Question

Psychology professor R.O. Dent requests funding for an experiment on compulsive thrill-seeking behavior in guinea pigs, in which the subject is to be attached to the end of a spring and whirled around in a horizontal circle. The spring has relaxed length b, and obeys Hooke’s law with spring constant k. It is stiff enough to keep from bending significantly under the guinea pig’s weight. (a) Calculate the length of the spring when it is undergoing steady circular motion in which one rotation takes a time T. Express your result in terms of k, b, T, and the guinea pig’s mass m. √ (b) The ethics committee somehow fails to veto the experiment, but the safety committee expresses concern. Why? Does your equation do anything unusual, or even spectacular, for any particular value of T? What do you think is the physical significance of this mathematical behavior?

Answer 1

x = F/k ... extension of the spring

But F is the centrifugal force on the poor piggy

F = mrw²

r is radius and w = 2 pi / T .... angular velocity

F = mb(6.28/T)²

x = 39.44mb/kT²

Total length of the spring = b + x

= b + 39.44mb/kT²

⁼ b(1 + 39.44m/kT²)

b)

As the length varies with square of periodic time, the changes are drastic.

Also, the equation assumes the radius to be 'b; Whereas, if exact value of radius is taken, as 'b+x', it will turn into an even higher order equation of T.

Physically, speaking. As the spring will expand, the centrifugal force will increase. As a result, the spring will extend even further. Again the force will increase and this cyclic behaviour will keep the spring extending until it finally breaks.