Imagine that two space explorers discovered a new inhabitable planet in another

Question

Imagine that two space explorers discovered a new inhabitable planet in another galaxy. In order to measure the gravitational acceleration of the newly discovered planet, they set up a pendulum on the surface of the planet. (a) The first explorer measures that the length of the pendulum is 1.93 m and that the pendulum swings with a period of 2.95 seconds when the amplitude of the pendulums swing is very small. Based on these measurements, what is the gravitational acceleration of the planet? kg m/s^2 (b) In order to confirm the first explorer?s results, the second explorer set up his own pendulum, and measured the period T of his pendulum by changing the length L of the pendulum. He found that the plot of T^2 vs. L is a straight line with a positive slope. From the value of the slope, he concluded that his measurement results are in full agreement with those of the first explorer. What is the value of the slope (in T^2 vs. L plot of the second explorer)? S^2/m

Explanation / Answer

time period of a simple pendulum=T=2*pi*sqrt(L/g)

where L=length of the pendulum

g=acceleration due to gravity

part a: given that T=2.95 seconds

L=1.93 m

==>2.95=2*pi*sqrt(1.93/g)

==>0.4695=sqrt(1.93/g)

==>1.93/g=0.4695^2=0.22043

==>g=1.93/0.22043=8.7556 m/s^2

part b:

as T=2*pi*sqrt(L/g)

==>T^2=4*pi^2*L/g

then slope of T^2 vs L plot is T^2/L=4*pi^2/g

as g found out to be 8.7556 m/s^2

slope=4*pi^2/8.7556

=4.509 s^2/m

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