Question

A bob of mass m is attached to one end of the rod of negligible mass and length r, the other end of which is pivoted freely at a fixed center O as shown in the figure. What initial speed must be given to the object to reach the top of the circle? (Hint: Use law of conservation of energy). Is this speed less or greater than the speed obtained in section 4.2.9?

Answer 1

To get the vertical speed given to the object to reach the top of the circle, the law of conservation of energy can be used at points (1) and (2)

Total energy at 1 = Total energy at 2

∴ Potential energy at point 1 = 0

from eqn (i)

`0 + 1/2mv_1^2 = 2mgr + 1/2mv_2^2`

= `1/2v_1^2 = 2gr + 1/2v_2^2`

= `v_1^2 - v_2^2 = 4gr` ...........................(2)

In this case, the bob of mass m is connected with a rod of negligible mass, so the velocity of the bob at the highest point can be equal to zero i.e. v₂ = 0

∴ eqn. (2) becomes `v_1^2 = 4gr, v_1 = sqrt(4gr)  ms^-1 "or"  v_1 = 2sqrt(gr)  ms^-1` 

The speed of the bob obtained here is lesser than the speed obtained in section 4.2.9. It is only because the string is replaced by a massless rod here.