A particle moves according $\mathrm{x} = \left(6 \mathrm{cm}\right) \cos \left(\frac{\mathrm{\pi }}{2}\mathrm{t}\right)$. Average velocity of the particle in the time interval between t = 0 to t = 3 s is

A particle performs S.H.M. with frequency f. Frequency of its velocity and acceleration are respectively :

Force acting on a body free to move on X-axis is given by F = $-{\mathrm{kx}}^{\mathrm{n}}$ where k is a positive constant. For which value of n motion of the body is not oscillatory?

In a forced oscillation, when the system oscillates under the action of the driving force $\mathrm{F} = {\mathrm{F}}_0 \sin \mathrm{\omega t}$ in addition to its internal restoring force, the particle oscillates with a frequency equal to

The  potential energy of a harmonic oscillator at equilibrium is 15 J and average kinetic energy is 5 J. Total energy at any instant is

A particle is executing S.H.M. Graph of its displacement with the position is shown in the figure. Maximum acceleration of the particle is

A uniform spring has a force constant K. It is cut into two pieces of lengths ${\mathrm{l}}_1 \mathrm{and} {\mathrm{l}}_2$, such that ${\mathrm{l}}_1 = {\mathrm{nl}}_2$. Time period of oscillation of a mass m with spring of length ${\mathrm{l}}_1$ is

A particle is executing SHM along a straight-line path. Then which of the following statement is correct?

Two pendulums A and B of length 144 cm and 100 cm are initially in the same phase at their equilibrium positions. After how much complete oscillations of B, they come in the same phase?

The time period of the spring-mass system depends upon

A particle is moving along X-axis (x in meter) and force acting on particle varies as F = -5x + 5N, then

In the following questions, a statement of assertion (A) is followed by a statement of the reason (R).

A: The periodic time of a hard spring is more as compared to the soft spring.

R: The spring constant of hard spring is less.

The phase difference between displacement and acceleration of a particle in a simple harmonic motion is:

Which of the following examples represent periodic motion?

Which of the following examples represent simple harmonic motion?

The figure depicts four x-t plots for the linear motion of a particle.

Which of the following is true?

$y={\sin }^3\omega t is$:

A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. (Take the direction from A to B as the positive direction.) The signs of velocity, acceleration, and force on the particle when it is at 3 cm away from A going towards B are:

Which of the following relationships between the acceleration 'a' and the displacement 'x' of a particle involves simple harmonic motion?

$\begin{array}{rcl}1. a& =& 0.7x\\ 2. a& =& -200x^2\\ 3. a& =& -10x\\ 4. a& =& 100 x^3\end{array}$

A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body?

A spring having a spring constant of 1200 N/m is mounted on a horizontal table as shown in the figure. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released. The frequency of oscillations will be
    

$\begin{array}{l}1. 3.0 {\mathrm{s}}^{-1}\\ 2. 2.7 {\mathrm{s}}^{-1}\\ 3. 1.2 {\mathrm{s}}^{-1}\\ 4. 3.2 {\mathrm{s}}^{-1}\end{array}$

A spring having a spring constant of 1200 N/m is mounted on a horizontal table as shown in the figure. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released. The maximum acceleration of the mass is:

$$\begin{gathered} \left(1\right) 6.0 m s^{-2} \\ \left(2\right) 8.0 m s^{-2} \\ \left(3\right) 3.3 m s^{-2} \\ \left(4\right) 5.1 m s^{-2} \end{gathered}$$

The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is its maximum speed?

The acceleration due to gravity on the surface of the moon is 1.7 m $s^{-2}$. What is the time period of a simple pendulum on the surface of the moon if its time period on the surface of the earth is 3.5 s? (g on the surface of the earth is 9.8 m $s^{-2}$ )

What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity?

$$\begin{gathered} \left(1\right) 100 s^{-1} \\ \left(2\right) 0 \\ \left(3\right) 150 s^{-1} \\ \left(4\right) None of these \end{gathered}$$

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?

$$\begin{gathered} \left(1\right) \mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\sqrt{{\mathrm{g}}^2+{\displaystyle \frac{{\mathrm{v}}^4}{{\mathrm{R}}^2}}}}} \\ \left(2\right) \mathrm{T}=4\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{{\mathrm{g}}^2+{\displaystyle \frac{{\mathrm{v}}^4}{{\mathrm{R}}^2}}}} \\ \left(3\right) \mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{{\mathrm{g}}^2+{\displaystyle \frac{{\mathrm{v}}^3}{{\mathrm{R}}^2}}}} \\ \left(4\right) \mathrm{T}=4\mathrm{\pi }\sqrt{\frac{\mathrm{l}}{\sqrt{{\mathrm{g}}^2+{\displaystyle \frac{{\mathrm{v}}^4}{{\mathrm{R}}^4}}}}} \end{gathered}$$

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:

$$\begin{gathered} \left(1\right) \mathrm{T}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{\mathrm{h\rho }}{{\mathrm{\rho }}_{\mathrm{l}}\mathrm{g}}} \\ \left(2\right) \mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{h\rho }}{{\mathrm{\rho }}_{\mathrm{l}}\mathrm{g}}} \\ \left(3\right) \mathrm{T}=2\mathrm{\pi }\sqrt{\frac{{\mathrm{h\rho }}_{\mathrm{l}}}{\mathrm{\rho g}}} \\ \left(4\right) \mathrm{T}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{{\mathrm{h\rho }}_{\mathrm{l}}}{\mathrm{\rho g}}} \end{gathered}$$

(assuming pressure-volume variations of air to be isothermal)

$$\begin{gathered} \left(1\right) \mathrm{T}=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{\mathrm{Vm}}{{\mathrm{Ba}}^3}} \\ \left(2\right) \mathrm{T}=\mathrm{\pi }\sqrt{\frac{\mathrm{Vm}}{{\mathrm{Ba}}^2}} \\ \left(3\right) \mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{Vm}}{{\mathrm{Ba}}^2}} \\ \left(4\right) \mathrm{T}=3\mathrm{\pi }\sqrt{\frac{\mathrm{Vm}}{{\mathrm{Ba}}^2}} \end{gathered}$$

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:

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:

$$\begin{gathered} \left(1\right) 2.1 \mathrm{Nm} {\mathrm{rad}}^{-1} \\ \left(2\right) 0.6 \mathrm{Nm} {\mathrm{rad}}^{-1} \\ \left(3\right) 3.2 \mathrm{Nm} {\mathrm{rad}}^{-1} \\ \left(4\right) 1.9 \mathrm{Nm} {\mathrm{rad}}^{-1} \end{gathered}$$

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?
$$\begin{gathered} \left(1\right) -3{\mathrm{\pi }}^2 {\mathrm{ms}}^{-2} \\ \left(2\right) 5{\mathrm{\pi }}^2 {\mathrm{ms}}^{-2} \\ \left(3\right) -5{\mathrm{\pi }}^2 {\mathrm{ms}}^{-2} \\ \left(4\right) 3{\mathrm{\pi }}^2 {\mathrm{ms}}^{-2} \end{gathered}$$