By what factor must one change the weight suspended vertically from a spring coil in order to five times its period of simple harmonic motion?

A) 1/25
B) 0.2
C) 5.0
D) 25.0
E) 1.0

Question

By what factor must one change the weight suspended vertically from a spring coil in order to five times its period of simple harmonic motion?
​
A) 1/25
B) 0.2
C) 5.0
D) 25.0
E) ​1.0

Quizplus · Accepted Answer

The period of simple harmonic motion is given by T=2π√(m/k), where m is the mass of the object and k is the spring constant. If we want to increase the period by a factor of 5, we need to solve for the new mass that will give us the new period. Let's call this mass m'.

5T=2π√(m'/k)
(5T/2π)^2=m'/k
m'=k(5T/2π)^2

Now, since we are changing the weight (and keeping the spring constant the same), we need to find the ratio of the new weight to the old weight that will give us this new mass:

m'/m=(k(5T/2π)^2)/(km/1)=(5T/2π)^2

So, we need to find the square root of this ratio to find the factor by which we need to change the weight:

√(m'/m)=√((5T/2π)^2)=25

Therefore, we need to increase the weight by a factor of 25 to increase the period of simple harmonic motion by a factor of 5. Answer: D) 25.0.

By What Factor Must One Change the Weight Suspended Vertically