Answer to Question #122955 in Physics for Higgins

Question

A grandfather clock has a pendulum exactly 1.00 m long and kept perfect time. A
spoiled brat broke the pendulum. When it was repaired, the pendulum was 98.0 cm.
1) What was the original period?
2) What is the new period?
3) Did the repaired clock lose time or gain time?
4) What would be the accumulated error after one day of operation?

Answer 1

1) The original period was

$$T_0=2\pi\sqrt{\frac{l_0}g}=2\text{ s}.$$

2) The new period is

$$T=2\pi\sqrt{\frac{l}g}=1.99\text{ s}.$$

3) The repaired clock gains time: 1000 oscillations became not 2000 but 1990 s. 10 second faster!

4) Find the accumulated error after one day of operation. In 24 hours there are 86400 seconds. The normal pendulum must do

$$n_0=t_0/T_0=24\cdot3600/2=43200\text{ oscillations}.$$

43200 oscillations on the new pendulum show time of

$$t=n_0T=43200\cdot1.99=85968\text{ seconds}.$$

We are 432 seconds, or 7 min 12 s faster now!