Frame of Reference

Q. The time period of mass suspended from a spring is T. If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be

1. $\frac{T}{4}$
2. T
3. $\frac{T}{2}$
4. 2T

Q. A body rolls down an inclined plane without slipping. The kinetic energy of rotation is $50\%$ of its translational kinetic energy. The body is :

1. Solid sphere
2. Solid cylinder
3. Ring
4. Hollow cylinder

Q.

In the given figure, a body of mass $M$ is held between two massless spring on a smooth inclined plane. The free end of the spring are attached to firm supports. If each spring has spring constant $K$, the frequency of oscillation of the given body:

1. $\frac{1}{2\pi }\sqrt{\frac{2K}{Mg\sin \alpha }}$
2. $\frac{1}{2\pi }\sqrt{\frac{K}{Mg\sin \alpha }}$
3. $\frac{1}{2\pi }\sqrt{\frac{2K}{M}}$
4. $\frac{1}{2\pi }\sqrt{\frac{K}{2M}}$

Q. A uniform cylinder of mass $m$ and radius $R$ rolls without slipping down a slope of angle $\theta$ with the horizontal. The cylinder is connected to a spring having spring constant $K$ while the other end of the spring is connected to a rigid support at $P$. The cylinder is released when the spring is unstretched. The maximum distance that the cylinder travel along the plane is

1. $\frac{3mg\sin \theta }{4K}$
2. $\frac{mg\sin \theta }{K}$
3. $\frac{4mg\sin \theta }{3K}$
4. $\frac{2mg\sin \theta }{K}$

Q. A solid sphere as shown is rolling without slipping. What is the maximum distance travelled by sphere on the inclined plane ?
Given $g=10{\text{m/s}}^2$

1. $12\text{m}$
2. $14\text{m}$
3. $16\text{m}$
4. $18\text{m}$

Q. The centre of a wheel rolling on a plane surface moves with a speed $v_0$. A particle on the rim of the wheel at the same level as the centre will be moving at a speed $\sqrt{x}{v}_0$. Then the value of $x$ is.

Q. When $W$ joule of work is done on a flywheel, its frequency of rotation increases from ${\nu }_1\text{Hz}$ to ${\nu }_2\text{Hz}$. The moment of inertia of the flywheel about its axis of rotation is given by

1. $\frac{W}{2{\pi }^2({\nu }_2^2-{\nu }_1^2)}$
2. $\frac{W}{2{\pi }^2({\nu }_2^2+{\nu }_1^2)}$
3. $\frac{W}{4{\pi }^2({\nu }_2^2-{\nu }_1^2)}$
4. $\frac{W}{4{\pi }^2({\nu }_2^2+{\nu }_1^2)}$

Q. A block of mass $2.0\text{kg}$ slides on a rough surface. At $t=0$, its speed is $2.0\text{m/s}$. It stops after covering a distance of $20\text{cm}$ because of the friction exerted by the surface on it. Find the work done by friction.

1. $-8\text{J}$
2. $-4\text{J}$
3. $-5\text{J}$
4. $-7\text{J}$

Q. A rod of mass M and length $L$ is standing on frictionless horizontal surface vertically. With slight disturbance it starts to fall. When rod is making angle of ${37}^{\circ }$ with vertical,

1. Speed of center of mass is $\frac{gL}{3}$
2. Angular velocity of rod is $\sqrt{\frac{15g}{13L}}$
3. Velocity of center of mass is vertically downwards
4. Speed of point of contact with ground is $\sqrt{\frac{12gL}{65}}$

Q. The time period of a mass suspended from a spring is $T$. If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be

1. $\frac{T}{4}$
2. $T$
3. $2T$
4. $\frac{T}{2}$

Q. A long block $A$ is at rest on a smooth horizontal surface. A small block $B$, whose mass is half of $A$, is placed on $A$ at one end and projected along $A$ with some velocity $u$. The coefficient of friction between the block is $\mu$.

1. The blocks will reach a final common velocity $\frac{u}{3}$
2. The work done against friction is two - thirds of the initial kinetic energy of B.
3. Before the blocks reach a common velocity, the acceleration of A relative to B is $\frac{2}{3}\mu g$
4. Before the blocks reach a common velocity the acceleration of A relative to B is $\frac{3}{2}\mu g$

Q. A block of mass m rigidly attached with a spring k is compressed through a distance A. If the block is released, the period of oscillation of the block for a complete cycle is equal to

1. 
2. 
3. 
4. 

Q. Two satellites of the same mass are lauched in the same orbit around the earth so as to rotate opposite to each other. If they collide inelastically and stick together as wreckage, the total energy of the system just after collision is

1. $-\frac{2GMm}{r}$
   
2. $-\frac{GMm}{r}$
   
3. $\frac{2GMm}{r}$
   
4. $\frac{GMm}{2r}$
   
   

Q. A particle of mass $m$ is attached to three identical springs $A,B$ and $C$ each of force constant $k$ as shown in figure. If the particle of mass $m$ in is pushed slightly against the spring $A$ and released, then the time period of oscillation is

1. $2\pi \sqrt{\frac{2m}{k}}$
2. $2\pi \sqrt{\frac{m}{2k}}$
3. $2\pi \sqrt{\frac{m}{k}}$
4. $2\pi \sqrt{\frac{m}{3k}}$

Q. A light rod of length $L$ can revolve in a vertical circle around point $O$. The rod carries two equal masses of mass $m$ each such that one mass is connected at the end of the rod and the second mass is fixed at the middle of the rod. $u$ is the velocity imparted to the end $P$ to deflect the rod to the horizontal position. Again mass $m$ in the middle of the rod is removed and mass at end $P$ is doubled. Now $v$ is the velocity imparted to end $P$ to deflect it to the horizontal position. Then ${\left(\frac{u}{v}\right)}^2$ is

Q. Find out the distance travelled by the block as shown in figure. If the initial speed of the block is $V$ and $\mu$ is the friction coefficient between the surface of block and ground.

1. $\frac{V^2}{2\mu g}$
2. $\frac{V^2}{\mu g}$
3. $\frac{V^2}{3\mu g}$
4. $\frac{V^2}{5\mu g}$

Q. A mass $m$ is hung on an ideal massless spring. Another equal mass is connected to the other end of the spring. The whole system is at rest. At $t=0$, $m$ is released and the system falls freely under gravity. Assume that natural length of the spring is $L_0$, its initial stretched length is $L$ and the acceleration due to gravity is $g$.
What is distance between masses as function of time ?

1. $L_0+2(L+2L_0)\cos \sqrt{\frac{2k}{m}}t$
2. $L_0+(L-L_0)\cos \sqrt{\frac{2k}{m}}t$
3. $L_0+(L-L_0)\sin \sqrt{\frac{2k}{m}}t$
4. $L_0+(L-L_0)\cos \sqrt{\frac{k}{m}}t$

Q. A particle of mass $m$ undergoing oscillation about $x=0$ has varying potential energy field given by $U\left(x\right)=\frac{1}{2}k{x}^2-V_0\text{cos}\left(\frac{x}{a}\right)$ where $V_0,k,a$ are constants and $x$ is its displacement. If the amplitude of oscillation is much smaller than $a$, the time period is given by

1. $2\pi \sqrt{\frac{m{a}^2}{k{a}^2+V_0}}$
2. $2\pi \sqrt{\frac{m}{k}}$
3. $2\pi \sqrt{\frac{m{a}^2}{V_0}}$
4. $2\pi \sqrt{\frac{m{a}^2}{k{a}^2-V_0}}$

Q. A particle of mass $m$ is projected with a velocity $v=k{v}_e(k<1)$ from the surface of the earth. $(v_e=$ escape velocity$)$. The maximum height above the surface reached by the particle is -

$R$ is the radius of earth.

1. $\frac{R{k}^2}{1-k^2}$
2. $R{\left(\frac{k}{1-k}\right)}^2$
3. $R{\left(\frac{k}{1+k}\right)}^2$
4. $\frac{R^{2}k}{1+k}$

Q. A rod of mass M and length $L$ is standing on frictionless horizontal surface vertically. With slight disturbance it starts to fall. When rod is making angle of ${37}^{\circ }$ with vertical,

1. Speed of center of mass is $\frac{gL}{3}$
2. Angular velocity of rod is $\sqrt{\frac{15g}{13L}}$
3. Velocity of center of mass is vertically downwards
4. Speed of point of contact with ground is $\sqrt{\frac{12gL}{65}}$

Q. A satellite is moving with a constant speed $v$ in circular orbit around the earth. An object of mass $m$ is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of ejection, the kinetic energy of the object is

1. $2m{v}^2$
2. $m{v}^2$
3. $\frac{1}{2}m{v}^2$
4. $\frac{3}{2}m{v}^2$

Q. A ball of mass $4\text{kg}$ is moving in 3 dimensional plane and velocity of ball w.r.t an observer is ${\overrightarrow{V}}_1=-3\hat{i}+4\hat{j}+5\hat{k}$ and velocity of observer w.r.t ground is given as ${\overrightarrow{V}}_2=2\hat{i}+2\hat{j}-4\hat{k}$. Find out kinetic energy of ball w.r.t ground.

1. $100\text{J}$
2. $38\text{J}$
3. $76\text{J}$
4. $19\text{J}$

Q. A block is moving with a velocity of $10\text{m/s}$ and the observer is moving in the opposite direction at a speed of $20\text{m/s}$. Find the ratio of the kinetic energy of block w.r.t ground to the kinetic energy of block w.r.t observer.

1. value of mass is necessary to determine the ratio.
2. $\frac{1}{3}$
3. $\frac{1}{9}$
4. $\frac{1}{2}$

Q. Two charges $+5\mu C$ and $-5\mu C$ separated by $4mm$ form an electric dipole. The dipole is placed in a uniform electric field of $4\times {10}^{5}N/C$ . The work done in rotating the electric dipole through ${180}^o$ , if it starts from the positions of $\theta =0$ is

1. $4mJ$
2. $12mJ$
3. $16mJ$
4. $8mJ$

Q. A mass $M$ is suspended form a spring of negligible mass. The spring is pulled a little and then released so that the mass executes simple harmonic oscillations with a time period $T$. If the mass is increased by $m$ then the time period becomes $\left(\frac{5}{4}T\right)$. Find the ratio $\frac{m}{M}$.

Q. A car is moving along a straight horizontal road with a speed $v_0$. If the coefficient of friction between the tyres and the road is $\mu$, the shortest distance in which the car be stopped is?

1. $\frac{v_0^2}{\mu g}$
2. $\frac{v_0^2}{2\mu g}$
3. ${\left(\frac{v_0}{\mu g}\right)}^2$
4. $\frac{v_0^2}{3\mu g}$

Q. A block of mass $5kg$ is on a rough horizontal surface and is at rest. Now a force of $24N$ is imparted to it with negligible impulse. If the coefficient of kinetic friction is $0.4$ and $g=9.8m/s^2$, then the acceleration of the block is

Q.

A particle moves on a rough horizontal ground with some initial velocity say v₀. If (3/4)th of its kinetic energy is lost in friction in time t₀, then coefficient of friction between the particle and the ground is.

1. $v_0/2g{t}_0$

2. $v_0/4g{t}_0$

3. $3v_0/4g{t}_0$

4. $v_0/g{t}_0$

Q. A wagon of mass $M$ has a block of mass $m$ attached to it as shown in the figure. The coefficient of friction between the block and wagon is $\mu$. The minimum acceleration of the wagon so that the block $m$ does not fall is

1. $\frac{g}{\mu }$
2. $\frac{\mu }{g}$
3. $\mu g$
4. $\frac{M\mu g}{m}$

Q. Acceleration of block $A$ varies with time as shown in figure the value of coefficient of kinetic friction between $A$ and $B$ is
the value of coefficient of kinetic friction between block $A$ and $B$ is