Force Law for Simple Harmonic Motion

Which of the following cannot be in the same direction for a simple harmonic oscillation?

In S.H.M., the restoring force is always directed _____ (away from/ towards) the mean position.

In case of simple harmonic motion, the restoring force is proportional to the _____.

Figure shows spring + block + pulley system which are light. The time period of mass would be

The equation of $\mathrm{S}\mathrm{H}\mathrm{M}$ of a particle is $\mathrm{a}+4{\mathrm{\pi }}^2\mathrm{x}=0$ where a is instantaneous linear acceleration at displacement $\mathrm{x}$ . The frequency of motion is -

A uniform thin rod has a mass $1\mathrm{ }\mathrm{k}\mathrm{g}$ and carries a mass $2.5\mathrm{ }\mathrm{k}\mathrm{g}$ at $\mathrm{B}$ . The rod is hinged at $\mathrm{A}$ and is maintained in the horizontal position by a spring having a spring constant $18\mathrm{ }\mathrm{k}\mathrm{N} {\mathrm{m}}^{-1}$ at $\mathrm{C}$ as shown in figure. The angular frequency of oscillation is nearly -

Two identical simple pendulums $\mathrm{A}$ and $\mathrm{B}$ are fixed at same point. They are displaced by very small angles $\mathrm{\alpha }$ and $\mathrm{\beta }\left(\mathrm{\beta }>\mathrm{\alpha }\right)$ and released from rest. Find the time after which $\mathrm{B}$ reaches its initial position for the first time. Collisions are elastic and length of strings is $\mathcal{l}$ .

Two particles are executing simple harmonic motion of same amplitude $A$ and frequency $\mathrm{\omega }$ along the $x-$axis. Their mean positions are separated by distance $x_0 (x_0>A)$. If the maximum separation between them is $(x_0+A)$, then the phase difference between their motions is,

A simple pendulum is released from $A$ shown. If $m$ and $l$ represent the mass of the bob and Length of the pendulum, the gain in kinetic energy at $B$ is

A rod of mass $m$ and length $\ell$ can rotate in horizontal plane free about a vertical axis passing through one of its end. Its mid point and the other end are attached to two spring of equal spring constants $k$. The springs are fixed to rigid supports as shown in the diagram. The rod is pushed slightly through a small angle in one direction and released. Find the frequency of its oscillation.

Out of the following functions representing motion of a particle which represents SHM

(1) $\mathit{\text{y}}=\text{sin}\omega \mathit{\text{t}}-\text{cos}\omega \mathit{\text{t}}$

(2) $\mathit{\text{y}}={\text{sin}}^3\omega \mathit{\text{t}}$

(3) $y=5\text{cos}\left(\frac{3\pi }{4}-3\omega t\right)$

(4) $\mathit{\text{y}}=1+\omega \mathit{\text{t}}+{\omega }^2{\mathit{\text{t}}}^2$

What is the time period of a simple pendulum with string length $l$ and acceleration due to gravity $g$?

A wooden cylinder of mass $20 \mathrm{g}$ and area of cross-section $1 {\mathrm{cm}}^2$, having a piece of lead of mass $60 \mathrm{g}$ attached to its bottom floats in water. The cylinder is depressed and then released. Show that it will execute S.H.M. Find the frequency of oscillations.