Question

There are two correct answers for this question:

A simple pendulum and a spring-mass pendulum both have identical frequencies. Which changes will result in the spring-mass system having twice the period of the pendulum?

a) Quadruple the mass of the simple pendulum

b) Replace the spring with one half the spring constant and double the mass

c) Make the string on the pendulum four times smaller and make the pendulum 4 times more massives

d) Double the mass in the mass-spring pendulum

Answer 1

Period of a simple pendulum is given by:

and period of the spring mass system is:

what we want is: T' = 2T

so,

=>

squaring both sides gives:

m/k = 4L/g

=> g = 4L/(m/k) = 4Lk/m

which we know, is a constant (g = 9.8 m/s²)

so, if k is reduced by a factor of 2, the mass of the spring-mass system must be decreased by the same factor of 2 to satisfy the above equation.

also, the length L can be decreased by a factor of 4 and the mass can be decreased by the same factor.