Question

You have a pendulum of 50 cm in length, and a spring that’s able to oscillate back & forth with k = 20 N/m. What mass (in kg) should you attach to the spring to get them to oscillate at the same frequency?

Answer 1

We all know how a pendulum swings. Even if one doesn't know, he or she can do one thing. If he or she stands in front of a swing then his or her point of view will be FRONT VIEW. To see the real picture, he or she should go to any of the sides of the swing (SIDE VIEW) and in that case, he or she should be able to see the swinging motion of a pendulum.

Now the period of swing of a simple pendulum mainly depends on its length(whose value is given here), gravity(whose value is known to us)and to a small extent on the maximum angle that the pendulum swings away from the vertical. So basically it is independent of the mass of the bob.

Now, if the amplitude is limited to small swings the time period T(the time taken for a complete cycle) of the pendulum

is,

where l=length of the pendulum=50cm=0.5m

g=9.81m/s^2

So, T=1.418 sec

Now, we all know

where f=frequency

So, f= 1/1.418 =0.705 Hz

Now, let come to the case of the spring. Spring also behaves like a simple pendulum. But unlike the previous case, here mass affects period and hence the frequency of the motion.

Here, the formula for the time period:

where m=mass which is attached to the spring

k=spring constant

Now, according to the given problem, the frequency of both cases remains the same.

So, here also T=1.418 sec

So