A ring of radius 3a is fixed rigidly on a tab | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $R = 3a$ be the radius of the fixed ring and $r = a$ be the radius of the small rolling ring.When the small ring is at the horizontal position

Vedclass · Accepted Answer

$\sqrt {2ga} $

Vedclass · Answer

(A) Let $R = 3a$ be the radius of the fixed ring and $r = a$ be the radius of the small rolling ring.When the small ring is at the horizontal position $A$,the height of its center from the lowest point is $h = R - r = 3a - a = 2a$.By the law of conservation of energy,the potential energy lost by the small ring equals the kinetic energy gained at the lowest point.Potential energy lost,$\Delta U = mgh = mg(2a) = 2mga$.At the lowest point,the kinetic energy $K$ consists of translational kinetic energy and rotational kinetic energy: $K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$.For a ring rolling without slipping,$I = mr^2$ and $\omega = v/r$.Substituting these,$K = \frac{1}{2}mv^2 + \frac{1}{2}(mr^2)(v/r)^2 = \frac{1}{2}mv^2 + \frac{1}{2}mv^2 = mv^2$.Equating energy: $2mga = mv^2$.Therefore,$v^2 = 2ga$,which gives $v = \sqrt{2ga}$.

Similar Questions

Two identical solid cylinders run a race starting from rest at the top of an inclined plane. If one cylinder slides and the other rolls,which of the following statements is true?

Suppose a body of mass $M$ and radius $R$ is allowed to roll on an inclined plane without slipping from its topmost point $A$. The acceleration of the body down the plane is given by (where $\beta = 1 + \frac{I}{MR^2}$):

$A$ solid cylinder and a solid sphere,having same mass $M$ and radius $R$,roll down the same inclined plane from the top without slipping. They start from rest. The ratio of the velocity of the solid cylinder to that of the solid sphere,with which they reach the ground,will be:

$A$ ring,sphere,and disc are rolling down an inclined plane from the same height. Find the wrong statement: (where $t$ is the time of descent,$a$ is the acceleration,and $v$ is the speed at the bottom).

Vedclass Test Series

Exam Paper Generator

Online Exam Module