Oscillations Te...

uniform rod of mass m and length I is suspended about its end. Time period is

The system shown in lying on a smooth horizontal surface. Mass $$m$$ is constrained to move along the line $$3$$. Springs $$1$$ and $$2$$ at right angles with respect to one another are symmetrically arranged with respect to $$3$$ which is an elastic cord. The force constants of $$1,2$$ and $$3$$ are $$k$$ each and they are all just taut in the condition shown. For small oscillations of $$m$$ along $$3$$, time period is

A block of mass $$m$$ is suspended from one end of a light spring as shown. The origin $$O$$ is considered at distance equal to natural length of spring from ceiling and vertical downward direction as positive $$Y-axis.$$ When the system is in equilibrium a bullet of mass $$m / 3$$ moving in vertical upward direction with velocity $$v _ { 0 }$$ strikes the block and embeds into it. As a result, the block (with bullet embedded into it) moves up and starts oscillating.
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The differential equation representing the SHM of a particule is $$\cfrac {{9d^2y}} {dt^2} +4Y=0.$$The time period of the particle is fiven by

Two point masses of 3.0 kg and 6.0 kg are attached to opposite ends of horizontal sprig whose spring constant is 300 N$$m^{-1}$$ as shown in the figure. The natural vibration frequency of the system is approximately.

Two masses $${m_1}$$ and $${m_2}$$ are joined by a spring as shown. The system is dropped to the ground from a certain height. The spring will be :-

A horizontal force $$F$$ is applied to the lower end of a uniform thin rod of mass $$4kg$$ and length $$L = 50\ cm$$ as shown in the figure. The rod undergoes only translational motion along the smooth horizontal surface. If $$F = 60\ N$$ determine the angle $$\theta$$ for translation motion of the rod. $$(g =10\ m/s^{2})$$.

Two particles are executing identical simple harmonic motions described by the equations, $$x_1=a\cos \left(\omega t+\left(\dfrac{\pi}{6}\right)\right)$$ and $$x_2=a\cos \left(\omega t+\dfrac{\pi}{3}\right)$$. The minimum interval of time between the particles crossing the respective mean position is?

The potential energy of a particle executing SHM along the x-axis is given by $$U=U_0-U_0\cos ax$$. What is the period of oscillation?

At some instant, velocity and acceleration of a particle of mass '$$m$$' in SHM are '$$a$$' and '$$b$$' respectively. If its angular frequency is $$\omega,$$ then its maximum K.E. is