Other Examples of SHM

Q. A uniform disc of mass $m$ and radius $R$ is pivoted smoothly at its centre of mass. A light spring of stiffness $k$ is attached with the disc tangentially as shown in the figure. If the disc is rotated through a small angle and released, the angular frequency of oscillation of the disc is (in $\text{rad/s}$)
[(Take $m=5\text{kg}$ and $k=10\text{N/m}$]

Q. A uniform bar with mass $m$ lies symmetrically across two rapidly rotating fixed rollers A and B, with distance ${}^{\prime }{l}^{\prime }$ between the bar's centre of mass and each roller. The rollers whose direction of rotation are shown in figure slip against the bar with coefficient of friction $\mu$. Suppose the bar is displaced horizontally by a small distance ${}^{\prime }{x}^{\prime }$ and then released, find the time period of oscillation.

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

Q. A rigid rod of mass $m$, with a ball of mass $M$ attached to the free end is restrained to oscillate in a vertical plane as shown in the figure. Find the natural frequency of oscillation.

1. $f=\frac{1}{2\pi }\sqrt{\frac{3k}{27M+7m}}$
2. $f=\frac{1}{\pi }\sqrt{\frac{3k}{27m+7m}}$
3. $f=\frac{1}{2\pi }\sqrt{\frac{k}{27M+7m}}$
4. $f=\frac{1}{\pi }\sqrt{\frac{k}{27M+7m}}$

Q.

A trolley of mass M oscillates with some time period about its mean position when it is connected with an ideal spring of stiffness constant K.

If an unknown mass m is gently placed at the top of trolley, its time period is found to be T.Calculate the coefficient of static friction so that the upper mass does not slip even for sizeable amplitude A.

1. $\frac{KA}{mg}$

2. $\frac{KA}{(M+m)g}$

3. $\frac{KAm}{mg}$

4. None of these

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 long uniform rod of length L and mass M is free to rotate in a vertical plane about a horizontal axis through its one end 'O'. A spring of force constant k is connected vertically between one end of the rod and ground. When the rod is in equilibrium it is parallel to the ground.

What is the period of small oscillations that result when the rod is rotated slightly and released?

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

2. $T=2\pi \sqrt{\frac{M}{k}}$

3. $T=2\pi \sqrt{\frac{2M}{k}}$

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

Q. A uniform square plate of side $a$ is hinged at one of its corners as shown. It is suspended such that it can rotate about horizontal axis. Find out its time period for small oscillations about its equilibrium position.

1. $2\pi \sqrt{\frac{\sqrt{2}a}{6g}}$
2. $2\pi \sqrt{\frac{2\sqrt{2}a}{3g}}$
3. $2\pi \sqrt{\frac{2a}{g}}$
4. $2\pi \sqrt{\frac{a}{2g}}$

Q. A uniform disc of mass $m$ and radius $R=\frac{80}{23{\pi }^2}$ is pivoted smoothly at point $P$. If another uniform ring of mass $m$ and radius $R$ is welded at the lowest point of the disc, the time period (in seconds) of the SHM of the system is

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

Q.

A long uniform rod of length L and mass M is free to rotate in a vertical plane about a horizontal axis through its one end 'O'. A spring of force constant k is connected vertically between one end of the rod and ground. When the rod is in equilibrium it is parallel to the ground.

What is the period of small oscillations that result when the rod is rotated slightly and released?

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

2. $T=2\pi \sqrt{\frac{M}{k}}$

3. $T=2\pi \sqrt{\frac{2M}{k}}$

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

Q. Two cubical blocks of side length $a$ and $2a$ are stuck symmetrically as shown in the figure. The combined block is floating in water with the bigger block just submerged completely. The block is pushed down a little and released. Find the time period of its oscillations. Neglect viscosity.

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

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.

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. A rod of mass $m$ and length $l$ hinged at one end is connected by two springs of spring constants $k_1$ and $k_2$ so that it is horizontal at equilibrium. The angular frequency of the system is (in $\text{rad/s}$)

[$\text{Take}l=1\text{m},b=\frac{1}{2\sqrt{2}}\text{m},k_1=16\text{N/m},k_2=1\text{N/m},m=\frac{1}{64}\text{kg}$]

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. 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. Find the period of the free oscillations of the arrangement shown below, if mass $M_1$ is pulled down a little.
[Force constant of the spring is $k$ and mass of the fixed pulley is negligible]

1. $T=2\pi \sqrt{\frac{M_1+M_2}{k}}$
2. $T=2\pi \sqrt{\frac{M_1+4M_2}{k}}$
3. $T=2\pi \sqrt{\frac{M_2+4M_1}{k}}$
4. $T=2\pi \sqrt{\frac{M_2+3M_1}{k}}$

Q. The period of oscillation of a simple pendulum of length $L$ suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination $\alpha$, is given by

1. $2\pi \sqrt{\frac{L}{g\cos \alpha }}$
2. $2\pi \sqrt{\frac{L}{g\sin \alpha }}$
3. $2\pi \sqrt{\frac{L}{g}}$
4. $2\pi \sqrt{\frac{L}{g\tan \alpha }}$

Q. A uniform metre scale of length $1\text{m}$ is balanced on a fixed semi-circular cylinder of radius $30\text{cm}$ 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 $(\text{Take}g=10{\text{ms}}^{-2})$

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

Q. A rod of mass $m$ and length $l$ hinged at one end is connected by two springs of spring constants $k_1$ and $k_2$ so that it is horizontal at equilibrium. The angular frequency of the system is (in $\text{rad/s}$)

[$\text{Take}l=1\text{m},b=\frac{1}{2\sqrt{2}}\text{m},k_1=16\text{N/m},k_2=1\text{N/m},m=\frac{1}{64}\text{kg}$]

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. $\frac{\pi }{4}$