Please be patient as the PDF generation may take up to a minute.

1. Which of the following functions of time represent/s periodic motion? 
(\(\omega\) is any positive constant)

(a)	\((\sin\omega t+ \cos \omega t)\)
(b)	\((\sin\omega t+ \cos 2\omega t+ \sin4\omega t)\)
(c)	\(e^{-\omega t}\)
(d)	\(\mathrm{log(\omega t)}\)
(e)	\(\sin^{2}\omega t\)

Choose the correct option from the given ones:

1.	(a) only
2.	(a), (b), and (d) only
3.	(a), (b), and (e) only
4.	All of these

2. The time period of the following periodic motions respectively are:

(a)	\(\sin\omega t+ \cos\omega t\)
(b)	\(\sin\omega t+ \cos2\omega t+ \sin4\omega t\)

1. \(\dfrac{2 \pi}{\omega}   \text{ and }   \dfrac{2 \pi}{3 \omega}\)
2. \(\dfrac{2 \pi}{\omega} \text{ for both}\)
3. \(\dfrac{2 \pi}{3 \omega} \text{ for both}\)
4. \(\dfrac{3 \pi}{\omega}   \text{ and }   \dfrac{2 \pi}{\omega}\)

3.

Given the following simple harmonic motion.
(A) \(\sin\omega t - \cos\omega t\)
(B) \(\sin^{2}\omega t\)
The time period motions respectively are:
1. \(\dfrac{2 \pi}{\omega}  \text{ and }  \dfrac{\pi}{\omega}\)
2. \(\dfrac{2 \pi}{\omega}   \text{ and }\dfrac{\pi}{2 \omega}\)
3. \(\dfrac{2 \pi}{\omega}   \text{ and }\dfrac{4 \pi}{\omega}\)
4. \(\dfrac{\pi}{\omega}   \text{ and }   \dfrac{3 \pi}{\omega}\)

4.

Which of the following relationships between the acceleration \(a\) and the displacement \(x\) of a particle involves simple harmonic motion?
1. \(a =   0 . 7 x\)
2. \(a =   - 200 x^{2} \)
3. \(a =   - 10 x\)
4. \(a =   100   x^{3}\)

5.

The figure given below depicts two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution are indicated in the figures.  Equations of the \(x\)-projection of the radius vector of the rotating particle \(\mathrm P\) in each case are, respectively:

6. A body oscillates with SHM according to the equation (in SI units), \(x= 5\cos\left[2\pi t +\dfrac{\pi}{4}\right].\) At \(t = 1.5 \text{ s},\) acceleration of the body will be:
1. \(140 \text{ cm} / \text{s}^2 \)
2. \(160 \text{ m} / \text{s}^2 \)
3. \(140 \text{ m} / \text{s}^2 \)
4. \(14 \text{ m} / \text{s}^2\)

7.

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. \(\mathrm{T}=4 \pi \sqrt{\frac{1}{\sqrt{\mathrm{g}^2+\frac{\mathrm{v}^4}{\mathrm{R}^4}}}}\)

8.

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. \(\mathrm{T}=\frac{1}{2 \pi} \sqrt{\frac{\mathrm{h} \rho_1}{\rho \mathrm{g}}}\)

9.

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. \(\sqrt{\left( x_0^2+\frac{v_0^2}{\pi\omega^2}\right)} \)

10.

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. \(\mathrm{T}=3 \pi \sqrt{\frac{\mathrm{Vm}_{\mathrm{m}}}{\mathrm{Ba}^2}}\)

11.

The figure shows the circular motion of a particle. The radius of the circle, the period, the sense of revolution, and the initial position are indicated in the figure. The simple harmonic motion of the \({x\text-}\)projection of the radius vector of the rotating particle \(P\) will be:
                                   
1. \(x \left( t \right) = B\text{sin} \left(\dfrac{2 πt}{30}\right)\) 
2. \(x \left( t \right) = B\text{cos} \left(\dfrac{πt}{15}\right)\) 
3. \(x \left( t \right) = B\text{sin} \left(\dfrac{πt}{15} + \dfrac{\pi}{2}\right)\) \(\)
4. \(x \left( t \right) = B\text{cos} \left(\dfrac{πt}{15} + \dfrac{\pi}{2}\right)\)