Question

Two blocks of masses m and M are joined with an ideal spring of spring constant k and kept on a rough surface. The spring is initially unstretched and the coefficient of friction between the blocks and the horizontal surface is u. What should
be the maximum speed of the block of mass Msuch that the smaller block does not move?

Answer 1

Given:

Two blocks of masses m and M,

Ideal spring, spring constant  = k,

Coefficient of friction between the blocks and the horizontal surface =  u.

To Find:

The maximum speed of the block of mass M such that the smaller block does not move.

Solution:

Let V be the maximum speed of the block of mass M.

Lets assume the spring elongates by 'x' m.

Then Total energy of the system ,

• Kinetic energy of M = Potential energy loss due to friction+ Spring energy

• 1/2 MV² = uMgx + 1/2kx²

For the mass 'm' not to move,

Net force should be zero.

forces on m ,

• umg due to friction , acting
• spring force, kx .
• Therefore, kx  = umg
• x = umg/k

• (kx)² = (umg)²
• 1/2kx² = 1/2(umg)²/k

Therefore applying this in above equation:

• 1/2MV² = uMgumg/k + 1/2(umg)²/k
• MV² = 2(ug)²Mm/k + (umg)²/k

• V =ug $\sqrt{\frac{2m}{k}+\frac{m^{2} }{Mk} }$

The maximum speed of the block of mass M such that the smaller block does not move is  V = ug $\sqrt{\frac{2m}{k}+\frac{m^{2} }{Mk} }$ .