A ring and a disc roll on the horizontal surface without slipping with same linear velocity. If both have same mass and total kinetic energy of the ring is 4 J then total kinetic energy of the disc is

Options

A. 3 J
B. 4 J
C. 5 J
D. 6 J

Answer

A

Explanation

Total kinetic energy of the ring it rolls without slipping , `K_("ring") = K_(T) + K_(R) = (1)/(2) mv^(2) + (1)/(2) I , omega^(2)`
= `(1)/(2) mv^(2) + (1)/(2) mr^(2) xx (v^(2))/(r^(2)) = mv^(2) " " (because I_(Y) = mr^(2) and omega = (v)/(r))`
But `K_("ring") =4 J` (given) `therefore mv^(2) = 4 J " " ... (i)`
Similarly , `K_("disc") = (1)/(2) mv^(2) + (1)/(2) I_(d) omega^(2)`
`= (1)/(2) mv^(2) + (1)/(2) xx (mr^(2))/(2) xx (v^(2))/(r^(2)) ( because I_(d) = (mr^(2))/(2))`
`= (3)/(4) mv^(2) = (3)/(4) xx 4 J = 3J " " ` (Using (i))