A solid cylinder of mass m = 4 kg and radius R = 10 cm has two ropes wrapped around it, one near each end. The cylinder is held horizontally by fixing the two free ends of the cords to the hooks on the ceiling and both the cords are exactly vertical. The cylinder is released to fall under gravity. The linear acceleration of the cylinder is [n/(n+1)]g; then n = ?
Physics · Mixed Chapter
JEE Mains

Question

A solid cylinder of mass m = 4 kg and radius R = 10 cm has two ropes wrapped around it, one near each end. The cylinder is held horizontally by fixing the two free ends of the cords to the hooks on the ceiling and both the cords are exactly vertical. The cylinder is released to fall under gravity. The linear acceleration of the cylinder is [n/(n+1)]g; then n = ?

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Let upward tension in each string be T and downward acceleration be a. The translational equation is mg−2T=ma. Step 2: The two strings together exert rotational torque 2TR=Iα. Since the strings unwind without slipping, a=αR. Step 3: For a solid cylinder I=½mR², so 2T=(I/R²)a=(m/2)a. Step 4: Substitute into translation: mg−(m/2)a=ma, giving a=2g/3. Step 5: Comparing a=[n/(n+1)]g gives n=2.