Oscillations Te...

PE of a particle is $$\displaystyle U(x)=\frac{a}{x^2}-\frac{b}{x}$$. Find the time period of small oscillation

A hydrogen atom has mass $$1.68\times 10^{-27}$$kg. When attached to a certain massive molecule it oscillates with a frequency $$10^{14}$$ Hz and with an amplitude $$10^{-9}$$ cm. Find the force acting on the hydrogen atom.

A particle of mass $$m$$ moves according to the equation $$F=-amr$$ where $$a$$ is a positive constant$$, r$$ is radius vector. $$r=r_0\hat{i}$$ and $$v=v_0\hat{j}$$ at $$t=0$$. Describe the trajectory.

Two masses $$m_1$$ and $$m_2$$ are connected to a spring of spring constant $$K$$ at two ends. The spring is compressed by $$y$$ and released. The distance moved by $$m_1$$ before it comes to a stop for the first time is 

In the figure shown, a spring mass system is placed on a horizontal smooth surface in between two vertical rigid walls $$W_{1}$$ and $$W_{2}$$. One end of spring is fixed with wall $$W_{1}$$ and other end is attached with mass $$m$$ which is free to move. Initially, spring is tension free and having natural length $$l_{0}$$. Mass $$m$$ is compressed through a distance a and released. Taking the collision between wall $$W_{2}$$ and mass $$m$$ as elastic and $$K$$ as spring constant, the average force exerted by mass $$m$$ on wall $$W_{2}$$ in one oscillation of block is

A particle of mass is executing oscillations about the origin on the x-axis. Its potential energy is $$V(x) = k|x|^3$$, where $$k$$ is a positive constant. If the amplitude of oscillation is $$a$$, then its time period $$T$$ is proportional 

A particle oscillates simple harmonically with a period of 16 s. Two second after crossing the equilibrium position its velocity becomes 1 m/s. The amplitude is

Two spring-mass systems support equal mass and have spring constants $$\displaystyle K_{1}$$ and $$\displaystyle K_{2}$$. If the maximum velocities in two systems are equal then ratio of amplitude of 1st to that of 2nd is 

The equation of displacement of a particle executing simple harmonic motion is x = (5m) $$\displaystyle \sin \left [ (\pi s^{-1})t+\frac{\pi }{3} \right ]$$. Write down the amplitude, time period and maximum speed. Also find the velocity at t = 1s.

One end of an ideal spring is fixed to a wall at origin $$O$$ and axis of spring is parallel to $$x-$$axis. A block of mass $$m=1 \ kg$$ is attached to free end of the spring and it is performing $$SHM$$. Equation of position of the block in coordinate system shown in figure is $$x=10+3\sin 10t $$. Here, $$t$$ is in second and $$x$$ in $$cm$$. Another block of mass $$M=3\  kg$$, moving towards the origin with velocity $$30\  cm/s$$ collides with the block performing $$SHM$$ at $$t=0$$ and gets stuck to it calculate. 
(i) New amplitude of oscillation 
(ii) New equation for position of the combined body.