A simple pendulum of length l and having a bob of mass M is suspended in a car. The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?
1. \(\mathrm{T}=2 \pi \sqrt{\frac{1}{\sqrt{\mathrm{g}^2+\frac{\mathrm{v}^4}{\mathrm{R}^2}}}}\)
2. \(\mathrm{T}=4 \pi \sqrt{\frac{\mathrm{l}}{\mathrm{g}^2+\frac{\mathrm{v}^4}{\mathrm{R}^2}}}\)
3. \(\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{l}}{\mathrm{g}^2+\frac{\mathrm{v}^3}{\mathrm{R}^2}}}\)
4. \(\mathrm{T}=4 \pi \sqrt{\frac{1}{\sqrt{\mathrm{g}^2+\frac{\mathrm{v}^4}{\mathrm{R}^4}}}}\)

A cylindrical piece of cork of density $\mathrm{\rho }$ and base area A and height h floats in a liquid of density ${\mathrm{\rho }}_{\mathrm{l}}$. The cork is depressed slightly and then released. If the cork oscillates up and down simple harmonically then its period is
1. \(\mathrm{T}=\frac{1}{2 \pi} \sqrt{\frac{\mathrm{h} \rho}{\rho_1 \mathrm{~g}}}\)
2. \(\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{h} \rho}{\rho_{\mathrm{l}} \mathrm{g}}}\)
3. \(\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{h} \rho_1}{\rho \mathrm{g}}}\)
4. \(\mathrm{T}=\frac{1}{2 \pi} \sqrt{\frac{\mathrm{h} \rho_1}{\rho \mathrm{g}}}\)

An air chamber of volume V has a neck area of cross-section 'a' into which a ball of mass 'm' just fits and can move up and down without any friction (as shown in the figure). When the ball is pressed down a little and released, it executes SHM. The time period of oscillations is:

(assuming pressure-volume variations of air to be isothermal)
   
1. \(\mathrm{T}=\frac{1}{2 \pi} \sqrt{\frac{\mathrm{Vm}_{\mathrm{m}}}{\mathrm{Ba}^3}}\)
2. \(\mathrm{T}=\pi \sqrt{\frac{\mathrm{Vm}_{\mathrm{m}}}{\mathrm{Ba}^2}}\)
3. \(\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{Vm}_{\mathrm{m}}}{\mathrm{Ba}^2}}\)
4. \(\mathrm{T}=3 \pi \sqrt{\frac{\mathrm{Vm}_{\mathrm{m}}}{\mathrm{Ba}^2}}\)

A circular disc of mass 10 kg is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be 1.5 sec. The radius of the disc is 15 cm. The torsional spring constant of the wire is:
1. 2.1 Nm rad⁻¹
2. 0.6 Nm rad⁻¹
3. 3.2 Nm rad⁻¹
4. 1.9 Nm rad⁻¹

A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 sec. What is the acceleration of the body when the displacement is 5 cm?
1. −3π² ms⁻²
2. 5π² ms⁻²
3. −5π² ms⁻²
4. 3π² ms⁻²

A body describes simple harmonic motion with an amplitude of \(5~\text{cm}\) and a period of \(0.2~\text{s}\). What is the velocity of the body when the displacement is \(3~\text{cm}\)?
1. \(0.4\pi ~\text{m/s}\)
2. \(0\)
3. \(0.5\pi ~\text{m/s}\)
4. \(0.3\pi ~\text{m/s}\)

A mass attached to a spring is free to oscillate, with angular velocity \(\omega\), in a horizontal plane without friction or damping. It is pulled to a distance${}_{}$ \(x_0\) and pushed towards the centre with a velocity \(v_0\) at time \(t=0\). The amplitude of the resulting oscillations is:
1. \(\sqrt{\left(2x_0^2+\frac{v_0^2}{\omega^2}\right)} \)
2. \(\sqrt{\left(x_0^2+\frac{v_0^2}{\omega^2}\right)} \)
3. \(\sqrt{\left(x_0^2+\frac{v_0^2}{2\omega^2}\right)} \)
4. \(\sqrt{\left( x_0^2+\frac{v_0^2}{\pi\omega^2}\right)} \)

Identify the correct definition:

The displacement of a particle executing S.H.M. is given by x = 0.01sin100$\pi$(t + 0.05). The time period is:

(1) 0.01 s

(2) 0.02 s

(3) 0.1 s

(4) 0.2 s

A boy is swinging in a swing. If he stands, the time period will:

(1) First decrease, then increase

(2) Decrease

(3) Increase

(4) Remain same