Question

Derive the equation of the motion in a vertical circle with suitable illustration. State its practical applications.

Answer 1

Consider a body of mass M tied at the end of a string and whirled in a circle of radius 'r'. Let v₁ and v₂ be velocities of the body and T₁ and T₂ be the tensions in the string at the lowest point A and the highest point B, respectively. The velocities of the body at point 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 the Figure.

At the lowest point A, a part of the tension T₁, balances the weight of the body and the remaining part provides the necessary centripetal force. Therefore,

T₁ - Mg = mv₁²/r     ....(i)

At the highest point, the tension in the string and the weight of the body together provide the necessary centripetal force. Hence,

T₂ + Mg = mv₂²/r     ...(ii)

In case, the tension T₂ at the highest point is zero, the whole of the centripetal force will be provided by the weight of the body. Therefore, in such a case, the equation (ii), becomes:

0 + Mg = mv₂²/r or, v₂ = √{gr}     ...(iii)

This is the minimum velocity, the body should possess at the top so that string does not slack. In case, the velocity of the body at point B is less than √{gr}, 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 √{gr} at the top. Let us now find the minimum velocity the body should possesses at the lowest point in order to just loop the vertical circle. 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

i.e., 1/2Mv₁² = Mg(2r) + 1/2Mv₂²

⇒ v₁² = 4gr + v₂²

Since, v₂ = √{gr}, we have

v²₁ = 4gr + gr = 5gr

⇒ v₁ = √{5gr}     ...(iv)

The equation (iv) 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'.

Practical applications of motion in a vertical circle- If a bucket containing water is rotated along a vertical circle such that its velocity at the lowest point is equal to greater than √{5 gr}, the water will not spill even when bucket is at the highest point. For the same reason, the person in an aeroplane who is not tied to his seat, will not fall down, while looping the vertical circle. In circus, the motor cyclist is able to perform the feat of driving the motorcycle along a vertical circle inside a cage.