Question

A particle of mass m is fixed to one end of a light rigid rod of length l and rotated in a vertical ciorcular path about its other end. The minimum speed of the particle at its highest point must be

Answer 1

Given :

• mass of particle = m
• length of rod = l

To find :

• Minimum speed of particle at highest point.

Solution:

1. When a rigid rod with an object tied to its end is rotated in a circular  path, the tension on the rod and weight of the object is adjusted so as to provide enough centripetal force.
2. Centripetal force is given by,
3. $\frac{mv^{2} }{l} = T - mgcos\alpha$   where α is angular displacement, m is mass of rod, T is tension on rod.
4. At the highest point, minimum velocity is obtained and tension on string is 0 and α is π.
5. cos π = -1 and T = 0. Substituting in formula we get,
6. $v = \sqrt{gl}$

Th answer is $\sqrt{gl}$

Answer 2

Hence minimum velocity at highest point is Zero.

Explanation:

Given data:

• Mass of particle = m
• Length of rod =  l
• To find: Minimum speed of particle = ?

Solution:

Let the velocity at B be v and at A be   = √ 4gR

Energy conservation at A and B,  1 / 2  m(  √ 4gR  )  2 +0 =  1 / 2  mv^2  + mg(2R)

⟹ v = 0

Thus minimum velocity at highest point is Zero.