In this example we will use pendulum motion to actually measure the acceleration

Question

In this example we will use pendulum motion to actually measure the acceleration of gravity on a different planet. An astronaut on the surface of Mars measures the frequency of oscillation of a simple pendulum consisting of a ball on the end of a string. He finds that the pendulum oscillates with a period of 1.5 s. But the acceleration due to gravity on Mars is less than that on earth, gMars=0.38gearth. Later, during a journey to another planet, the astronaut finds that his simple pendulum oscillates with a period of 0.92 s. What planet is he now on?

SOLUTION

SET UP Each planet has a different value of the gravitational acceleration g near its surface. The astronaut can measure g at his location, and from this he can determine what planet he's on. First we use the information about Mars to find the length L of the string that the astronaut is swinging. Then we use that length to find the acceleration due to gravity on the unknown planet.

SOLVE First we use the Mars data to determine L. We solve the equation for the period on Mars for L:

TMarsL===2LgMarsT2gMars42=T2(0.38gearth)42=(1.5s)2(0.38×9.8m/s2)420.21m

Now we use this value of L with T=0.92s to solve for the acceleration due to gravity on the unknown planet, gplanet:

gplanet=42LT2=42(0.21m)(0.92s)2=9.8m/s2

He is on planet earth!

REFLECT In principle, an astronaut can use this method to measure the acceleration due to gravity of any planet or satellite while traveling around the solar system.

Part A - Practice Problem:

The acceleration due to gravity near the surface of a distant planet is about 4.1 m/s2 . If the astronaut arrives there and swings the same pendulum bob, what will be the period of the pendulum?

Express your answer in seconds to two significant figures.

Explanation / Answer

Time period on mass Tm =1.5 s

Accleration due to gravity g =9.8 m/s^2

Acceleration due to mass gm = 0.38g =0.38*9.8 = 3.724 m/s^2

Acceleration due to planet gs = 4.1 m/s^2

T = 2pi [L/g]1/2

Ts/Tm = [gm/gs]1/2

Ts = (Tm) [gm/gs]1/2 = (1.5) (3.724/4.1)1/2

Ts = 1.4 s

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