A man having a wrist watch and a pendulum clock rises on a $TV$ tower. The wrist watch and pendulum clock per chance fall from the top of the tower. Then
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(d) Function of wrist watch depends upon spring action so it is not effected by gravity but pendulum clock has time period, $T = 2\pi \sqrt {\frac{l}{g}} $. During free fall effective acceleration becomes zero, so time period comes out to be infinity i.e. the clock stops.
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An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $M$. The piston and the cylinder have equal cross sectional area $A$. When the piston is in equilibrium, the volume of the gas is $V_0$ and its pressure is $P_ 0$. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency
$A$ particle of mass m is constrained to move on $x$ -axis. $A$ force $F$ acts on the particle. $F$ always points toward the position labeled $E$. For example, when the particle is to the left of $E, F$ points to the right. The magnitude of $F$ is a constant $F$ except at point $E$ where it is zero. The system is horizontal. $F$ is the net force acting on the particle. The particle is displaced a distance $A$ towards left from the equilibrium position $E$ and released from rest at $t = 0.$ Find minimum time it will take to reach from $x = - \frac{A}{2}$ to $0$.
A particle performs $SHM$ on $x-$ axis with time period of $0.5\,sec,$ such that it's velocity is zero at $x = -3\,cm$ and at $x = 9\,cm$. It was located at $x = 0$ and moving in negative $'x'$ at $t = 0$. The equation of $SHM$ of the particle is
A simple pendulum has time period $T_1$. The point of suspension is now moved upward according to equation $y = k{t^2}$ where $k = 1\,m/se{c^2}$. If new time period is $T_2$ then ratio $\frac{{T_1^2}}{{T_2^2}}$ will be
A particle free to move along the $x-$axis has potential energy given by $U(x) = k[1 - \exp {( - x)^2}]$ for $ - \infty \le x \le + \infty $, where k is a positive constant of appropriate dimensions. Then
Equations ${y_1} = A\sin \omega t$ and ${y_2} = \frac{A}{2}\sin \omega t + \frac{A}{2}\cos \omega t$ represent $S.H.M.$ The ratio of the amplitudes of the two motions is
The position co-ordinates of a particle moving in a $3-D$ coordinates system is given by $x = a\,cos\,\omega t$ , $y = a\,sin\,\omega t$ and $z = a\omega t$ The speed of the particle is