Oscillations Te...

A mass $$m$$ is undergoing SHM in the vertical direction about the mean position $$y_{0}$$ with amplitude A and angular frequency $$\omega$$. At a distance $$y$$ from the mean position, the mass detaches from the spring. Assume that the spring contracts and does not obstruct the motion of $$m$$. Find the distance $$y^{*}$$ (measured from the mean position) such that the height $$h$$ attained by the block is maximum. $$(\mathrm{A}\mathrm{\omega})^{2}>g$$

A particle performs simple harmonic motion with amplitude A. Its speed is tripled at the instant that it is at a distance $$\displaystyle\frac{2A}{3}$$ from equilibrium position. The new amplitude of the motion is.

A small block is connected to one end of a massless spring of unstretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at t=0. lt then executes simple harmonic motion with angular frequency $$\omega =\frac { \pi }{ 3 } rad/s$$.
Simultaneously at t=0, a small pebble is projected with speed v from point P at an angle of 45 as shown in the figure. Point P is at a horizontal distance of 10 m from O. If the pebble hits the block at t = 1 s, the value of v is
(take $$g = 10 m/{ s }^{ 2 }$$)

A particle is executing SHM along a straight line. Its velocities at distances $$x_1$$ and $$x_2$$ from the mean position are $$V_1$$ and $$V_2$$, respectively. Its time period is:

Average velocity of a particle executing SHM in one complete vibration is : 

A simple pendulum performs simple harmonic motion about $$x = 0$$ with an amplitude $$'a'$$ and time period $$'T'$$. The speed of the pendulum at $$x = \dfrac{a}{2}$$ will be:

A particle is describing simple harmonic motion. If its velocities are $$v_{1}$$ and $$v_{2}$$ when the displacements from the mean position are $$y_{1}$$ and $$y_{2}$$ respectively, then its time period is

If maximum speed of a particular in SHM is given by $$\displaystyle { V }_{ m }$$. What is its average speed?

The velocity vector v and displacement vector x of a particle executing SHM are related as $$\dfrac{vdv}{dx}=-\omega^2x$$ with the initial condition $$V=v_0$$ at $$x=0$$. The velocity v, when displacement is x, is?

In SHM the net force towards mean position is related to its displacement (x) from mean position by the relation