# Problem on torsional constant

I came to find out through books and actual experiment that the torsional constant of a wire in a torsional pendulum is directly inverse to its length
I find the torsional constant using the rotational version of hooke's law, t=k*theta.
I have always thought that a greater length should give a greater torque (thus a higher torsional constant) because it has a higher momentum.
I tried to search for more specific explainations as to why it is a inverse relation on the internet, but without much help. The books I got in the libraries are either too simple or too advaned for me.
Any explainations would be greatly appreciated. Thx!

Doc Al
Mentor
Think of it like this. The torsional constant is defined as the amount of torque needed to get a certain angular twist: $k = \tau/\theta$. So assume a given wire of length L has a constant k. It requires a torque $\tau$ to produce a twist of $\theta$. What if I only needed a twist of $\theta/2$? Would you agree that I only need half the torque? (I presume you would.)

Now consider a wire of length 2L. You can think of it as being composed of two wires of length L in series. What the torsional constant of this composite wire? If I want a net twist of $\theta$, realize that each half of the wire only gets a twist of $\theta/2$. Thus the same twist ($\theta$) requires only half the torque. Thus the net torsional constant of a wire of length 2L is 1/2 the constant of a wire of length L. Make sense?

Note that this is the same thing that happens with springs put in series. Say I have two springs of spring constant k. If I hook them in series, what's the spring constant of the composite double spring? Figure it out the same way as I did above and you'll find that the double spring has a spring constant of k/2.

Thx a lot for you help, I got the hang of it now