Answer to Question #123206 in Astronomy | Astrophysics for Hem narayan Sah

Question

Because your spaceship has an engine failure, you crash-land with an emergency capsule at the equator of a nearby planet. The planet is very small and the surface is a desert with some stones and small rocks laying around. You needwater to survive. However,water is only available at the poles of the planet. You find the following items in your emergency capsule:
 Stopwatch
 Electronic scale
 2m yardstick
 1 Litre oil
 Measuring cup
Describe an experiment to determine your distance to the poles by using the available items.
Hint: As the planet is very small, you can assume the same density everywhere.

Answer 1

At the equator, the acceleration due to gravity is the smallest. At the poles, it is the greatest because no centripetal force acts on bodies there. Therefore, measure the weight of the cup filled with some fixed volume of oil (say 500 ml). The weight at the equator will be

$$P_e=(m_\text{cup}+m_\text{oil})g-\frac{(m_\text{cup}+m_\text{oil})v^2}{R},\\\space\\ P_e=(m_\text{cup}+\rho V)g-\frac{4(m_\text{cup}+\rho V)\pi^2R}{T^2},\\\space\\ R=[(m_\text{cup}+\rho V)g-P_e]\frac{T^2}{4(m_\text{cup}+\rho V)\pi^2}.$$

So, to find the radius of the planet, you need to know the true mass of the cup (should be written in the bottom), the volume and density of oil, how long it takes for the planet to make one revolution (use your stopwatch to measure the period).

To make your findings of the radius more precise, conduct the experiment with different volumes of oil. You will get a function and it will be easy to find the radius.

What you don't know yet is the acceleration due to gravity $g$. You can find it by making a physical pendulum. Make the yardstick oscillate around one of its ends and measure the period:

$$T=2\pi\sqrt{\frac{I}{mgL}}.$$

The moment of inertia of the stick rotating around one of its ends:

$$I=\frac{mL^2}{3},\\\space\\ T=2\pi\sqrt{\frac{L}{3g}}.$$

You know the length, have stopwatch, find g:

$$g=\frac{4\pi^2L}{3T^2}$$

Finally, multiply the radius by $\pi/2$ to find the distance to the poles.