COE in SHM

Q.

The pulley shown in figure has a moment of inertia I about its axis and mass m. Find the time period of vertical oscillation of its centre of mass. The spring has spring constant k and the string does not slip over the pulley.

1. $2\pi \sqrt{\frac{(\frac{I}{r^2}+m)}{4k}}$

2. $2\pi \sqrt{\frac{4k}{(\frac{I}{r^2}+m)}}$

3. $\sqrt{\frac{(\frac{I}{r^2}+m)}{4k}}$

4. none of these

Q.

Consider the situation shown in figure. Show that if the blocks are displaced slightly in opposite directions and released, they will execute simple harmonic motion. Calculate the time period.

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

2. $2\pi \sqrt{\frac{2m}{k}}$

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

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

Q. A particle is performing SHM. When the displacement is one half of its amplitude, find what fraction of total energy are the kinetic and potential energies of the particle?

1. $\frac{3}{4},\frac{1}{4}$
2. $\frac{3}{8},\frac{5}{8}$
3. $\frac{1}{3},\frac{2}{3}$
4. $\frac{2}{3},\frac{1}{3}$

Q. In simple harmonic motion:

1. Potential energy and kinetic energy may not be equal in mean position.
2. Potential energy and kinetic energy may be equal in extreme position.
3. Potential energy may be zero at extreme position.
4. Kinetic energy plus potential energy oscillates simple harmonically.

Q. A particle is performing SHM. When the displacement is one half of its amplitude, find what fraction of total energy are the kinetic and potential energies of the particle?

1. $\frac{3}{4},\frac{1}{4}$
2. $\frac{3}{8},\frac{5}{8}$
3. $\frac{1}{3},\frac{2}{3}$
4. $\frac{2}{3},\frac{1}{3}$

Q.

The spring shown in figure is kept I a stretched position with extension $x_0$ when the system is released. Assuming the horizontal surface to be frictionless, find the frequency of oscillation.

1. $\frac{1}{2\pi }\sqrt{\frac{kMm}{(M+m)}}$

2. $\frac{1}{2\pi }\sqrt{\frac{k(M+m)}{Mm}}$

3. $2\pi \sqrt{\frac{kMm}{(M+m)}}$

4. $2\pi \sqrt{\frac{k(M+m)}{Mm}}$

Q.

In the system as shown, mass of the block is 1kg and spring constant is 25 $\frac{N}{m}$. The maximum kinetic energy of the block is 50J.

1. The amplitude of oscillation is 2m

2. At half of the amplitude, the kinetic energy is 37.5 J

3. At half of the amplitude, potential energy of the spring is 12.5 J

4. At half of the amplitude, potential energy of the spring is 20 J

Q. The total energy of a harmonic oscillator of mass $2\text{kg}$ is $9\text{J}$. If its potential energy at mean position is $5\text{J}$, its $\text{K.E.}$ at the mean position will be :

1. $9\text{J}$
2. $14\text{J}$
3. $4\text{J}$
4. $11\text{J}$

Q. A particle starts SHM from the mean position $x=0$. Its amplitude is $a$ and total energy is $E$. When the particle is at a position $5\text{cm}$ from the mean position, its kinetic energy is $\frac{3E}{4}$. The value of $a$ (in $\text{cm}$) is
[Assume, The potential energy of the particle is zero at the mean position]

Q. A particle of mass m is executing oscillations about the origin on the x-axis. Its potential energy is $V\left(x\right)=k|x{|}^3$, where k is a positive constant. If the amplitude of oscillation is a, then its time period T is

1. Proportional to $\frac{1}{\sqrt{a}}$
2. Independent of a
3. Proportional to $\sqrt{a}$
4. Proportional to $a^{\frac{3}{2}}$