The displacement of a particle is represented by the equation $y=3 \cos \left(\frac{\pi}{4}-2 \omega t\right)$. The motion of the particle is
The displacement of a particle is represented by the equation $y=\sin ^3 \omega t$. The motion is
The relation between acceleration and displacement of four particles are given below
Which, one of the particle is exempting simple harmonic motion?
Motion of an oscillating liquid column in a U-tube is
A particle is acted simultaneously by mutually perpendicular simple harmonic motion $$x=a \cos \omega t$$ and $$y=a \sin \omega t$$. The trajectory of motion of the particle will be
The displacement of a particle varies with time according to the relation $y=a \sin \omega t+b \cos \omega t$.
Four pendulums A, B, C and D are suspended from the same
elastic support as shown in figure. $A$ and $C$ are of the same length, while $B$ is smaller than $A$ and $D$ is larger than $A$. If $A$ is given a transerse displacement,
Figure shows the circular motion of a particle. The radius of the circle, the period, sense of revolution and the initial position are indicated on the figure. The simple harmonic motion of the $x$-projection of the radius vector of the rotating particle $P$ is
The equation of motion of a particle is $x=a \cos (\alpha t)^2$. The motion is
A particle executing SHM has a maximum speed of $30 \mathrm{~cm} / \mathrm{s}$ and a maximum acceleration of $60 \mathrm{~cm} / \mathrm{s}^2$. The period of oscillation is
When a mass $m$ is connected individually to two springs $S_1$ and $S_2$, the oscillation frequencies are $v_1$ and $v_2$. If the same mass is attached to the two springs as shown in figure, the oscillation frequency would be
The rotation of earth about its axis is
Motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower point is
Displacement versus time curve for a particle executing SHM is shown in figure. Choose the correct statements.
Which of the following statements is/are true for a simple harmonic oscillator?
The displacement-time graph of a particle executing SHM is shown in figure. Which of the following statement is/are true ?
A body is performing SHM, then its
A particle is in linear simple harmonic motion between two points. $A$ and $B, 10 \mathrm{~cm}$ apart (figure.) take the direction from $A$ to $B$ as the positive direction and choose the correct statements.
AO = OB = 5 cm
BC = 8 cm
Displacement versus time curve for a particle executing SHM is shown in figure. Identify the points marked at which (i) velocity of the oscillator is zero, (ii) speed of the oscillator is maximum.
Displacement versus time curve for a particle executing SHM is shown in figure. Identify the points marked at which (i) velocity of the oscillator is zero, (ii) speed of the oscillator is maximum.
Two identical springs of spring constant $k$ are attached to a block of mass $m$ and to fixed supports as shown in figure. When the mass is displaced from equilibrium position by a distance $x$ towards right, find the restoring force.
What are the two basic characteristics of a simple harmonic motion?
When will the motion of a simple pendulum be simple harmonic?
What is the ratio of maximum acceleration to the maximum velocity of a simple harmonic oscillator?
What is the ratio between the distance travelled by the oscillator in one time period and amplitude?
In figure, what will be the sign of the velocity of the point $P^{\prime}$, which is the projection of the velocity of the reference particle $P . P$ is moving in a circle of radius $R$ in anti-clockwise direction.
Show that for a particle executing SHM, velocity and displacement have a phase difference of $\pi / 2$.
Draw a graph to show the variation of $\mathrm{PE}, \mathrm{KE}$ and total energy of a simple harmonic oscillator with displacement.
The length of a second's pendulum on the surface of earth is 1 m . What will be the length of a second's pendulum on the moon?
Find the time period of mass $M$ when displaced from its equilibrium position and then released for the system shown in figure.
Show that the motion of a particle represented by $y=\sin \omega t-\cos \omega t$ is simple harmonic with a period of $2 \pi / \omega$.
Find the displacement of a simple harmonic oscillator at which its PE is half of the maximum energy of the oscillator.
A body of mass $m$ is situated in a potential field $U(x)=U_0(1-\cos \alpha x)$ when, $U_0$ and $\alpha$ are constants. Find the time period of small oscillations.
A mass of 2 kg is attached to the spring of spring constant $50 \mathrm{Nm}^{-1}$. The block is pulled to a distance of 5 cm from its equilibrium position at $x=0$ on a horizontal frictionless surface from rest at $t=0$. Write the expression for its displacement at anytime $t$.
Consider a pair of identical pendulums, which oscillate with equal amplitude independently such that when one pendulum is at its extreme position making an angle of $2^{\circ}$ to the right with the vertical, the other pendulum makes an angle of $1^{\circ}$ to the left of the vertical. What is the phase difference between the pendulums?
A person normally weighing 50 kg stands on a massless platform which oscillates up and down harmonically at a frequency of $2.0 \mathrm{~s}^{-1}$ and an amplitude 5.0 cm . A weighing machine on the platform gives the persons weight against time.
(a) Will there be any change in weight of the body, during the oscillation?
(b) If answer to part (a) is yes, what will be the maximum and minimum reading in the machine and at which position?
A body of mass $m$ is attached to one end of a massless spring which is suspended vertically from a fixed point. The mass is held in hand, so that the spring is neither stretched nor compressed. Suddenly, the support of the hand is removed. The lowest position attained by the mass during oscillation is 4 cm below the point, where it was held in hand.
(a) What is the amplitude of oscillation?
(b) Find the frequency of oscillation?
A cylindrical $\log$ of wood of height $h$ and area of cross-section $A$ floats in water. It is pressed and then released. Show that the log would execute SHM with a time period.
$$T=2 \pi \sqrt{\frac{m}{A \rho g}}$$
where, $m$ is mass of the body and $\rho$ is density of the liquid.
One end of a $V$-tube containing mercury is connected to a suction pump and the other end to atmosphere. The two arms of the tube are inclined to horizontal at an angle of $45^{\circ}$ each. A small pressure difference is created between two columns when the suction pump is removed. Will the column of mercury in $V$-tube execute simple harmonic motion? Neglect capillary and viscous forces. Find the time period of oscillation.
A tunnel is dug through the centre of the earth. Show that a body of mass $m$ when dropped from rest from one end of the tunnel will execute simple harmonic motion.
A simple pendulum of time period $1 s$ and length $l$ is hung from a fixed support at 0 . Such that the bob is at a distance $H$ vertically above $A$ on the ground (figure) the amplitude is $\theta_0$ the string snaps at $\theta=\theta_0 / 2$. Find the time taken by the bob to hit the ground. Also find distance from $A$ where bob hits the ground. Assume $\theta_0$ to be small, so that $\sin \theta_0 \simeq \theta_0$ and $\cos \theta_0 \simeq 1$
