a) Two pendulums consist of a very thin string at the end of which hangs a mass which is much larger than the mass of the rod. In pendulum #1, the rod is made of copper. In pendulum #2, the rod is made of invar. What is the percentage in the period of the pendulums when the temperature increases by 16 degrees C?

For Copper?
For Invar?
b) The period of the pendulums are the time unit in a clock. How many seconds will the clock go slower in one day when the temperature increases by 16 degrees C?
For Copper?
For Invar.
Thanks.

a. for copper: 2.65; 1.6%; 0.014%; 0.0014%

for invar: 0.00012%; 0.00072%; 0.38%; 0.58%
This is what I have done so far:
Change in L for copper= 17*10^-6*10^3*16=
0.272m=27.2cm
Change in L for invar= 0.9*10^3*16= 0.0144m= 1.44cm
b.For copper: 12 sec; 18sec; 23 sec; 1382 sec.
For invar: 0.1 sec; 0.6 sec; 3 sec; 328 sec
I am having a problem figuring the percentages and time factors. Thanks for your help.

a) To determine the percentage change in the period of the pendulums when the temperature increases by 16 degrees Celsius, we need to use the formula for the period of a simple pendulum:

T = 2π√(L/g)

where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

For both pendulums, the only variable affected by the temperature change is the length of the pendulum, L. The change in length for each pendulum can be calculated using the coefficient of linear expansion (α) for the respective materials.

Copper has a coefficient of linear expansion, αcopper, while invar has a coefficient of linear expansion, αinvar. The change in length for each pendulum due to the temperature increase can be calculated using the formula:

ΔL = αΔT

where ΔT is the change in temperature.

Using the change in length, ΔL, we can calculate the new length, Lnew, by adding the change in length to the original length, L0.

Once we have the new lengths for both pendulums, we can substitute them into the pendulum period formula to find the new periods, Tnew. The percentage change in the period can then be calculated using the equation:

Percentage Change = ((Tnew - T0) / T0) * 100

where T0 is the original period.

b) To determine the number of seconds the clock will go slower in one day when the temperature increases by 16 degrees Celsius, we need to calculate the difference in periods for each pendulum due to the temperature change.

The change in the period, ΔT, can be calculated using the formula:

ΔT = Tnew - T0

Since the period is measured in seconds, we will have the difference in periods in seconds as well.