Motion in a Vertical Plane Consider a body of mass m tied at the end of a string and whirled in a vertical circle of radius r. Let v1 & v2 be velocities of the body and T1 and T2 be tensions in the string at the lowest point A and the highest point B respectively. The velocities of the body at points A and B will be directed along tangents to the circular path at these points while tensions in the string will always act towards the fixed point O as shown in figure. At the lowest point A, a part of tension T, balances the weight of the body and the remaining part provides the necessary centripetal force. Therefore.
T1 – mg = \(\frac{m v_{1}^{2}}{r}\) ……….. (1)
At the highest point, the tension in the string and the weight of the body together provide the necessary centripetal force.
Let us now find the out minimum velocity, the body should possess at the lowest point, so that the string does not slack when it is at the highest point. The body is then said to just loop the vertical circle. It is obvious that the velocity at the lowest point will be minimum. When the velocity at the highest point is also minimum.
Minimum velocity at the highest point :
From equation (2), it follows that the velocity at the highest point will be minimum when the tension in the string at the highest point is zero.
T2 = 0 . ………… (3)
n that case, the whole of the centripetal force will be provided by the weight of body. Therefore in such a case, the equation (2) becomes :
0 + mg = \(\frac{m v_{2}^{2}}{r}\)
or v2 = \(\sqrt{g r}\) …………… (4)
This is the minimum velocity the body should possess at the top, so that it can just loop the vertical circle without the slackening of the string. In case the velocity of the body at point B is less than \(\sqrt{g r}\). The string will slack and the body will not loop the circle. Therefore, a body will just loop the vertical circle if it possesses velocity equal to \(\sqrt{g r}\) at the top.
Minimum velocity at the lowest point : According to the principle of conservation of energy.
K.E. of the body at point A = (P.E. + K.E) of the body at point B
\(\frac{1}{2} m v_{1}^{2}=m g(2 r)+\frac{1}{2} m v_{2}^{2}\)
or \(v_{1}^{2}=4 g r+v_{2}^{2}\)
As said earlier, when the velocity at the highest point is minimum. The velocity at the lowest point will also be minimum.
Setting v2 = \(\sqrt{g r}\), we have
\(v_{1}^{2}\) = 4gr + gr
or v1 = \(\sqrt{5 g r}\) ……….. (5)
The equation (5) gives the magnitude of the velocity at the lowest point with which body can safely go round the vertical circle of radius r or can loop the circle of radius r. Let us find out the tension in the string at the lowest point in such a case.
In equation (1),
setting v1 = \(\sqrt{5 g r}\), we have
T1 - mg = \(\frac{m \times(\sqrt{5 g r})^{2}}{r}\)
or T1 – mg = 5mg
or T1 = 6mg
