A mass m fall on spring constant k and negligible mass from a height h. Assuming it sticks to the pan and executes simple harmonic motion, the maximum height upto which the pan will rise is:
When the mass m
falls on the spring from a height, let the length of the spring
compressed be x.
Total distance by which the mass m falls is: (h+x)
Loss of PE of the mass is equal to the energy stored in the
spring.
∴mg(h+x)=12kx2
∴x2−(2mgk)x−2mghk=0
∴x=2mgk±√(2mgk)2+8mghk)2
∴x=mgk±√(mgk)2+2mghk)=mgk(1+√(1+2khmg))
We ignore the -ve sign, since
it will give a negative x which is not a physical situation.
Thus,
the maximum height to which the mass will rise is: mgk[1+√(1+2khmg)]