Question

How does the adding spin to the string(we twisted the string) of the sipmle pendulum affect to its period (T). Show some equations and work to prove if there's any affect to the period of pendulum.

Answer 1

The time period of a simple pendulum is given by :-

T = 2π × √(l/g) where, l = length of the string

g = acceleration due to gravity.

Now, if we talk about twisting the string of the simple pendulum, you will observe that the length of the string will decrease with twisting. As a result, the time period will also DECREASE as time period 'T' is directly proportional to the length 'l' of the pendulum.

Now, in addition to the simple harmonic motion, the pendulum will also have a torsional pendulum-like motion due to the twisted string. In the ideal case, both these motions should be separate and completely independent of each other resulting in two different Lagrangian Equations which depend on different variables. What I mean by this is that the equation for simple harmonic motion is :

= - (g/l) sin() where is the angle that string makes with the vertical. This would result in the time period being

T = 2π × √(l/g).

On the other hand, the torsional motion will result in the equation of motion being :

= -* where = restoring torque due to twisted string, = torsional constant and, = angle by which the string has been twisted. This would result in the time period of rotation of the simple pendulum being: T = 2 *(I/) where I = moment of inertia of the pendulum bob.

Now, these two motions will be independent of each other. It will be like having two separate simple harmonic motions and hence, there are two separate time periods, one for the regular swinging of the pendulum and the other one for the rotational motion of the pendulum due to twisted spring. This is all true, in the ideal case, where the string is massless, taut, inextensible and the tension is uniformly distributed all over the string.

But, in the real world scenario, the twisted string will affect the simple harmonic motion as it will offer more tension and will increase the length of the pendulum by gradually untwisting. All these changes can be measured experimentally but not theoretically because of the two motions being separate in the ideal case and so, it will difficult to find the connection between them.

Hope this helps!!!