What is the force constant of a spring cut in half, and would the frequency of SHM differ from the frequency of the same mass and energy?
If a uniform spring is cut in half, what is the force constant of each half, and how would the frequency of SHM using a half-spring differ from the frequency using the same mass and the same energy?
When a spring is cut in half, it would take twice as much force to stretch it by the same length. Now we know from the equation F = -kx that k = -F/x where F is the force required to stretch a spring by a distance x and k is the spring constant. Now if the force constant is k initially and on the spring being cut it changes to k’, we see that k’=-2F/m = 2k. Therefore we get the new spring constant as twice the initial spring constant.
The frequency of oscillation of a spring with spring constant k is given by (1/2*pi)*sqrt (k/m). Here as k has become 2k the frequency increases by a factor of sqrt 2. The new frequency is sqrt 2 times the old frequency of oscillation.