A spring is attached vertically to the ceiling of a lift and the lower end of spring is connected with a block of mass \(2~\text{kg}\). If the lift starts accelerating upwards with an acceleration \(2~\text{m/s}^2,\) then find the amplitude of SHM, while the spring constant is \(100~\text{N/m}\):
1. \(8~\text{cm}\)
2. \(1~\text{cm}\)
3. \(2~\text{cm}\)
4. \(4~\text{cm}\)

The motion of the particle is started at t = 0 and the equation of motion is given by $x = 8 \sin \left(100t + \frac{\mathrm{\pi }}{6}\right)$, where x is in cm and t is in seconds. When will the particle come to rest for the first time?

1.  $\frac{\mathrm{\pi }}{300} \mathrm{s}$

2.  $\frac{\mathrm{\pi }}{200} \mathrm{s}$

3.  $\frac{\mathrm{\pi }}{100} \mathrm{s}$

4.  $\frac{\mathrm{\pi }}{400} \mathrm{s}$

What will be the frequency of oscillation of a simple pendulum, if the length of the pendulum is equal to the radius of earth?

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

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

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

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

The graph of potential energy \((U)\) versus displacement \((x)\) is shown. Which of the following describes the oscillation about the mean position, \(x = 0\text{?}\)

1.		2.
3.		4.

The time period of a body under S.H.M. is ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ when restoring forces ${\mathrm{F}}_1$ and ${\mathrm{F}}_2$ respectively act on it. What will be the time period of S.H.M. when both the forces act simultaneously on it?

1.  $\frac{{\mathrm{T}}_1{\mathrm{T}}_2}{{\mathrm{T}}_1 + {\mathrm{T}}_2}$

2.  ${\mathrm{T}}_1 + {\mathrm{T}}_2$

3.  $\sqrt{\frac{{\mathrm{T}}_1 + {\mathrm{T}}_2}{{\mathrm{T}}_1{\mathrm{T}}_2}}$
4. \(\frac{\mathrm{T_1 T_2}}{\sqrt{\mathrm{T^2_1+T^2_2}}}\)

A particle is subjected to two mutually perpendicular SHM such that x = 2sin$\mathrm{\omega }$t and

$\mathrm{y} = 2 \sin \left[\mathrm{\omega t} + \frac{\mathrm{\pi }}{2}\right]$. The path of the particle will be

1.  An ellipse

2.  A straight line

3.  A parabola

4.  A circle

When a periodic force \(\vec{F_1}\) acts on a particle, the particle oscillates according to the equation \(x=A\sin\omega t\). Under the effect of another periodic force \(\vec{F_2}\), the particle oscillates according to the equation \(y=B\sin(\omega t+\frac{\pi}{2})\). The amplitude of oscillation when the force (\(\vec{F_1}+\vec{F_2}\)) acts are:

A particle is executing SHM with amplitude A and angular frequency $\mathrm{\omega }$. The time taken by the particle to move from x = 0 to x = A/2 is

1.  $\frac{\mathrm{\pi }}{12 \mathrm{\omega }}$

2.  $\frac{\mathrm{\pi }}{6 \mathrm{\omega }}$

3.  $\frac{\mathrm{\pi }}{3 \mathrm{\omega }}$

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

A spring-block system oscillates with a time period \(T\) on the earth's surface. When the system is brought into a deep mine, the time period of oscillation becomes \(T'.\) Then one can conclude that:
1. \(T'>T\)
2. \(T'<T\)
3. \(T'=T\)
4. \(T'=2T\)