Other Examples of SHM

Q. The ends of a rod of length $l$ and mass $m$ are attached to two identical springs as shown in figure. The rod is free to rotate about its centre ${}^{\prime }{\text{O}}^{\prime }$. The rod is depressed slightly at end $\text{A}$ and released. The time period of the resulting oscillation is

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

Q.

A quantity $x$ is given by $\left(\frac{IF{v}^2}{W{L}^4}\right)$ in terms of moment of inertia $I$, force $F$, velocity $v$, work $W$ and Length $L$. The dimensional formula for $x$ is same as that of:

1. Coefficient of viscosity

2. Energy density

3. Force constant

4. Planck’s constant

Q. A wooden block performs SHM on a frictionless surface with frequency $v_0$. The block carries a charge + Q on its surface. If now a uniform electric field E is switched on as shown, then the SHM of the block will be

1. Of the same frequency and with shifted mean position
2. Of the same frequency and with the same mean position
3. Of changed frequency and with the same mean position
4. Of changed frequency and with shifted mean position

Q.

A particle of mass m is suspended from a ceiling through a string of length L. The particle moves in a horizontal circle of radius r. Find (a) the speed of the particle and (b) the tension in the string. Such a system is called a conical pendulum.

1. Speed = rg tan

   Tension =

2. Speed =

   Tension = mg

3. None of these

4. Speed =

   Tension =

Q. An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $M$. The piston and cylinder have equal cross sectional area $A$. When the piston is in equilibrium, the volume of the gas is $V_0$ and its pressure is $P_0$. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency

1. $\frac{1}{2\pi }\frac{V_{0}M}{A^2\gamma }(P_0+\frac{Mg}{A})$
2. $\frac{1}{2\pi }\sqrt{\frac{A^2\gamma }{M{V}_0}(P_0+\frac{Mg}{A})}$
3. $\frac{1}{2\pi }\sqrt{\frac{M{V}_0}{A\gamma (P_0+\frac{Mg}{A})}}$
4. $\frac{1}{2\pi }\frac{A\gamma }{V_{0}M}(P_0+\frac{Mg}{A})$

Q. A disc of radius $R$ and mass $M$ is pivoted at the rim and is set for small oscillations. If simple pendulum has to have the same period as that of the disc, the length of the simple pendulum should be

1. $\frac{5}{4}R$
2. $\frac{2}{3}R$
3. $\frac{3}{4}R$
4. $\frac{3}{2}R$

Q. The adiabatic bulk modulus of a diatomic gas at atmospheric pressure is

1. $2.8\times {10}^5{\text{Nm}}^{-2}$
2. $1{\text{Nm}}^{-2}$
3. $1.4\times {10}^4{\text{Nm}}^{-2}$
4. $1.4\times {10}^5{\text{Nm}}^{-2}$

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

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

Q. A uniform chain of mass $m$ and length $l$ hangs on a thread and touches the surface of the table by its lower end. The thread breaks suddenly, find the force exerted by the table on the chain when half of its length has fallen on the table. The fallen part does not form a heap.

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

Q. A smooth inclined plane having an angle of inclination of ${30}^{\circ }$ with the horizontal has a $2.5\text{kg}$ mass held by a spring which is fixed at the upper end. If the mass is taken $2.5\text{cm}$ up along the surface of the inclined plane, the tension in the spring reduces to zero. If the mass is now released, the angular frequency of oscillation is:

1. $7\text{rad/s}$
2. $14\text{rad/s}$
3. $0.7\text{rad/s}$
4. $1.4\text{rad/s}$

Q. The ends of a rod of length $l$ and mass $m$ are attached to two identical springs as shown in figure. The rod is free to rotate about its centre ${}^{\prime }{\text{O}}^{\prime }$. The rod is depressed slightly at end $\text{A}$ and released. The time period of the resulting oscillation is

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

Q. A smooth piston of mass $m$ and area of cross-section $A$ is in equilibrium with the gas in the jar, when the pressure of the gas is $P_0$. Find the angular frequency of oscillation of the piston, assuming adiabatic change of state of the gas.

1. $\omega =\sqrt{\frac{\gamma P_0{A}^2}{2m{V}_0}}$
2. $\omega =\sqrt{\frac{\gamma P_0{A}^2}{m{V}_0}}$
3. $\omega =\sqrt{\frac{2\gamma P_0{A}^2}{m{V}_0}}$
4. $\omega =\sqrt{\frac{P_0{A}^2}{m{V}_0}}$

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 simple pendulum of length $l$ is displaced so that its taught string is horizontal and then released. A uniform bar pivoted at one end is simultaneously released from its horizontal position. If their motions are synchronous, what is the length of the bar?

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

Q. A uniform semicircular ring having mass $m$ and radius $r$ is hanging at one of its ends freely as shown in the figure. The ring is slightly disturbed so that it oscillates in its own plane. The time period of oscillation of the ring is

1. $2\pi \sqrt{\frac{r}{g(1+\frac{1}{{\pi }^2})}}$
2. $2\pi \sqrt{\frac{r}{g{(1-\frac{4}{{\pi }^2})}^{\frac{1}{2}}}}$
3. $2\pi \sqrt{\frac{r}{g{(1-\frac{2}{{\pi }^2})}^{\frac{1}{2}}}}$
4. $2\pi \sqrt{\frac{2r}{g{(1+\frac{4}{{\pi }^2})}^{\frac{1}{2}}}}$

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

Q. A man weighing $60\text{kg}$ stands on the horizontal platform fixed to a spring balance. The platform starts executing simple harmonic motion of amplitude $0.1\text{m}$ and frequency $\frac{2}{\pi }\text{Hz}$. Which of the following statements is correct?
[Assume the man is always in contact with the platform and spring balance has only vertical motion]

1. The spring balance reads the weight of man as $60\text{kg}$.
2. The spring balance reading fluctuates between $60\text{kg}$ and $70\text{kg}$.
3. The spring balance reading fluctuates between $50\text{kg}$ and $60\text{kg}$.
4. The spring balance reading fluctuates between $50\text{kg}$ and $70\text{kg}$.

Q. A simple pendulum of period T has a metal bob which is negatively charged. If it is allowed to oscillate above a positively charged metal plate, its period will

1. Remains equal to T
2. Less than T
3. Greater than T
4. Infinite

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.

The left block in figure collides in-elastically with the right block and sticks to it. Find the amplitude of the resulting simple harmonic motion.

1. 

2. 

3. 

4. 

Q.

Suppose a tunnel is dug along a diameter of the earth. A particle is dropped from a point, a distance h directly above the tunnel. The motion of the particle as seen from the earth is

1. simple harmonic

2. parabolic

3. on a straight line

4. periodic

Q. A spherical cavity of radius $\frac{R}{2}$ is removed from a solid sphere of radius $R$ and the sphere is placed on a rough horizontal surface as shown in the figure. The sphere is given a gentle push and the friction is large enough to prevent slipping. Find the time period of oscillation of the sphere.

1. $2\pi \sqrt{\frac{20R}{17g}}$
2. $2\pi \sqrt{\frac{177R}{10g}}$
3. $2\pi \sqrt{\frac{4R}{7g}}$
4. $2\pi \sqrt{\frac{147R}{17g}}$

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

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

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

Q.

A metal ring of mass m and radius R is placed on a smooth horizontal table and is set rotating about its own axis in such a way that each part of the ring moves with a speed v. Find the tension in the ring.

1. N

2. N

3. N

4. N

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