How Much Time Does a Shortened Pendulum Clock Lose in 24 Hours?

[bleedblue1234]

Homework Statement

Pendulum clocks are typically made so the period of the pendulum is 1 second or 2 seconds, but they don't have to be. Suppose a grandfather clock uses a pendulum that is 85 centimeters long. The clock is accidentally broken, and when repaired, the length is shorter by 2.0 millimeters. Every 24 hours of correct time, the "repaired" clock will be off how much?

Homework Equations

Tp = 2(pi)sqrt(L/g)

The Attempt at a Solution

So i found the period of the pendulum for both clocks but now I am stuck as to how to find the error...

 

[mgb_phys]

You have the periods of the 85 and 84.8 cm pendulums.
Whats the percentage difference?
eg If the wrong period is 5% shorter then the clock will be 5% faster and will record 5% more time

 

[bleedblue1234]

.001177... so it will be the percent difference times the number of minutes in a day?

so 1.70 minutes?

 

[bleedblue1234]

verification anyone?

 

[mgb_phys]

That's correct.
Another better way to do it though (and probably what you will learn) is to look at the order of the equation

In the pendulum equation the length is a sqrt() so the ratio of change in time is sqrt() the ratio of change in length, ie a 4times change in lentgth gives a 2x change in period.

So in this case the change in time is 1-sqrt(84.8/85), which when multiplied by 24*60 gives 1.7
Then a double check, shorter pendulums go fatser, so the 84.8 means a quicker rate and time is lost