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Give the location of the centre of mass of mass of (i) sphere, (ii) cylinder, (in) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body ?
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In the
molecule, the
separation between the nuclei of the two atoms is about
. Find the approximate
location of the CM of the molecule, given that the chlorine atom is about 35.5
times as massive as a hydrogen atom and nearly all the mass of an atom is
concentrated in all its nucleus.
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A child sits stationary at one end of a long trolley moving uniformly with speed v on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, then what is the effect of the speed of the centre of mass of the (trolley + child) system ?
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Show that the area of the triangle
contained between the sectors
and
is one half of
the magnitude of
.
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Show that
is equal in magnitude
to the volume of the parallelepiped formed on the three vectors,
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Find the components along the
axes of the angular
momentum
of a particle,
whose position vector is
with
components
and momentum is
with components
and
. Show that if the
particle moves only in the
plane,
the angular momentum has only a z-component.
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Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the vector angular momentum of the two particle system is the same whatever be the point about which the angular momentum is taken.
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A non-uniform bar of weight W is
suspended at rest by two strings of negligible weights as shown in Fig. 7.100.
The angles made by the strings with the vertical are 36.9° and 53.1°
respectively. The bar is 2 m long. Calculate the distance d of the centre of
gravity of the bar from its left end.
Fig. 7.100
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A car weighs 1800 kg. The distance between its front and back axles is 1.8 m. Its centre of gravity is 1.05 m behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.
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(a) Find the moment of inertia of a sphere
about a tangent to the sphere, given the moment of inertia of the sphere about any
of its diameters to be
where
M is the mass of the sphere and R is the radius of the sphere. [Central
Schools 11]
(b) Given the moment of inertia of a
disc of mass M and radius R about any of its diameters to be
find its moment
of inertia about an axis normal to the disc and passing through a point on its
edge.
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Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time ?
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A solid cylinder of mass 20 kg rotates
about its axis with angular speed
.
The radius of the cylinder is 0.25 m. What is the kinetic energy associated with
the rotation of the cylinder ? What is the magnitude of angular momentum of the
cylinder about its axis ?
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(i) A child stands at the centre of turntable with his two arms out stretched. The turntable is set rotating with an angular speed of40 rpm How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to 2/3 times the initial value ? Assume that the turntable rotates without friction.
(ii) Show that the child's new kinetic energy of rotation more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy ?
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A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is angular acceleration of the cylinder if the rope is pulled with a force of 30 N ? What is the linear acceleration of the rope ? Assume that there is no slipping.
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To maintain a rotor at a uniform angular
speed of
an engine needs
to transmit a torque of 180 Nm.
What is the power required by the engine
? Assume that the engine is 100% efficient.
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From a uniform disc of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.
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A metre stick is balanced on a knife edge at its centre.
When two coins, each of mass 5 g are put one on top of the other at the 12.0 cm mark, the stick is found to be balanced at 45.0 cm.
What is the mass of the metre stick ?
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A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination, (a) Will it reach the bottom with the same speed in each case ? (b) Will it take longer to roll down one plane than the other ? (c) If so, which one and why ?
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A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/s.
How much work has to be done to stop it ?
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The oxygen molecule has a mass
and a moment of
inertia of
about an axis
through its centre perpendicular to the line joining the two atoms.
Suppose the mean speed of such a molecule
in a gas is 500 m/ s and that its kinetic energy of rotation is two thirds of
its kinetic energy of translation. Find the average angular velocity of the molecule.
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A solid cylinder rolls up an inclined plane of angle of inclination 30°. At the bottom of the inclined plane the centre of mass of the cylinder has a speed of 5 m/s.
(a) How far will the cylinder go up the plane ?
(b) How long will it take to return to the bottom ?
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As shown in Fig. 7.105, the two sides of
a step ladder BA and CA are 1.6 m long and hinged at A. A rope DE, 0.5 m is tied
half way up. A weight 40 kg is suspended from a point F, 1.2 m from B along the
ladder BA. Assuming the floor to be frictionless and neglecting the weight of
the ladder, find the tension in the rope and forces exerted by the floor on the
ladder
(Take
).
Fig. 7.105
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A man stands on a rotating platform,
with his arms stretched horizontally holding a 5kg weight in each hand. The angular
speed of the platform is 30 revolutions per minute. The man then brings his
arms close to his body with the distance of each weight from the axis changing
from 90 cm to. 20 cm. The moment of inertia of the man together with the platform
maybe taken to be constant and equal to
.
Ans.
(a) What is his new angular speed ? (Neglect friction)
(b) Is kinetic energy conserved in the
process ? If not/from where does the change come about ?
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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 1.0m wide and weighs 12 kg. It is hinged at
one end and rotates about a vertical axis practically without friction, find the
angular speed of the door just after the bullet embeds into it.
(Hint.
The moment of inertia of the door about the vertical axis at one end is )
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Two discs of moments of inertia
and
about their respective
axes (normal to the disc and passing through the centre), and rotating with
angular speed
and
are brought into
contact face to face with their axes of rotation coincident.
(i) What is the angular speed of the
two-disc system? (ii) Show that the kinetic energy of the combined system is
less than the sum of the initial kinetic energies of the two discs. How do you account
for this loss in energy ? Take
.
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(a) Prove the theorem of perpendicular axes.
(b) Prove the theorem of parallel axes.
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Prove the result that the velocity
of translation
of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an
inclined plane of a height h is given by
using dynamical consideration (i.e., by
consideration of forces and torques). Note k is the radius of gyration of the
body about its symmetry axis, and R is the radius of the body. The body starts
from rest at the top of the plane.
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A disc rotating about its axis with angular speed coq is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is R What are the linear velocities of the points A B and C on the disc shown in Fig. 7.108 ? Will the disc roll in the direction indicated ?
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Explain why friction is necessary to make the disc in Fig. 7.108 given in previous question roll in the direction indicated.
(i) Give the direction of frictional force at B, and the sense of frictional torque, before perfect rolling begins.
(ii) What is the force of friction after perfect rolling begins ?
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A solid disc and a ring, both of radius
10 cm are placed on a horizontal table simultaneously, with initial angular
speed equal to
.
Which of the two will start to roll earlier ? The co-efficient of kinetic
friction is
.
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A solid cylinder of mass 10 kg and
radius 15 cm is rolling perfectly on a plane of inclination 30°. The
coefficient of static friction,
.
(i) Find the force of friction acting on the cylinder, (ii) What is the work
done against friction during rolling ? (in) If the inclination 6 of the plane
is increased, at what value of
does
the cylinder begin to skid, and not roll perfectly ?
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question_answer32)
For
which of the following does the centre of mass lie outside the body?
(a)
A pencil (b) A shotput
(c)
A dice (d) A bangle.
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question_answer33)
Which
of the following points is the likely position of the centre of mass of the
system shown in Fig.?
Hollow
(a) A (b)
B
(c)
C (d) D
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question_answer34)
A
particle of mass m is moving in yz-plane with a uniform velocity \[\upsilon
\] with its trajectory running parallel to +ve y-axis and intersecting z-axis
at \[z=a\] (Fig.).
The change in its angular momentum about the origin as it bounces elastically
from a wall at \[y=\] constant
is:
(a) \[m\upsilon
a\,\,\hat{e}\,x\] (b) \[2m\upsilon a\,\,\hat{e}x\]
(c) \[ym\upsilon
\,\,\hat{e}x\] (d) \[2ym\upsilon \,\,\hat{e}x\]
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question_answer35)
When
a disc rotates with uniform angular velocity, which of the following is not
true?
(a)
The sense of rotation remains same.
(b)
The orientation of the axis of rotation remains same.
(c)
The speed of rotation is non-zero and remains same.
(d)
The angular acceleration is non-zero and remains same.
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question_answer36)
A
uniform square plate has a small piece Q of an irregular shape removed and
glued to the centre of the plate leaving a hole behind (Fig.). The moment of
inertia about the z-axis is then
(a) Increased
(b) decreased
(c) the same
(d) changed in
unpredicted manner
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question_answer37)
In
problem 5, the C.M. of the plate is now in the following quadrant of \[x-y\]
plane,
(a)
I (b) II
(c)
III (d) IV
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question_answer38)
The
density of a non-uniform rod of length 1m is given by \[\rho
\,(x)\,=\,a(1+\,b{{x}^{2}})\] where a and b are constants and \[0\le \,x\le \,1.\]
The centre of
mass of the rod will be at
(a)
\[\frac{3(2+b)}{4(3+\,b)}\] (b) \[\frac{4(2+b)}{3(3+\,b)}\]
(c)
\[\frac{3(3+b)}{4(2+\,b)}\] (d) \[\frac{4(3+b)}{3(2+\,b)}\]
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question_answer39)
A
Merry-go-round, made of a ring-like platform of radius R and mass M, is
revolving with angular speed \[\omega \]. A person of mass M is standing on it.
At one instant, the person jumps off the round, radially away from the centre
of the round (as seen from the round). The speed of the round afterwards is
(a)
\[2\omega \] (b) \[\omega \]
(c)
\[\frac{\omega }{2}\] (d) O
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question_answer40)
Choose
the correct alternatives:
(a)
For a general rotational motion, angular momentum L and angular velocity \[\omega
\] need
not be parallel.
(b)
For a rotational motion about a fixed axis, angular momentum L and angular
velocity \[\omega \] are
always parallel.
(c)
For a general translational motion momentum p and velocity \[\upsilon \]
are always
parallel.
(d)
For a general translational motion, acceleration a and velocity \[\upsilon \]
are always
parallel.
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question_answer41)
Figure
shows two identical particles 1 and 2, each of mass m, moving in opposite
directions with same speed \[\upsilon \] along parallel lines. At a particular
instant, \[{{r}_{1}}\] and
\[{{r}_{2}}\] are
their respective position vectors drawn from point A which is in the plane of
the parallel lines.
Choose
the correct options:
(a)
Angular momentum I1 of particle 1 about A is \[{{l}_{1}}=m\upsilon
{{r}_{1}}e\]
(b)
Angular momentum l2 of particle 2 about A is \[{{l}_{2}}=m\upsilon
{{r}_{2}}e\]
(c)
Total angular momentum of the system about A is \[l=m\upsilon
({{r}_{1}}+{{r}_{2}})e\]
(d) Total angular
momentum of the system about A is \[l=m\upsilon ({{d}_{2}}-{{d}_{1}})\otimes \]
e
represents a unit vector coming out of the page.
\[\otimes
\] represents
a unit vector going into the page.
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question_answer42)
The
net external torque on a system of particles about an axis is zero. Which of
the following are compatible with it?
(a)
The forces may be acting radially from a point on the axis.
(b)
The forces may be acting on the axis of rotation.
(c)
The forces may be acting parallel to the axis of rotation.
(d)
The torque caused by some forces may be equal and opposite to that caused by
other forces.
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question_answer43)
Figure shows a
lamina in \[x-y\] plane.
Two axes z and z' pass perpendicular to its plane. A force F acts in the plane
of lamina at point P as shown. Which of the following are true ? (The point P
is closer to z'-axis than the z-axis.)
(a)
Torque \[\tau \] caused by F about z axis is along \[-\hat{k}\].
(b)
Torque \[\tau '\] caused
by F about z' axis is along \[-\hat{k}\].
(c)
Torque \[\tau \] caused by F about z axis is greater in magnitude than that
about z axis.
(d)
Total torque is given be \[\tau =\,\tau +\,\tau '\].
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question_answer44)
With
reference to Fig. of a cube of edge a and mass m, state whether the following
are true or false. (O is the centre of the cube.)
(a) The moment of
inertia of cube about z-axis is \[{{I}_{z}}={{I}_{x}}+\,{{I}_{y}}\]
(b) The moment of
inertia of cube about z? is \[I{{'}_{z}}={{I}_{z}}+\frac{m{{a}^{2}}}{2}\]
(c) The moment of
inertia of cube about z? is \[={{I}_{z}}=\frac{m{{a}^{2}}}{z}\]
(d) \[{{I}_{x}}={{I}_{y}}\]
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question_answer45)
The
centre of gravity of a body on the earth coincides with its centre of mass for
a 'small' object whereas for an 'extended' object it may not. What is the
qualitative meaning of 'small' and 'extended' in this regard? For which of the
following the two coincides?
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question_answer46)
Why
does a solid sphere have smaller moment of inertia than a hollow cylinder of
same mass and radius, about an axis passing through their axes of symmetry?
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question_answer47)
The
variation of angular position q,
of a point on a rotating rigid body, with time t is shown in Fig. Is the body
rotating clockwise or anti-clockwise?
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question_answer48)
A
uniform cube of mass m and side a is placed on a frictionless
horizontal surface. A vertical force F is applied to the edge as shown in Fig.
Match the following (most appropriate choice):
(a) \[mg/4\,<F<mg/2\]
|
(i) Cube will move up.
|
(b) \[F>mg/2\]
|
(ii) Cube will not exhibit motion.
|
(c) \[F>mg\]
|
(iii) Cube will begin to rotate and slip
at A.
|
(d) \[F=mg/4\]
|
(iv) Normal reaction effectively at a/3
from A, no motion.
|
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question_answer49)
A
uniform sphere of mass m and radius R is placed on a rough horizontal surface
(fig.) The sphere is struck horizontal at a height h from the floor. Match the
following:
(a) \[h=R/2\]
|
(i) Sphere rolls without
slipping with a constant velocity and no loss of energy.
|
(b) \[h=R\]
|
(ii) Sphere spring clockwise,
loses energy by friction.
|
(c) \[h=3R/2\]
|
(iii) Sphere spins
anticlockwise, lose energy by friction.
|
(d) \[h=7R/5\]
|
(iv) Sphere has only a
translation motion, looses energy by friction.
|
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question_answer50)
The
vector sum of a system of non-collinear forces acting on a rigid body is given
to be non-zero. If the vector sum of all the torques due to the system of
forces about a certain point is found to be zero, does this mean that it is
necessary zero about any arbitrary point?
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question_answer51)
A
wheel in uniform motion about an axis passing through its centre and
perpendicular to its plane is considered to be in mechanical (translational
plus rotational) equilibrium because no net external force or torque is
required to sustain its motion. However, the particles that constitute die
wheel do experience a centripetal acceleration directed towards die centre. How
do you reconcile this fact with the wheel being in equilibrium? How would you
set a half-wheel into uniform motion about an axis passing through the centre
of mass of the wheel and perpendicular to its plane? Will you require external
forces to sustain the motion?
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question_answer52)
A
door is hinged at one end and is free to rotate about a vertical axis (Fig.).
Does its weight cause any torque about this axis? Give reason for your answer.
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question_answer53)
\[(n-1)\]
equal point masses each of mass m are placed at die vertices of a regular
n-polygon. The vacant vertex has a position vector a with respect to the centre
of the polygon. Find the position vector of centre of mass.
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question_answer54)
Find
the centre of mass of a uniform (a) half-disc, (b) quarter-disc.
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question_answer55)
Two
discs of moments of inertia I1 and I2 about their
respective axes (normal to the disc and passing through the centre), and
rotating with angular speed \[{{\omega }_{1}}\] and \[{{\omega }_{2}}\] are
brought into contact face to face with their axes of rotation coincident.
(a)
Does the law of conservation of angular momentum apply to the situation? Why?
(b)
Find the angular speed of the two-disc system.
(c)
Calculate the loss in kinetic energy of the system in the process.
(d)
Account for this loss.
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question_answer56)
A
disc of radius R is rotating with an angular speed \[{{\omega }_{0}}\] about a horizontal
axis. It is placed on a horizontal table. The coefficient of kinetic friction
is \[{{\mu }_{k}}\].
(a)
What was the velocity of its centre of mass before being brought in contact
with the table?
(b)
What happens to the linear velocity of a point on its rim when placed in
contact with the table?
(c)
What happens to the linear speed of the centre of mass when disc is placed in
contact with the table?
(d)
Which force is responsible for the effects in (b) and (c).
(e)
What condition should be satisfied for rolling to begin?
(f)
Calculate the time taken for the rolling to begin.
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question_answer57)
Two
cylindrical hollow drums of radii R and 2R, and of a common height h,
are rotating with angular velocities \[\omega \] (anti clockwise) and \[\omega
\] (clockwise), respectively. Their axes, fixed are parallel and in a
horizontal plane separated by \[(3R+\delta )\]. They are now brought in contact
\[(\delta \,\to 0)\].
(a)
Show the frictional forces just after contact.
(b)
Identify forces and torques external to the system just after contact.
(c)
What would be the ratio of final angular velocities when friction ceases?
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question_answer58)
A
uniform square plate S (side c) and a uniform rectangular plate R (sides b, a)
have identical areas and masses.
Show
that
(i) \[{{I}_{xR}}/{{I}_{xS}}\,<\,1;\] (ii) \[{{I}_{yR}}/{{I}_{yS}}>1;\]
(iii) \[{{I}_{zR}}/{{I}_{zS}}>1\].
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question_answer59)
A
uniform disc of radius R, is resting on a table on its rim. The coefficient of
friction between disc and table is \[\mu \]. Now the disc is pulled with a
force F as shown in the figure. What is the maximum value of F for which the
disc rolls without slipping?
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