then a and b are related as :          [JEE MAIN 11-04-2015]

A)
a = b

B)
a = 2b

C)
2a = b

D)
3a = 2b

• question_answer56) A particle of mass 2 kg is on a smooth horizontal table and moves in a circular path of radius 0.6 m. The height of the table from the ground is 0.8 m. If the angular speed of the particle is 12 rad ${{\text{s}}^{\text{-1}}}\text{,}$the magnitude of its angular momentum about a point on the ground right under the centre of the circle is : [JEE MAIN 11-04-2015]

A)
$\text{8}\text{.64}\,\text{kg}\,{{\text{m}}^{\text{2}}}{{\text{s}}^{\text{-1}}}$

B)
$\text{11}\text{.52}\,\text{kg}\,{{\text{m}}^{\text{2}}}{{\text{s}}^{\text{-1}}}$

C)
$\text{14}\text{.4}\,\text{kg}\,{{\text{m}}^{\text{2}}}{{\text{s}}^{\text{-1}}}$

D)
$\text{20}\text{.16}\,\text{kg}\,{{\text{m}}^{\text{2}}}{{\text{s}}^{\text{-1}}}$

• question_answer57)  A particle of mass m is moving along the side of a square of side 'a', with a uniform speed u in the x-y plane as shown in the figure:                                                                [JEE MAIN - I 3-4-2016] Which of the following statement is false for the angular momentum $\vec{L}$ about the origin?

A)
$\vec{L}=\frac{m\upsilon }{\sqrt{2}}R\hat{k}$ when the particle is moving from D to A

B)
$\vec{L}=-\frac{m\upsilon }{\sqrt{2}}R\hat{k}$ when the particle is moving from A to B

C)
$\vec{L}=m\upsilon \left[ \frac{R}{\sqrt{2}}-a \right]\hat{k}$ when the particle is moving from C to D

D)
None of these

• question_answer58)  A roller is made by joining together two cones at their vertices O. It is kept on two rails AB and CD which are placed asymmetrically (see figure), with its axis perpendicular to CD and its centre O at the centre of line joining AB and CD (see figure). It is given a light push so that it starts rolling with its centre O moving parallel to CD in the direction shown. As it moves, the roller will tend to:-                                                                                              [JEE MAIN - I 3-4-2016]

A)
turn left and right alternately.

B)
turn left.

C)
turn right.

D)
go straight.

• question_answer59)  A cubical block of side 30cm is moving with velocity $2m{{s}^{-1}}$on a smooth horizontal surface. The surface has a bump ata point O as shown in figure. The angular velocity (in rad/s) of the block immediately after it hits the bump, is:                                                                                                                [JEE ONLINE 09-04-2016]

A)
9.4

B)
6.7

C)
5.0

D)
13.3

• question_answer60)  In the figure shown/ ABC is a uniform wire. If centre of mass wire lies vertically below point A, then $\frac{BC}{AB}$ [JEE ONLINE 10-04-2016]

A)
1.85

B)
1.5

C)
3

D)
1.37

• question_answer61) The moment of inertia of a uniform cylinder of length$\ell$and radius R about its perpendicular bisector is I. What is the ratio $\ell /R$such that the moment of inertia is minimum?                                                          [JEE Main 2017]

A)
1

B)
$\frac{3}{\sqrt{2}}$

C)
$\sqrt{\frac{3}{\sqrt{2}}}$

D)
$\frac{\sqrt{3}}{2}$

• question_answer62)  A slender uniform rod of mass M and length $\ell$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle$\theta$ with the vertical is:    [JEE Main 2017]

A)
$\frac{3g}{2\ell }\cos \theta$

B)
$\frac{2g}{3\ell }\cos \theta$

C)
$\frac{3g}{2\ell }\sin \theta$

D)
$\frac{2g}{3\ell }\sin \theta$

• question_answer63)  A uniform disc of radius R and mass M is free to rotate only about its axis. A string is wrapped over its rim and a body of mass m is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is - [JEE Online 08-04-2017]

A)
$\frac{2Mg}{2m+M}$

B)
$\frac{2Mg}{2M+m}$

C)
$\frac{2mg}{2M+m}$

D)
$\frac{2mg}{2m+M}$

• question_answer64) Moment of inertia of an equilateral triangular lamina ABC, about the axis passing through its centre and perpendicular to its plane is I0 as shown in the figure. A cavithy DEF is cut out from the lamina, where D, E, F are the mid points of the sides. Moment of inertia of the remaining part of lamina about the same axis is -                    [JEE Online 08-04-2017]

A)
$\frac{15}{16}{{I}_{0}}$

B)
$\frac{3{{I}_{0}}}{4}$

C)
$\frac{7}{8}{{I}_{0}}$

D)
$\frac{31{{I}_{0}}}{32}$

• question_answer65)  A circular hole of radius $\frac{R}{4}$ is made in a thin uniform disc having mass M and radius R, as shown in figure. The moment of inertia of the remaining portion of the disc about an axis passing through the point $\text{O}$ and perpendicular to the plane of the disc is                                                             [JEE Online 09-04-2017]

A)
$\frac{219{{R}^{2}}}{256}\,$

B)
$\frac{237M{{R}^{2}}}{512}$

C)
$\frac{197M{{R}^{2}}}{256}\,$

D)
$\frac{19M{{R}^{2}}}{512}$

• question_answer66)  The machine as shown has 2 rods of length 1 m connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has a roller that rolls along the floor in a slot. As the roller goes back and forth, a 2 kg weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a -  [JEE Online 09-04-2017]

A)
speed which is $\frac{3}{4}th$ of that of the roller when the weight is 0.4 m above the ground

B)
constant speed

C)
decreasing speed

D)
increasing speed

• question_answer67)  Seven identical circular planar disks, each of mass M and radius R are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is: [JEE Main Online 08-04-2018]

A)
$\frac{\text{73}}{\text{2}}\text{M}{{\text{R}}^{\text{2}}}$

B)
$\frac{\text{181}}{\text{2}}\text{M}{{\text{R}}^{\text{2}}}$

C)
$\frac{\text{19}}{\text{2}}\text{M}{{\text{R}}^{\text{2}}}$

D)
$\frac{\text{55}}{\text{2}}\text{M}{{\text{R}}^{\text{2}}}$

• question_answer68)  From a uniform circular disc of radius R and mass 9 M, a small disc of radius $\frac{R}{3}$ is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is:                                                  [JEE Main Online 08-04-2018]

A)
$\text{10 M}{{\text{R}}^{2}}$

B)
$\frac{37}{9}\text{ M}{{\text{R}}^{2}}$

C)
$\text{4 M}{{\text{R}}^{2}}$

D)
$\frac{40}{9}\text{ M}{{\text{R}}^{2}}$

• question_answer69)  A uniform rod AB is suspended from a point X, at a variable distance $x$ from $A$, as shown. To make the rod horizontal, a mass $m$ is suspended from its end$A$. A set of $(m,x)$ values is recorded. The appropriate variables that give a straight line, when plotted, are:                                                    [JEE Online 15-04-2018]

A)
$m,\frac{l}{x}$

B)
$m,\frac{l}{{{x}^{2}}}$

C)
$m,x$

D)
$m,{{x}^{2}}$

• question_answer70)  A force of $40N$acts on a point $B$ at the end of an L-shaped object, as shown in the figure. The angle $0$ that will produce maximum moment of the force about point $A$ is given by:                        [JEE Online 15-04-2018]

A)
$\tan \theta =\frac{1}{4}$

B)
$\tan \theta =2$

C)
$\tan \theta =\frac{1}{2}$

D)
$\tan \theta =4$

• question_answer71)  A think rod$MN$, free to rotate in the vertical plane about the fixed end$N$, is held horizontal. When the end $M$ is released the speed of this end, when the rod makes an angle $\alpha$ with the horizontal, will be proportional to: (see figure)                                                                                           [JEE Online 15-04-2018 (II)]

A)
$\sqrt{\cos \alpha }$

B)
$\cos \alpha$

C)
$\sin \alpha$

D)
$\sqrt{\sin \alpha }$

• question_answer72)  A thin uniform bar of length $L$ and mass $8m$lies on a smooth horizontal table. Two point masses $m$ and $2m$ moving in the same horizontal plane from opposite sides of the bar with speeds $2v$ and $v$ respectively. The masses stick to the bar after collision at a distance $\frac{L}{3}$ and $\frac{L}{6}$ respectively from the centre of the bar. If the bar starts rotating about its centre of mass as a result of collision, the angular speed of the bar will be: [JEE Online 15-04-2018 (II)]

A)
$\frac{v}{6L}$

B)
$\frac{6v}{5L}$

C)
$\frac{3v}{5L}$

D)
$\frac{v}{5L}$

• question_answer73)  A thin circular disk is in the$xy$ plane as shown in the figure. The ratio of its moment of inertia about z and z axes will be                                                                                                [JEE Main Online 16-4-2018]

A)
$1:2$

B)
$1:4$

C)
$1:3$

D)
$1:5$

• question_answer74) If the angular momentum of a planet of mass m, moving around the Sun in a circular orbit is L, about the center of the Sun, its areal velocity is:                                                                          [JEE Main 09-Jan-2019 Morning]

A)
$\frac{L}{m}$

B)
$\frac{L}{2m}$

C)
$\frac{4L}{m}$

D)
$\frac{2L}{m}$

• question_answer75)  Two masses m and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length l. The rod is suspended by a thin wire of torsional constant k at the centre of mass of the rod-mass system (see figure). Because of torsional constant k, the restoring torque is $\tau \,\,=\,\,k\theta$ for angular displacement $\theta$. If the rod is rotated by ${{\theta }_{0}}$ and released, the tension in it when it passes through its mean position will be:                                                                                                                [JEE Main 09-Jan-2019 Morning]

A)
$\frac{2\,k\,{{\theta }_{0}}^{2}}{l}$

B)
$\frac{k\,{{\theta }_{0}}^{2}}{l}$

C)
$\frac{3k\,{{\theta }_{0}}^{2}}{l}$

D)
$\frac{k\,{{\theta }_{0}}^{2}}{2\,l}$

• question_answer76)  A rod of length 50 cm is pivoted at one end. It is raised such that if makes an angle of $30{}^\circ$from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in rad ${{s}^{-1}}$) will be ($g=10\text{ }m{{s}^{-2}}$)                                                                           [JEE Main 09-Jan-2019 Evening]

A)
$\sqrt{\frac{30}{2}}$

B)
$\frac{\sqrt{20}}{3}$

C)
$\frac{\sqrt{30}}{2}$

D)
$\sqrt{30}$

• question_answer77) To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is u, the torque, applied by the machine on the mop is - [JEE Main 10-Jan-2019 Morning]

A)
$\mu \,FR/2$

B)
$\mu \,FR/3$

C)
$\mu \,FR/6$

D)
$\frac{2}{3}\mu \,FR$

• question_answer78) A homogeneous solid cylindrical roller of radius R and mass M is pulled on a cricket pitch by a horizontal force. Assuming rolling without    slipping, angular acceleration of the cylinder is-     [JEE Main 10-Jan-2019 Morning]

A)
$\frac{F}{2mR}$

B)
$\frac{2\,F}{3mR}$

C)
$\frac{3\,F}{2mR}$

D)
$\frac{F}{3mR}$

• question_answer79)  A rigid massless rod of length 3l has two masses attached at each end as shown in the figure. The rod is pivoted at point P on the horizontal axis (see figure).  When released  from initial horizontal position, its instantaneous angular acceleration will be-                                                                        [JEE Main 10-Jan-2019 Evening]

A)
$\frac{g}{13l}$

B)
$\frac{g}{2l}$

C)
$\frac{g}{3l}$

D)
$\frac{7g}{3l}$

• question_answer80)  Two identical spherical balls of mass M and radius R each are stuck on two ends of a rod of length 2R and mass M (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is:                                                                                           [JEE Main 10-Jan-2019 Evening]

A)
$\frac{17}{15}M{{R}^{2}}$

B)
$\frac{137}{15}M{{R}^{2}}$

C)
$\frac{209}{15}M{{R}^{2}}$

D)
$\frac{152}{15}M{{R}^{2}}$

• question_answer81)  An equilateral triangle ABC is cut from a thin solid sheet of wood.                 [JEE Main 11-Jan-2019 Morning] (See figure) D, E and F are the mid-points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of the triangle is ${{I}_{0}}$. If the smaller triangle DEF is removed from ABC, the moment of inertia of the remaining figure about the same axis is I. Then

A)
$I=\frac{15}{16}{{I}_{0}}$

B)
$I=\frac{9}{16}{{I}_{0}}$

C)
$I=\frac{3}{4}{{I}_{0}}$

D)
$I=\frac{{{I}_{0}}}{4}$

• question_answer82)  A string is wound around a hollow cylinder of mass 5 kg and radius 0.5 m. If the string is now pulled with a horizontal force of 40 N, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string) [JEE Main 11-Jan-2019 Evening]

A)
$10\text{ }rad/{{s}^{2}}$

B)
$\text{20 }rad/{{s}^{2}}$

C)
$\text{12 }rad/{{s}^{2}}$

D)
$\text{16 }rad/{{s}^{2}}$

• question_answer83)  A circular disc ${{D}_{1}}$ of mass M and radius R has two identical discs ${{D}_{2}}$ and ${{D}_{3}}$of the same mass M and radius R attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis OO, passing through the centre of ${{D}_{1}}$, as shown in the figure, will be-                                                                                                                          [JEE Main 11-Jan-2019 Evening]

A)
$3M{{R}^{2}}$

B)
$\frac{4}{5}M{{R}^{2}}$

C)
$\frac{2}{3}M{{R}^{2}}$

D)
$M{{R}^{2}}$

• question_answer84) The magnitude of torque on a particle of mass 1 kg is 2.5 N m about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5 m, the angle between the force and the position vector is (in radians) [JEE Main 11-Jan-2019 Evening]

A)
$\frac{\pi }{3}$

B)
$\frac{\pi }{8}$

C)
$\frac{\pi }{4}$

D)
$\frac{\pi }{6}$

• question_answer85)  The position vector of the centre of mass $\vec{r}cm$of an asymmetric uniform bar of negligible area of cross-section as shown in figure is-                       [JEE Main 12-Jan-2019 Morning]

A)
$\vec{r}\,cm=\frac{11}{8}L\hat{x}+\frac{3}{8}L\,\hat{y}$

B)
$\vec{r}\,cm=\frac{13}{8}L\hat{x}+\frac{5}{8}L\,\hat{y}$

C)
$\vec{r}\,cm=\frac{3}{8}L\hat{x}+\frac{11}{8}L\,\hat{y}$

D)
$\vec{r}\,cm=\frac{5}{8}L\hat{x}+\frac{13}{8}L\,\hat{y}$

• question_answer86) Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also J, is- [JEE Main 12-Jan-2019 Morning]

A)
14 cm

B)
16 cm

C)
12 cm

D)
18 cm

• question_answer87) A long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical axis, the liquid rises up near the wall. If the radius of vessel is 5 cm and its rotational speed is 2 rotations per second, then the difference in the heights between the centre and the sides, in cm, will be-                              [JEE Main 12-Jan-2019 Evening]

A)
0.4

B)
2.0

C)
0.1

D)
1.2

• question_answer88) The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of x from it, is I(x). Which one of the graphs represents the variation of I(x) with x correctly?                        [JEE Main 12-Jan-2019 Evening]

A)

B)

C)

D)

• question_answer89)  A thin circular plate of mass M and radius R has its density varying as $\rho (r)={{\rho }_{0}}r$with ${{\rho }_{0}}$ as constant and r is the distance from its centre. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is$I=aM{{R}^{2}}.$ The value of the coefficient a is:                                          [JEE Main 8-4-2019 Morning]

A)
$\frac{3}{2}$

B)
$\frac{1}{2}$

C)
$\frac{3}{5}$

D)
$\frac{8}{5}$

• question_answer90)  A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights ${{h}_{sph}}$and ${{h}_{cyl}}$on the incline. The ratio $\frac{{{h}_{sph}}}{{{h}_{cyl}}}$is given by :-                                    [JEE Main 8-4-2019 Afternoon]

A)
$\frac{14}{15}$

B)
$\frac{4}{5}$

C)
1

D)
$\frac{2}{\sqrt{5}}$

• question_answer91)  A uniform rectangular thin sheet ABCD of mass M has length a and breadth b, as shown in the figure. If the shaded portion HBGO is cut-off, the coordinates of the centre of mass of the remaining portion will be :- [JEE Main 8-4-2019 Afternoon]

A)
$\left( \frac{2a}{3},\frac{2b}{3} \right)$

B)
$\left( \frac{5a}{3},\frac{5b}{3} \right)$

C)
$\left( \frac{3a}{4},\frac{3b}{4} \right)$

D)
$\left( \frac{5a}{12},\frac{5b}{12} \right)$

• question_answer92)  A rectangular solid box of length 0.3 m is held horizontally, with one of its sides on the edge of a platform of height 5m. When released, it slips off the table in a very short time $\tau =0.01s,$ remaining essentially horizontal. The angle by which it would rotate when it hits the ground will be (in radians) close to :- [JEE Main 8-4-2019 Afternoon]

A)
0.02

B)
0.28

C)
0.5

D)
0.3

• question_answer93) A solid sphere of mass 'M' and radius 'a' is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field at distance '3a' from the centre will be :           [JEE Main 9-4-2019 Morning]

A)
$\frac{2GM}{9{{a}^{2}}}$

B)
$\frac{GM}{3{{a}^{2}}}$

C)
$\frac{GM}{9{{a}^{2}}}$

D)
$\frac{2GM}{3{{a}^{2}}}$

• question_answer94) The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane. : (i) a ring of radius R, (ii) a solid cylinder of radius $\frac{R}{2}$ and (iii) a solid sphere of radius $\frac{R}{4}$. If in each case, the speed of the centre of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is :         [JEE Main 9-4-2019 Morning]

A)
4 : 3 : 2

B)
14 : 15 : 20

C)
10 : 15 : 7

D)
2 : 3 : 4

• question_answer95) A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $\theta ,$where $\theta$ is the angle by which it has rotated, is given as $k{{\theta }^{2}}.$If its moment of inertia is I then the angular acceleration of the disc is :                                                       [JEE Main 9-4-2019 Morning]

A)
$\frac{k}{2I}\theta$

B)
$\frac{k}{I}\theta$

C)
$\frac{k}{4I}\theta$

D)
$\frac{2k}{I}\theta$

• question_answer96) Moment of inertia of a body about a given axis is 1.5 kg ${{m}^{2}}$. Initially the body is at rest. In order to produce a rotational kinetic energy of 1200 J, the angular acceleration of $20\text{ }rad/{{s}^{2}}$ must be applied about the axis for a duration of :-                                                                           [JEE Main 9-4-2019 Afternoon]

A)
2 s

B)
5s

C)
2.5 s

D)
3 s

• question_answer97) A particle of mass 'm' is moving with speed '2v' and collides with a mass '2m' moving with speed 'v' in the same direction. After collision, the first mass is stopped completely while the second one splits into two particles each of mass 'm', which move at angle$45{}^\circ$with respect to the original direction. The speed of each of the moving particle will be :                                                                                             [JEE Main 9-4-2019 Afternoon]

A)
$\text{v/}\left( 2\sqrt{2} \right)$

B)
$2\sqrt{2}\text{v}$

C)
$\sqrt{2}\text{v}$

D)
$\text{v}/\sqrt{2}$

• question_answer98) A thin smooth rod of length L and mass M is rotating freely with angular speed${{\omega }_{0}}$about an axis perpendicular to the rod and passing through its center. Two beads of mass m and negligible size are at the center of the rod initially. The beads are free to slide along the rod. The angular speed of the system , when the beads reach the opposite ends of the rod, will be:                                                                                       [JEE Main 9-4-2019 Afternoon]

A)
$\frac{M{{\omega }_{0}}}{M+3m}$

B)
$\frac{M{{\omega }_{0}}}{M+m}$

C)
$\frac{M{{\omega }_{0}}}{M+2m}$

D)
$\frac{M{{\omega }_{0}}}{M+6m}$

• question_answer99) A thin disc of mass M and radius R has mass per unit area $\sigma (r)=k{{r}^{2}}$ where r is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is : [JEE Main 10-4-2019 Morning]

A)
$\frac{M{{R}^{2}}}{6}$

B)
$\frac{M{{R}^{2}}}{3}$

C)
$\frac{2M{{R}^{2}}}{3}$

D)
$\frac{M{{R}^{2}}}{2}$

• question_answer100)  A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5 s, is close to:                                                                                    [JEE Main 10-4-2019 Afternoon]

A)
$4.0\times {{10}^{6}}Nm$

B)
$2.0\times {{10}^{5}}Nm$

C)
$1.6\times {{10}^{5}}Nm$

D)
$7.9\times {{10}^{6}}\text{ }Nm$

• question_answer101) The time dependence of the position of a particle of mass $m=2$is given by $\vec{r}(t)=2t\hat{i}-3{{t}^{2}}\hat{j}.$ Its angular momentum, with respect to the origin, at time $t=2$is :                   [JEE Main 10-4-2019 Afternoon]

A)
$36\hat{k}$

B)
$-34(\hat{k}-\hat{i})$

C)
$48(\hat{i}+\hat{j})$

D)
$-48\hat{k}$

• question_answer102) A solid sphere of mass M and radius R is divided into two unequal parts. The first part has a mass of $\frac{7M}{8}$and is converted into a uniform disc of radius 2R. The second part is converted into a uniform solid sphere. Let ${{I}_{1}}$ be the moment of inertia of the disc about its axis and ${{I}_{2}}$be the moment of inertia of the new sphere about its axis. The ratio ${{I}_{1}}/{{I}_{2}}$ is given by :                                           [JEE Main 10-4-2019 Afternoon]

A)
185

B)
65

C)
285

D)
140

• question_answer103)  61. A circular disc of radius b has a hole of radius a at its centre (see figure). If the mass per unit area of the disc varies as $\left( \frac{{{\sigma }_{0}}}{r} \right),$then the radius of gyration of the disc about its axis passing through the centre is : [JEE Main Held on 12-4-2019 Morning]

A)
$\frac{a+b}{2}$

B)
$\frac{a+b}{3}$

C)
$\sqrt{\frac{{{a}^{2}}+{{b}^{2}}+ab}{2}}$

D)
$\sqrt{\frac{{{a}^{2}}+{{b}^{2}}+ab}{3}}$

• question_answer104) A person of mass M is, sitting on a swing of length L and swinging with an angular amplitude ${{\theta }_{0}}$. If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance $\ell (\ell <<L),$is close to:                                 [JEE Main Held on 12-4-2019 Morning]

A)
$Mg\ell$

B)
$Mg\ell (1+{{\theta }_{0}}^{2})$

C)
$Mg\ell (1-{{\theta }_{0}}^{2})$

D)
$Mg\ell \left( 1+\frac{{{\theta }_{0}}^{2}}{2} \right)$

• question_answer105) A uniform rod of length$\ell$is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends. If the tension generated in the rod due to rotation is T(x) at a distance x from the axis, then which of the following graphs depicts it most closely?                  [JEE Main Held on 12-4-2019 Morning]

A)

B)

C)

D)

• question_answer106)  A smooth wire of length $2\pi r$is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed $\omega$about the vertical diameter AB, as shown in figure, the bead is at rest with respect to the circular ring at position P as shown. Then the value of ${{\omega }^{2}}$ is equal to : [JEE Main 12-4-2019 Afternoon]

A)
$\left( g\sqrt{3} \right)/r$

B)
$\frac{\sqrt{3}g}{2r}$

C)
$2g/r$

D)
$2g/\left( r\sqrt{3} \right)$

• Three particles of masses 50 g, 100 g and 150 g are placed at the vertices of an equilateral triangle of side 1 m (as shown in the figure).
The (x, y) coordinates of the centre of mass will be :
[JEE Main 12-4-2019 Afternoon]

A)
$\left( \frac{7}{12}m,\frac{\sqrt{3}}{8}m \right)$

B)
$\left( \frac{\sqrt{3}}{4}m,\frac{5}{12}m \right)$

C)
$\left( \frac{7}{12}m,\frac{\sqrt{3}}{4}m \right)$

D)
$\left( \frac{\sqrt{3}}{8}m,\frac{7}{12}m \right)$

• question_answer108) The radius of gyration of a uniform rod of length l, about an axis passing through a point $\frac{l}{\text{4}}$ away from the centre of the rod, and perpendicular to it, is:                                   [JEE MAIN Held on 07-01-2020 Morning]

A)
$\sqrt{\frac{\text{7}}{\text{48}}}l$

B)
$\frac{\text{1}}{\text{8}}l$

C)
$\frac{1}{4}l$

D)
$\sqrt{\frac{\text{3}}{\text{8}}}l$

• question_answer109)  As shown in the figure, a bob of mass m is tied by a massless string whose other end portion is wound on a fly wheel (disc) of radius r and mass m. When released from rest the bob starts falling vertically. When it has covered a distance of h, the angular speed of the wheel will be: [JEE MAIN Held on 07-01-2020 Morning]

A)
$r\sqrt{\frac{3}{2gh}}$

B)
$r\sqrt{\frac{3}{4gh}}$

C)
$\frac{1}{r}\sqrt{\frac{4gh}{3}}$

D)
$\frac{1}{r}\sqrt{\frac{2gh}{3}}$

• question_answer110)  Three point particles of masses 1.0 kg, 1.5 kg and 2.5 kg are placed at three corners of a right angle triangle of sides 4.0 cm, 3.0 cm and 5.0 cm as shown in the figure. The center of mass of system is at a point [JEE MAIN Held on 07-01-2020 Morning]

A)
1.5 cm right and 1.2 cm above 1 kg mass

B)
2.0 cm right and 0.9 cm above 1 kg mass

C)
0.9 cm right and 2.0 cm above 1 kg mass

D)
0.6 cm right and 2.0 cm above 1 kg mass

• question_answer111) Mass per unit area of a circular disc of radius a depends on the distance r from its centre as$\sigma \,(r)=A+Br$. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is: [JEE MAIN Held on 07-01-2020 Evening]

A)
$2\pi {{a}^{4}}\,\left( \frac{A}{4}+\frac{B}{5} \right)$

B)
$\pi {{a}^{4}}\,\left( \frac{A}{4}+\frac{aB}{5} \right)$

C)
$2\pi {{a}^{4}}\,\left( \frac{A}{4}+\frac{aB}{5} \right)$

D)
$2\pi {{a}^{4}}\,\left( \frac{aA}{4}+\frac{B}{5} \right)$

• question_answer112) An elevator in a building can carry a maximum of 10 persons, with the average mass of each person being 68 kg. The mass of the elevator itself is 920 kg and it moves with a constant speed of 3 m/s. The frictional force opposing the motion is 6000 N. If the elevator is moving up with its full capacity, the power delivered by the motor to the elevator $(g=10\text{ }m/{{s}^{2}})$must be at least                                     [JEE MAIN Held on 07-01-2020 Evening]

A)
56300 W

B)
66000 W

C)
48000 W

D)
62360 W

• question_answer113)  Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is$\mu =0.4,$ the maximum possible value of $100\times \frac{b}{a}$ for box not to topple before moving is _________.    [JEE MAIN Held on 07-01-2020 Evening]

• question_answer114)  The coordinates of centre of mass of a uniform Flag shaped lamina (thin flat plate) of mass 4 kg. (The coordinates of the same are shown in figure) are                            [JEE MAIN Held On 08-01-2020 Morning]

A)
$\left( 0.75\text{ }m,\text{ }1.75\text{ }m \right)$

B)
$\left( 1.25\text{ }m,\text{ }1.50\text{ }m \right)$

C)
$\left( 1\text{ }m,\text{ }1.75\text{ }m \right)$

D)
$\left( 0.75\text{ }m,\text{ }0.75\text{ }m \right)$

• question_answer115) Consider a uniform rod of mass M = 4 m and length l pivoted about its centre. A mass moving with velocity v making angle$\theta =\frac{\pi }{4}$to the rods long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is                [JEE MAIN Held On 08-01-2020 Morning]

A)
$\frac{4}{7}\frac{V}{l}$

B)
$\frac{3}{7}\frac{V}{l}$

C)
$\frac{3}{7\sqrt{2}}\frac{V}{l}$

D)
$\frac{3\sqrt{2}}{7}\frac{V}{l}$

• question_answer116) A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of 5.00 cm/s. Its kinetic energy is                                                  [JEE MAIN Held on 08-01-2020 Evening]

A)
$6.25\times {{10}^{4}}J$

B)
$1.13\times {{10}^{3}}J$

C)
$8.75\times {{10}^{\,4}}\text{ }J$

D)
$8.75\times {{10}^{3}}\text{ }J$

• question_answer117) A particle of mass m is dropped from a height h above the ground. At the same time another particle of the same mass is thrown vertically upwards from the ground with a speed of$\sqrt{2gh}$. If they collide head-on completely inelastically, the time taken for the combined mass to reach the ground, in units of $\sqrt{\frac{h}{g}}$ is                                                                                                 [JEE MAIN Held on 08-01-2020 Evening]

A)
$\sqrt{\frac{1}{2}}$

B)
$\sqrt{\frac{3}{4}}$

C)
$\frac{1}{2}$

D)
$\sqrt{\frac{3}{2}}$

• question_answer118)  As shown in fig. when a spherical cavity (centred at O) of radius 1 is cut out of a uniform sphere of radius R (centred at C), the centre of mass of remaining (shaded) part of sphere is at G, i.e. on the surface of the cavity. R can be determined by the equation                                    [JEE MAIN Held on 08-01-2020 Evening]

A)
$\left( {{R}^{2}}\text{+}R+1 \right)\left( 2R \right)=1$

B)

C)

D)

• question_answer119)  Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d. The ratio  of moment of inertia of the system about an axis passing the centroid and about center of any of the spheres  and perpendicular to the plane of the triangle is: [JEE MAIN Held on 09-01-2020 Morning]

A)

B)

C)

D)

• question_answer120)  One end of a straight uniform 1 m long bar is pivoted on horizontal table. It is released from rest when it makes an angle  from the horizontal (see figure). Its angular speed when it hits the table is given as , where n is an integer. The value of n is ______ [JEE MAIN Held on 09-01-2020 Morning]

• question_answer121) A rod of length L has non-uniform linear mass density given by, where a and b are constants and. The value of x for the centre of mass of the rod is at            [JEE MAIN Held on 09-01-2020 Evening]

A)

B)

C)

D)

• question_answer122)  A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see fig.), A massless string is warapped over its rim and two blocks of masses  and  are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when descents by a distance h is [JEE MAIN Held on 09-01-2020 Evening]

A)

B)

C)

D)