Two blocks of masses m and M are connected by means of a metal wire of cross sectional area A passing over a frictionless fixed pulley as shown in the figure. The system is then released. If M = 2m, then the stress produced in the wire is

Problem Statement

Two blocks of masses m and 2m are connected by means of a metal wire of cross-sectional area A, passing over a frictionless fixed pulley. The system is then released. If M = 2m, determine the stress produced in the wire.

Solution

1. Understanding the System:There are two blocks, one of mass m and the other of mass 2m. They are connected by a wire with a cross-sectional area A, and the system passes over a frictionless pulley. The stress in the wire depends on the tension created when the system is released.
2. Force Analysis:When released, both blocks experience gravitational forces. The heavier block (2m) will accelerate downward, pulling the lighter block (m) upward. The tension in the wire must counterbalance these forces.
3. Using Newton’s Second Law:For the block of mass 2m:2mg - T = 2maFor the block of mass m:T - mg = maBy solving these equations, we find the tension T, which relates to the forces acting on both masses.
4. Calculating the Acceleration (a):Adding the two equations eliminates T and gives:2mg - mg = 3ma → a = g/3
5. Determining the Tension (T):Substituting a = g/3 into either equation, say T - mg = ma:T = mg + ma = mg + m(g/3) = 4mg/3
6. Finding the Stress in the Wire:Stress is defined as the force (tension) divided by the cross-sectional area:Stress = T / A = (4mg/3) / A = 4mg / (3A)

Final Answer

The stress produced in the wire when two blocks of masses m and 2m are connected and the system is released is: 4mg / (3A).