Question

An apple weighs 1.15 N. When you hang it from the end of a long spring of force constant 1.59 N/m and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. (Because the angle is small, the back and forth swings do not cause any appreciable change in the length of the spring.)

What is the unstretched length of the spring (i.e., without the apple attached)?

Answer 1

2.167 m

Step-by-step explanation:

As we know that linear frequency is proportional to angular frequency, ω = 2πf, the pendulum angular frequency is half the bounce angular frequency

√[ g / L ] = ½√[ k / m ]

4g / L = k / m

L = 4mg / k = 4 (1.15 N) / (1.59 N/m) = 2.89 m

The above is length of the spring with the apple attached. When you attached the apple, the spring stretched a distance

So, F = ks ⇒ s = F / k

s = mg / k = (1.15 N) / (1.59 N/m) = 0.723 m

So before you attached the apple, the spring had a length of:

2.89 - 0.723 = 2.167 m