Question

Gillian buys a pendulum clock at a discount store and discovers when she gets it home that it loses 6.00 minutes each day.

Answer 1

Answer:

a) The pendulum must be shortened to keep accurate time

b) It should be shortened by 0.0082 m or 8.2 mm

Explanation:

Simple Pendulum

A simple pendulum is a system with a point mass that is suspended from a weightless string to a fixed point. It describes a harmonic motion because the oscillations repeat regularly, and kinetic energy is transformed into potential energy, and vice versa.

The equation for the period of a simple pendulum is

$\displaystyle T = 2\pi \sqrt{\frac{L}{ g}}$

where L is the length of the sting and g is the acceleration of gravity .

Note that if we increase the length, the period will also increase, and the oscillations are slower, i.e. take longer to complete

a) Gillian's pendulum is running slow because it loses 6 minutes each day. As shown above, the longer the string, the slower the oscillations, thus he needs to shorten the string to make it move faster and keep up in time .

b) We know the period is 2 seconds. That will give us the actual length of the pendulum, solving the above equation for L .

$\displaystyle L = \frac{gT^2}{4\pi^2}$

$\displaystyle L = \frac{(9.8)2^2}{4\pi^2}$

$\displaystyle L = 0.9929\ m$

The new period should be less than the original. We know that actually, the pendulum's mechanisms make a period of 2 seconds in a measured time of

3600*24+6*60=86760 seconds each day .

This should be shortened to the correct time of 3600*24=86400 seconds per day, so the new period should be

$T'=2\ sec*(86400/86760)=1.9917\ sec$

Which yields to a new length  of

$\displaystyle L' = \frac{gT'^2}{4\pi^2}$

$\displaystyle L' = \frac{(9.8)(1.9917)^2}{4\pi^2}$

$L'=0.9847\ m$

Difference of lengths = 0.9929 - 0.9847=0.0082 m

$\boxed{\text{It should be shortened by 0.0082 m or 8.2 mm}}$