Other Examples of SHM

Q.

A thin circular ring of radious R and mass M is hinged at its periphery in a vertical plane.If it is slightly displaced from equilibrium position and then released then the time period of oscillation of ring will be;

1. 

2. 

3. 

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

Q.

$1kg$mass has $K.E.$ of $1J,$ when its speed is:

1. $0.45m{s}^{-1}$

2. $1m{s}^{-1}$

3. $1.4m{s}^{-1}$

4. $4.4m{s}^{-1}$

Q. A light rod of length $l$ pivoted at O is connected with two springs of stiffness $k_1\&k_2$ at a distance of $a$ & $l$ from the pivot respectively. A block of mass $m$ attached with the spring $k_2$ is kept on a smooth horizontal surface. Find the angular frequency of small oscillations of the block $m$.

1. $\omega =\sqrt{\frac{k_1^2{a}^2}{m(k_1{a}^2+k_2{l}^2)}}$
2. $\omega =\sqrt{\frac{k_2^2{a}^2}{m(k_1{a}^2+k_2{l}^2)}}$
3. $\omega =\sqrt{\frac{k_1{k}_2{a}^2}{m(k_1{a}^2+k_2{l}^2)}}$
4. None of these

Q. What is the time period of a pendulum formed by pivoting a meter stick so that it is free to rotate about a horizontal axis passing through the $75\text{cm}$ mark?

1. $1.00\text{sec}$
2. $1.25\text{sec}$
3. $1.53\text{sec}$
4. $2.5\text{sec}$

Q. An $L-$ shaped system of two identical rods of mass $m$ and length $l$ each, are resting on a peg $P$ as shown in the figure. Find the period of oscillations if the system is displaced slightly in its plane by a small angle $\theta$ ?

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

Q. A thin uniform vertical rod of mass $m$ and length $l$ is hinged about the point $O$ as shown in the figure. The combined stiffness of springs is equal to $k$. Considering the springs to be light, find the angular frequency of small oscillations.

1. $\omega =\sqrt{\frac{3k}{m}+\frac{3g}{l}}$
2. $\omega =\sqrt{\frac{3k}{m}+\frac{g}{l}}$
3. $\omega =\sqrt{\frac{k}{m}+\frac{3g}{l}}$
4. $\omega =\sqrt{\frac{3k}{m}+\frac{3g}{2l}}$

Q.

Assume that a tunnel is dug across the earth (radius = R) passing through its centre. Find the time a particle takes to cover the length of the tunnel if it is projected into the tunnel with a speed of $\sqrt{gR}$

1. $\pi \sqrt{g}R$

2. $\frac{\pi }{2}\frac{\sqrt{g}}{R}$

3. $2\pi$

4. $\pi$

Q. A thin uniform vertical rod of mass $m$ and length $l$ is hinged about the point $O$ as shown in the figure. The combined stiffness of springs is equal to $k$. Considering the springs to be light, find the angular frequency of small oscillations.

1. $\omega =\sqrt{\frac{3k}{m}+\frac{3g}{l}}$
2. $\omega =\sqrt{\frac{3k}{m}+\frac{g}{l}}$
3. $\omega =\sqrt{\frac{k}{m}+\frac{3g}{l}}$
4. $\omega =\sqrt{\frac{3k}{m}+\frac{3g}{2l}}$

Q. Two uniform ropes having linear mass densities $m$ and $4m$ are joined to form a closed loop. The loop is hanging over a fixed frictionless small pulley with the lighter rope above and denser rope below, as shown in the figure. (Figure represents equilibrium position). Now if point $A$ (joint) is slightly displaced in downward direction and released, it is found that the loop performs SHM. Find the time period of oscillation.
$[\text{Take}l=\frac{150}{4{\pi }^2}\text{m},g=10{\text{m/s}}^2]$

1. $5\text{s}$
2. $15\text{s}$
3. $10\text{s}$
4. $25\text{s}$

Q.

The given setup is in equilibrium. Springs were in natural length before the force was applied. If I have to replace the above setup of 1 block and 2 springs by 1 block and 1 spring, what will be the spring constant of that new spring such that the block is displaced by the same amount as previously?

1. 

2. 

3. 

4. 

Q. A sphere of radius $R$ is half submerged in liquid of density $\rho$. If the sphere is slightly pushed down and released, it oscillates, then what is the frequency of its oscillation ?

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

Q. A wire is bent at an angle $\theta$. A rod of mass $m$ can slide along the bent wire without friction as shown in the figure. A soap film is maintained in the wire frame kept in a vertical position and the rod is in equilibrium as shown in the figure. If rod is displaced slightly in vertical direction, then the time period of small oscillations of the rod is :-

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

Q. In the given figure an L-shaped bar of mass $M$ is pivoted at one of its end, so that it can freely rotate in a vertical plane. Find the frequency of oscillation, if it is slightly disturbed from its equilibrium position.

1. $\frac{1}{2\pi }\sqrt{\frac{g}{L}}$
2. $\frac{1}{2\pi }\sqrt{\frac{3g\sqrt{10}}{4L}}$
3. $\frac{1}{2\pi }\sqrt{\frac{3g}{4L}}$
4. $\frac{1}{2\pi }\sqrt{\frac{3g\sqrt{5}}{4L}}$

Q. The rotational kinetic energy of two bodies of moments of inertia $9kg-m^2$ and $1kg-m^2$ are same. The ratio of their angular moments is

1. $1:3$
2. $3:1$
3. $1:9$
4. $9:1$

Q. In the arrangement shown in figure, pulleys are small and light and the springs are ideal. $K_1=25{\pi }^2\frac{N}{m},K_2=2K_1,K_3=3K_1$ and $K_4=4K_1$ are the force constants of the springs. Calculate the period of small vertical oscillations (in $\text{s}$) of the block of mass $m=3\text{kg}$

Q. A uniform metre scale of length 1 m is balanced on a fixed semi – circular cylinder of radius 30 cm as shown in figure. One end of the scale is slightly depressed and released. The time period (in seconds) of the resulting simple harmonic motion is $(Takeg=10m{s}^{-2})$

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

Q.

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance $\frac{R}{2}$ from the earth's center where R is the radius of the earth. the wall of the tunnel is frictionless. Find the time period.

1. $2\pi \sqrt{\frac{R}{M}}$

2. $2\pi \sqrt{\frac{R^3}{GM}}$

3. $\sqrt{\frac{2R^3}{GM}}$

4. $2\pi \sqrt{\frac{M}{R}}$

Q. Two identical rods $PQ$ and $RS$ each of length $L$ and mass $M$ are joined to form a T-shaped frame as shown in figure. The moment of inertia of given arrangement about an axis passing through $PQ$ is

1. $M{L}^2$
2. $\frac{4M{L}^2}{3}$
3. $\frac{2}{5}M{L}^2$
4. $\frac{M{L}^2}{3}$

Q. Find the period of small oscillations of the bob of mass $m$ shown in the figure. Given, mass of the rod is also $m$.

1. $T=2\pi \sqrt{\frac{2m}{3k}}$
2. $T=2\pi \sqrt{\frac{m}{3k}}$
3. $T=2\pi \sqrt{\frac{7m}{3k}}$
4. $T=2\pi \sqrt{\frac{4m}{3k}}$