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 such that both the cords are exactly vertical. The cylinder is released to fall under gravity. Find the tension in the cords when they unwind and the linear acceleration of the cylinder.

• A. $6.5\frac{m}{s^2},6.5N$
• B. $5.5\frac{m}{s^2},6.5N$
• C. $6.5\frac{m}{s^2},5.5N$
• D. $5.5\frac{m}{s^2},5.5N$

Answer 1

The correct option is A

$6.5\frac{m}{s^2},6.5N$

Let a = linear acceleration and $\alpha$ = angular acceleration of the cylinder.

for the linear motion of the cylinder : mg - 2T = ma

for the rotational motion : net torque = 1$\alpha$

Also, the linear acceleration of cylinder is same as the tangential acceleration of any point on its surface. A = R$\alpha$
Combining the three equations, we get : $mg=ma+\frac{m}{2}a$

$\Rightarrow a=\frac{2g}{3}=6.53\frac{m}{s^2}andT=\frac{mg-ma}{2}=6.53N$.