The viscosity of a fluid u, can be determined by measuring the terminal velocity \[{{V}_{T}}\] of a sphere when it descends in the fluid.
The fluid has a density \[{{\rho }_{f}}\] while the sphere has a density \[{{\rho }_{s}}\] and a diameter of d. The viscosity can then be calculated by the formula \[\mu =\frac{5({{\rho }_{s}}-{{\rho }_{f}})}{9{{V}_{T}}}{{d}^{2}}\]
A disc shaped body has two tight windings of light threads, one on the inner rim of radius \[R=1\text{ }m\]and the other on outer rim of radius 2R (see figure). It is kept on a horizontal surface and the ends of the two threads are pulled horizontally in opposite directions with force of equal magnitude\[F=20N\]. Mass of the body and its moment of inertia about an axis through centre 0 and perpendicular to the plane of the figure are \[M=4\text{ }kg\]and \[I=8\,kg{{m}^{2}}\] respectively. Find the kinetic energy of the body 2 seconds after the forces begin to act, if the surface is rough enough to ensure rolling without sliding.
The linear mass density of the string shown in the figure is \[\mu =1\,g/m.\]. One end of the string is tied to a prong of a tuning fork and the other end carries a block of mass M. The length of the string between the tuning fork and the pulley is\[L=2.0\text{ }m\]. When the tuning fork vibrates, the string resonates with it when mass M is either \[16\text{ }kg\]or\[25\text{ }kg\]. However, standing waves are not observed for any other value of M lying between \[16\text{ }kg\]and \[\text{25 }kg\]. Assume that end A of the string is practically at rest and calculate the frequency of the fork.
A ring of mass M and radius a lies on a smooth horizontal surface. An insect of mass m sitting on it starts crawling on the ring with a constant speed. The trajectory of the centre of the ring is
A)
a circle of radius \[\left( \frac{Ma}{m} \right)\]
doneclear
B)
a circle of radius \[\left( \frac{ma}{M+m} \right)\]
doneclear
C)
a circle of radius \[\left( \frac{ma}{M} \right)\]
A constant tangent force F acts at the top of a solid cylinder of mass m and radius R so that the cylinder rolls on a horizontal surface without slipping. Find the magnitude and the direction of the friction force exerted by the cylinder on the surface.
Three planets of same density and with radii \[{{R}_{1}},{{R}_{2}}\] and \[{{R}_{3}}\] such that \[{{R}_{1}}=2{{R}_{2}}=3{{R}_{3}}\] have gravitation fields on the surface \[{{g}_{1}},{{g}_{2}}\] and \[{{g}_{3}}\] and escape velocities \[{{v}_{1}},{{v}_{2}}\] and \[{{v}_{3}}\] respectively. Then
A body with an initial temperature \[{{\theta }_{i}}\] is allowed to cool in a surrounding which is at a constant temperature of \[{{\theta }_{0}}({{\theta }_{0}}<{{\theta }_{i}})\]. Assume that Newton's law of cooling is obeyed. Let k = constant. The temperature of the body after time t is best expressed by (where B is integrating constant)
A certain amount of an ideal monatomic gas undergoes a thermodynamic process such that \[V{{T}^{2}}\] constant (V = volume of gas and T = temperature of gas). Then under the process
A)
When heat is supplied to the gas its temperature will increase.
doneclear
B)
The coefficient of volume expansion of the gas equals\[\frac{-2}{T}\].
doneclear
C)
The molar heat capacity of the gas is\[2R\].
doneclear
D)
When heat is supplied to the gas its pressure increases.
A wooden disc of mass M and radius R has a single loop of wire wound on its circumference. It is mounted on a massless rod of length d. The ends of the rod are supported at its ends so that the rod is horizontal and disc is vertical. A uniform magnetic field \[{{B}_{0}}\] exists in vertically upward direction. When a current I is given to the wire one end of the rod leaves the support. Find least value of I.
Two identical thin rings, each of radius R are coaxially placed at a distance R apart. If \[{{Q}_{1}}\] and \[{{Q}_{2}}\] are, respectively, the charges uniformly spread on the two rings, the work done in moving a charge q from the centre of one ring to the centre of the other ring is
In the circuit shown, switch \[{{S}_{2}}\]is open and \[{{S}_{1}}\] is closed since long. Take \[E=20\text{ }V,\text{ }L=8.5\text{ }H\]and\[R=10\,\Omega \]. The rate of change of energy stored in the magnetic field inside the inductor, immediately after \[{{S}_{2}}\] is closed, is
The x-z plane separates two media A and B of refractive indices \[{{\mu }_{1}}=1.5\] and \[{{\mu }_{2}}=2\]. A ray of light travels from A to B. Its directions in the two media are given by unit vectors \[{{\vec{u}}_{1}}=a\hat{i}+b\hat{j}\] and \[{{\vec{u}}_{2}}=c\hat{i}+b\hat{j}\] then
If light of wavelength of maximum intensity emitted from surface at temperature T is used to cause photoelectric emission from a metallic surface, the maximum kinetic energy of the emitted electron is \[6\text{ }eV,\] which is 3 times the work function of the metallic surface. If light of wavelength of maximum intensity emitted from a surface at temperature \[{{T}_{2}}({{T}_{2}}=2{{T}_{1}})\] is used, the maximum kinetic energy of a, the photoelectrons emitted is
Two radioactive materials \[{{X}_{1}}\] and \[{{X}_{2}}\] have decay constants \[10\lambda \] and \[2\lambda ,\] respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of \[{{X}_{1}}\] to that of \[{{X}_{2}}\] will be \[1/e\] after a time
In the given circuit, the Zener diode has breakdown voltage \[{{V}_{z}}=3\] volt and the maximum power limit \[{{P}_{\max }}=18\,mW.\] Choose the correct option.
A)
If \[{{V}_{B}}=12\] volt, the power dissipated in Zener diode will exceed the maximum power limit, specified for it.
doneclear
B)
If \[{{V}_{B}}=12\] volt, the power dissipated in Zener diode will not exceed the maximum power limit, specified for it.
doneclear
C)
If \[{{V}_{B}}=15\]volt, the power dissipated in Zener diode will exceed the maximum power limit, specified for it.
doneclear
D)
If \[{{V}_{B}}=20\]volt, the power dissipated in Zener diode will not exceed the maximum power limit, specified for it.
The system shown is in equilibrium. All the strings and pulleys are massless. Instantaneous acceleration of A, B and C when string above B is cut are, respectively,
A particle is shifted from A to B and then from B to C where A, B and C are the midpoints of the corresponding faces of a cube of side 2 m. If a force \[\vec{F}=\left( 3\hat{i}+4\hat{j}-5\hat{k} \right)N\] N is continuously acting on the particle,
Charges \[{{Q}_{1}}\] and \[{{Q}_{2}}\] lie inside and outside respectively of a closed surface S. Let E be the field at any point on S and \[\phi \] be the flux of E over S. Now study the following statements.
(i) If \[{{Q}_{1}}\] changes, both E and \[\phi \] will change,
(ii) If \[{{Q}_{2}}\] changes, E will change but \[\phi \] will not change.
(iii) If \[{{Q}_{1}}=0\] and \[{{Q}_{2}}\ne 0,\] then \[E\ne 0\] but \[\phi =0\]
(iv) If \[{{Q}_{1}}\ne 0\] and \[{{Q}_{2}}=0,\] then \[E=0\]but \[\phi =0\].
Figure shows an infinitely long wire carrying an outward current\[{{I}_{1}}\]. The current is along Z axis. There is a curved wire carrying current\[{{I}_{2}}\]. The magnetic force on this wire between \[({{x}_{1}},{{y}_{1}})\] and \[({{x}_{2}},{{y}_{2}})\] is
A ball is projected from point O on the ground. It hits a smooth vertical wall AB at a height h and rebounds elastically. The ball finally lands at a point C on the ground. During the course of motion, the maximum height attained by the ball is H. It \[\frac{OA}{OC}=\frac{1}{3},\] find the ratio \[\frac{h}{H}.\]
A block of mass 2 kg is kept at origin at \[t=0\] and is having a velocity \[4\sqrt{5}m/s\] in positive x-direction. The only force acting on it is conservative and its potential energy is defined as \[U=-{{x}^{3}}+6{{x}^{2}}+15\] (SI units). Find the velocity (in m/s) when its acceleration is minimum after\[t=0\].
An observer is standing at a point 0, at a distance of 100 cm from a convex lens of focal length\[50\text{ }cm\]. A plane mirror is placed behind the lens at a distance of \[150\text{ }cm\]from the lens. The mirror now starts moving towards the right with a velocity of\[10\text{ }cm/s\]. What will be the magnitude of velocity (in cm/s) other own image as seen by the observer, at the moment when the mirror just starts moving?
A small sphere of mass 1 kg is moving with a velocity\[(6\hat{i}+\hat{j})m{{s}^{-1}}.\]. It hits a fixed smooth wall and rebound with velocity\[(4\hat{i}+\hat{j})m{{s}^{-1}}\]. What is the coefficient of restitution between the sphere and the wall?
In YDSE, distance between the slits and the screen is 2 m. Distance between slits is 2 mm and wavelength of light is 500 nm. A mica sheet of 100 urn thickness and refractive index \[1.5\]is placed after one of the slits. What is the number of fringes crossing a given point?
Ethylene is produced by \[{{C}_{4}}{{H}_{8}}\xrightarrow{\Delta }2{{C}_{2}}{{H}_{4}}\] The rate constant is \[2.3\times {{10}^{-4}}{{s}^{-1}}\]. Approximately in what time will the molar ratio of ethylene to cyclobutane in mixture attain the value equal to one? \[(\log \,2=0.3,\,\,\log 3=0.47)\]
The bond dissociation energy of \[B-F\]in \[B{{F}_{3}}\]is \[646\text{ }kJ\text{ }mo{{l}^{-1}}\]whereas that of \[C-F\]in \[C{{F}_{4}}\]is\[515\text{ }kJ\text{ }mo{{l}^{-1}}\]. The correct reason for higher \[B-F\] bond dissociation energy as compared to that of \[C-F\] is
A)
stronger a bond between B and F in \[B{{F}_{3}}\]as compared between C and F in\[C{{F}_{4}}\].
doneclear
B)
significant \[p\pi -p\pi \] interaction between B and F in \[B{{F}_{3}}\]whereas there is no possibility of such interaction between C and F in\[C{{F}_{4}}\].
doneclear
C)
lower degree of \[p\pi -p\pi \] interaction between B and F in \[B{{F}_{3}}\]than between C and F in\[C{{F}_{4}}\].
How many compounds will give Cannizzaro reaction? \[C{{H}_{3}}-C{{H}_{2}}OH;\] \[{{C}_{6}}{{H}_{5}}CHO;\] \[C{{H}_{3}}CHO;\] \[HCHO;\] \[{{(C{{H}_{3}})}_{3}}C.CHO;\] \[C{{H}_{3}}-CO-C{{H}_{3}}\]
In a cold climate, water gets frozen causing damage to the radiator of a car. Ethylene glycol is used as an antifreezing agent. Calculate the amount of ethyiene glycol to be added to 4 kg of water to prevent it from freezing at \[-6{}^\circ C\] (\[{{K}_{f}}\]for water\[=1.85\text{ }K\text{ }mol{{e}^{-1}}\text{ }kg\])
2 moles of an ideal gas at \[27{}^\circ C\] temperature is expanded reversibly from \[2\text{ }L\]to\[20\text{ }L\]. Calculate the entropy change in calories \[(R=2\,cal/mol\,K)\]
If a variate assumes the values 0, 1,2, ...., n with frequencies\[^{n}{{C}_{0}}{{,}^{n}}{{C}_{1}}{{,}^{n}}{{C}_{2}},...{{,}^{n}}{{C}_{n}}\] then mean square deviation about the value x = 0 is
If the tangent drawn at a point \[({{t}^{2}},2t)\]on the parabola \[{{y}^{2}}=4x\] is same as normal drawn at \[(\sqrt{5}cos\alpha ,2sin\alpha )\]on the ellipse\[\frac{{{x}^{2}}}{5}+\frac{{{y}^{2}}}{4}=1\], then which of the following is not true?
Let a, b, c be the three roots of the equation\[{{x}^{3}}+{{x}^{2}}-333x-1002=0\].If\[P={{a}^{3}}+{{b}^{3}}+{{c}^{3}},\]then the value of \[\frac{P}{2006}=\]
If \[\sin \theta +{{\sin }^{2}}\theta +{{\sin }^{3}}\theta =1\], then the value of \[{{\cos }^{6}}\theta -4{{\cos }^{4}}\theta +8{{\cos }^{2}}\theta \]must be