Question

Figure shows a smooth track, a part of which is a circle of radius R. A block of mass m is pushed against a spring of spring constant k fixed at the left end and is then released. Find the initial compression of the spring so that the block presses the track with a force mg when it reaches the point P, where the radius of the track is horizontal.

Answer 1

Let the string compression = x 
So the Potential energy = 1/2 k.x²
the released point potential energy is equal to change in Kinetic energy 
At the point P , Kinetic energy is converted to Potential energy.

Now Energy at  P = Kinetic energy + Potential energy.
$1/2 mv^2 + mgR$
 $1/2 mv^2+ mgR =1/2 kx^2$
$mv^2+2mgR =kx^2$  ---→(a)
Here the block press the track to the Normal force. which is equal to mg.
Now,
 mv² =mgR
[tex]mv^2/R = mg [/tex]
Put this value in eq (a)
$mgR+2mgR = kx^2$
$kx^2= 3mgR$
$x^2 = 3mgR/k$
$x = \sqrt{x(3mgR)/k }$

Hence the Initial compression is $x = \sqrt{x(3mgR)/k }$ 




Hope it Helps. :-)