Answer to Question #123129 in Physics for arden

Question

A 100KG barrel with a raduis of 50 cm has two ropes wrapped around it , the barrel is released from rest , causing the ropes to unwind and the barrel fall spinning toward the ground . what is the speed of the barrel after it has fallen a distance of 10m ? what is the tension in each rope ? assume that the barrels mass is uniformaly distributed and that the barrel rotates as a solid cylinder >

Answer 1

Apply law of conservation of energy. The moment of inertia of a solid cylinder is

$$I=\frac{1}{2}mr^2.$$

The initial potential energy will convert into kinetic energy of translational and rotational motion:

$$mgh=\frac{mv^2}{2}+\frac{I\omega^2}{2},$$

where

$$\omega=\frac{v}{r}.$$

Find the speed:

$$mgh=\frac{mv^2}{2}+\frac{(mr^2/2)(v/r)^2}{2}=\frac{3}{4}mv^2,\\\space\\ v=2\sqrt{\frac{gh}{3}}=2\sqrt{\frac{9.8\cdot10}{3}}=11.43\text{ m/s}.$$