A stone of mass m is tied to an elastic string of negligible mass and spring constant k. The unstretched length of the string is L and has negligible mass. The other end of the string is fixed to a nail at a point P. Initially the stone is at the same level as the point P. The stone is dropped vertically from point P.

(a) Find the distance y from the top when the mass comes to rest for an instant, for the first time.

(b) What is the maximum velocity attained by the stone in this drop?

(c) What shall be the nature of the motion after the stone has reached its lowest point?

It is clear from question that Ball will at least travel to natural length of spring and further it will stretch due its elastic nature of spring.

Let assume that distance travelled by stone of Mass = x

Now we know that it isn’t elastic collision, therefore Energy will conserve. By travelling a distance of Y it will gain Potential energy of spring which will store in spring whereas it will lose its potential energy.

Therefore loss in Potential energy =

And gain in Potential energy of spring =

Here we taking the difference as energy is gained by spring is for the extra distance it will travel from its natural length.

Equating both equations

Solving the quadratic equation we will get two solutions, the solution with positive will answer as we are dealing in positive domain of distance.

(b) Maximum will be attained when a particle isn’t experiencing any force. That is when a particle’s own gravitational force is equal to force due to elasticity of spring

Where p is distance from its natural length (No force is experienced before spring stretches out of its natural length)

Using Conservation of energy, we know that loss in potential energy will be gain in kinetic energy which will be gain by particle and gain in elastic energy of spring.

Equating p to,

And solving the quadratic equation and taking only positive sign for velocity as we dealing in positive domain, we get,

(c) Intuitively we know that particle will perform SHM, but we need to prove that. Therefore we need to reduce it to SHM equation.

We know that at particular distance y (from its initial position) we can write,

As the F_Net which will experienced by stone will due to gravitational force towards down and elastic force towards up. And we also know that acceleration is double derivative of distance travel therefore we can write,

To reduce more, we will assume that

Now double differentiating w w.r.t time we will get,

Substituting in above equation we get,

Therefore, its angular frequency is

For calculating its mean position we have to find a particular distance at which no force is acting,