Energy Conservation Method

Q. A uniform chain of length $3\text{m}$ and mass $3\text{kg}$ overhangs a smooth table with $2\text{m}$ laying on the table. If $k$ is the kinetic energy of the chain in joule as it completely slips off the table, then the value of $k$ is ______.

Take $g=10{\text{m/s}}^2$

Q. The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane
$\left(i\right)$ a ring of radius $R$,
$\left(ii\right)$a solid cylinder of radius $\frac{R}{2}$ and
$\left(iii\right)$ a solid sphere of radius $\frac{R}{4}$.
If in each case, the speed of the center of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is

1. $4:3:2$
2. $2:3:4$
3. $10:15:7$
4. $14:15:20$

Q. Two blocks A and B each of mass m are connected by a massless spring of natural length L and spring constant k. The blocks are initially resting on a smooth horizontal floor with the spring as its natural length, as shown in figure.

A third identical block C, also of mass m moves on the floor with a speed v along the line joining A and B collides with A, elastically, then

1. The kinetic energy of AB system at maximum compression of the spring is zero.
2. The kinetic energy of the AB system at maximum of the spring is .
3. The maximum compression of the spring is v .
4. The maximum compression of the spring is v

Q. A solid uniform cylinder of mass $M$ performs small oscillations in the horizontal plane if slightly displaced from its mean position as shown in the figure. Initially, the springs are at their natural lengths and the cylinder does not slip on the ground during oscillations due to friction between the ground and cylinder. The force constant of each spring is $k$. What is the time period of this oscillation?

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

Q. One end of a long metallic wire of length $l$ is tied to the ceiling. The other end is tied to a massless spring of spring constant $K$. A mass ($M$) hangs freely from the free end of the spring. The area of cross-section and Young's modulus of the wire are $a$ and $Y$ respectively. If the mass is slightly pulled down and released, it will oscillate with a frequency $f$ equal to

1. $\frac{1}{2\pi }\sqrt{\frac{Kl}{MYa}}$
2. $\frac{1}{2\pi }\sqrt{\frac{Ya}{Ml}}$
3. $\frac{1}{2\pi }\sqrt{\frac{YaK}{M(Ya+Kl)}}$
4. $\frac{1}{2\pi }\sqrt{\frac{K}{M}}$

Q. A uniform plank of mass $m=1\text{kg}$ and of sufficient length , which is free to move only in the horizontal direction is placed upon the top of a solid cylinder of mass $2m$ and radius $R$. The plank is attached to a fixed wall by means of a light spring of spring constant $k=7\text{N/m}$. Assuming , there is no slipping between cylinder and the plane system, and between cylinder and the ground. Find the Angular frequency of small oscillations of the system.

1. $2\text{rad/sec}$
2. $3\text{rad/sec}$
3. $1\text{rad/sec}$
4. $4\text{rad/sec}$

Q. Two spring systems have equal mass and spring constants $k_1$ and $k_2$. If the maximum velocities in two systems are equal, then the ratio of amplitude of first to that of second is,

1. $\sqrt{\frac{k_1}{k_2}}$
2. $\frac{k_1}{k_2}$
3. $\frac{k_2}{k_1}$
4. $\sqrt{\frac{k_2}{k_1}}$

Q. A solid cylinder (radius $r=0.2\text{m}$) rolls without slipping in a cylindrical trough (radius $R=0.8\text{m}$). The time period of small oscillations is
[Take $g=10{\text{m/s}}^2$]

1. $\frac{3\pi }{5}\text{sec}$
2. $\frac{\pi }{5}\text{sec}$
3. $\frac{\pi }{3}\text{sec}$
4. $\frac{2\pi }{5}\text{sec}$

Q. A uniform rod of mass M and length is hanging from its central point as shown. A small ball of equal mass M is attached to the lower end. Time period of small oscillation of the rod is

1. $2\pi \sqrt{\frac{l}{2g}}$
2. $2\pi \sqrt{\frac{l}{3g}}$
3. $2\pi \sqrt{\frac{l}{g}}$
4. $2\pi \sqrt{\frac{2l}{3g}}$

Q. A uniform plank of mass $m=1\text{kg}$ and of sufficient length , which is free to move only in the horizontal direction is placed upon the top of a solid cylinder of mass $2m$ and radius $R$. The plank is attached to a fixed wall by means of a light spring of spring constant $k=7\text{N/m}$. Assuming , there is no slipping between cylinder and the plane system, and between cylinder and the ground. Find the Angular frequency of small oscillations of the system.

1. $2\text{rad/sec}$
2. $1\text{rad/sec}$
3. $3\text{rad/sec}$
4. $4\text{rad/sec}$

Q. A uniform plank of mass $m=2kg$ free to move in the horizontal direction only, is placed at the top of a solid cylinder of mass $M=2kg$ and radius $R$. The plank is attached to a fixed wall by means of a light spring of spring constant $K=11N/m$. There is no slipping between the cylinder and the plank, and between the cylinder and the ground. The angular frequency $\left(rad/s\right)$ of small oscillation of the system

Q. A solid cylinder (radius $r=0.2\text{m}$) rolls without slipping in a cylindrical trough (radius $R=0.8\text{m}$). The time period of small oscillations is
[Take $g=10{\text{m/s}}^2$]

1. $\frac{3\pi }{5}\text{sec}$
2. $\frac{\pi }{5}\text{sec}$
3. $\frac{2\pi }{5}\text{sec}$
4. $\frac{\pi }{3}\text{sec}$

Q. Two blocks A and B, each of mass m, are connected by a massless spring of natural length L and spring constant k. the blocks are initially resting on a smooth horizontal floor with the spring at its natural length. A third identical block C, also of mass m, moving on the floor with a speed along the line joining A and B, collides with A (see figure) Then

1. The kinetic energy of A – B system at maximum compression of the spring is zero
2. The kinetic energy of A – B system at maximum compression of the spring is
3. The maximum compression of the spring is
4. The maximum compression of the spring is

Q. One end of a light spring of spring constant $200\text{N/m}$ is fixed to a block $\text{A}$ of mass $2\text{kg}$ placed on horizontal frictionless table. The other end of the spring is fixed to a wall. A smaller block $\text{B}$ of mass $3\text{kg}$ is placed on block $\text{A}$. The system is displaced by a small amount and released. What is the magnitude of friction coefficient $\mu$ at the surface of block $\text{A}$ and $\text{B}$ so that the upper block does not slip over lower block $?$
(Amplitude $\left(\text{A}\right)=0.1\text{m})$

1. $\mu =0.7$
2. $\mu =0.5$
3. $\mu =0.66$
4. $\mu =0.25$

Q. The frequency $f$ of vibrations of a mass $m$ suspended from a spring of spring constant $k$ is given by $f=C{m}^x{k}^y$, where $C$ is a dimensionless constant. The values of $x$ and $y$ are respectively:

1. $\frac{1}{2},\frac{1}{2}$
2. $-\frac{1}{2},-\frac{1}{2}$
3. $\frac{1}{2},-\frac{1}{2}$
4. $-\frac{1}{2},\frac{1}{2}$

Q. A person stands in contact against the wall of a cylindrical drum of radius $R$ rotating with an angular velocity $\omega$ . If the coefficient of friction between the wall and the person is $\mu$ , the minimum rotational speed of the cylinder which enables the person to remain stuck to the wall when the floor is suddenly removed is

1. ${\omega }_{min}=\sqrt{\frac{g}{\mu R}}$
2. ${\omega }_{min}=\sqrt{\frac{\mu R}{g}}$
3. ${\omega }_{min}=\sqrt{\frac{2g}{\mu R}}$
4. ${\omega }_{min}=\sqrt{\frac{gR}{\mu }}$

Q. A solid cylinder of mass m is attached to a horizontal spring with force constant k. The cylinder can roll without slipping along the horizontal plane. (See the accompanying figure.) Show that the center of mass of the cylinder executes simple harmonic motion with a period $T=2\pi \sqrt{\frac{3m}{2k}}$, if displaced from mean position.

Q. Give the equations of rotational motion.

Q. One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to a massless spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of cross – section and Young’s modulus of the wire are A and Y respectively. If the mass is slightly pulled down and released, it will oscillate with a time period given by

1. 
2. 
3. 
4. 

Q. The length of a second's pendulum at a place where g = 9.8m/s ${}^2$ is 90.2 cm. State whether true or false.

1. True
2. False

Q. A piston/cylinder contains water at ${500}^{o}C,3MPa$. It is cooled in a polytropic process to ${200}^{o}C,1MPa$. Find the specific work in the process.

1. $2155kJ$
2. $3089kJ$
3. $1255kJ$
4. $5152kJ$

Q. A 10 kg metal block is attached to a spring of spring constant $1000N{m}^{-1}$. A block is displaced from equilibrium position by 10 cm and released. The maximum acceleration of the block is

1. $10m{s}^{-2}$
2. $100m{s}^{-2}$
3. $200m{s}^{-2}$
4. $0.1m{s}^{-2}$

Q. In the given figure, a mass $M$ is attached to a horizontal spring, which is fixed on one side to a rigid support. The spring constant of the spring is $k$. The mass oscillates on a frictionless surface with time period $T$ and amplitude $A$. When the mass is in equilibrium position, as shown in the figure, another mass $m$ is gently fixed upon it then the new amplitude of oscillation will be :

1. $A\sqrt{\frac{M}{M+m}}$
2. $A\sqrt{\frac{M}{M-m}}$
3. $A\sqrt{\frac{M-m}{M}}$
4. $A\sqrt{\frac{M+m}{M}}$

Q. A spring connected between two blocks of mass 2 kg and 4 kg is compressed by 20 cm & released. After time $t_0$, the spring gains natural length. Find the work done by the spring on the 2kg block between t = 0 & $t=t_0$.

1. 20 J
2. 40 J
3. $\frac{20}{3}$ J
4. $\frac{40}{3}$ J

Q. A $2kg$ at the end of a string $1m$ long is whirled in a vertical circle at a constant speed. The speed of the stone is $4m/sec$. The tension is the string will be $52N$, when the stone is -

1. At the top of the circle
2. At the bottom of the circle
3. Halfway Down
4. None of the above

Q. A particle executes $SHM$ of period $1.2sec$ and amplitude $8cm$. Find the time it takes to travel $3cm$ from the positive extremity or its oscillation. $[{\cos }^{-1}\left(5/8\right)=0.9rad]$

1. $0.17sec$
2. $0.42sec$
3. $0.32sec$
4. $0.28sec$

Q. One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to mass less spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of cross-section and Young's modulus of the wire are A and Y respectively. If the mass is slightly pulled down and released, it will oscillate with a time period T equal to

1. 
2. 
3. 
4. 

Q. A disc of radius \(5~\text{cm}\) rolls on a horizontal surface with linear velocity \(v = 1 \hat{i}~\text{m/s}\) and angular velocity \(50 (-\hat{k})~\text{rad/sec}\) . Height of particle from ground on rim of disc which has velocity in vertical direction is (in \(\text{cm}\)) -

Q. Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is(are) correct?

1. Cylinder Q reaches the ground with larger angular speed
2. Cylinder P has larger linear acceleration than cylinder Q
3. Both cylinders reach the ground with same translational kinetic energy
4. Both cylinders P and Q reach the ground at the same time

Q. For the damped oscillator system shown in above figure, the block has a mass of $1.50kg$ and the spring constant is $8.00N/m$. The damping force is given by $-b\left(dx/dt\right)$, where $b=230g/s$. The block is pulled down $12.0cm$ and released. (a) Calculate the time required for the amplitude of the resulting oscillations to fall to one-third of its initial value. (b) How many oscillations are made by the block in this time?