Question

The system shown in Fig.  consists of two blocks  (masses m₁ and m₂), a spring (spring constant k), a massless rope and two massless pulleys. Block 1 lies on a rough surface (coefficient of kinetic friction µ). At the beginning of the motion (t = 0), the spring is unstretched and the position of block 1 is given by x₁ = 0. Determine the velocity x₁ as a function of the position x₁. 

Answer 1

Since we want to find the velocity as a function of the work - energy theorem 

T₁ − T₀ = U .

In moving from the initial position to an arbitrary position,the gravitational force , the force in the friction force perform work:

 The kinetic energies are given by

With the kinematic relation

x₂=2x₁

the work - energy theorem yields