Part 1 - After reading about Isaac Newton and his law of gravity, you may be challenged by this query. An astronaut lands on a planet that has 3 times the mass of the earth and has twice the earth's diameter. What would the astronaut weigh on the surface of this planet if his/her weight at the surface of the earth was 200 pounds? You need to include all your reasoning with your answer. Please do NOT derive the answer mathematically. Rather, solve it using reasoning and a knowledge of how gravity works.
We can make this problem much easier if you consider each effect separately.
Effects of Mass
First, the planet this astronaut lands on has 3 times the mass of the earth. This effect will automatically make the weight rise by a factor of 3. Why? If you consider the equation Newton derived:
F = G M1 M2 / d 2
If F is the force of gravity (weight) of the astronaut, what would happen if we kept everything else the same and let the mass of the planet (M1) increase by a factor of 3? You can see that this is straight forward ... the force must also increase by a factor of 3. This happens if you replace M1 (mass of earth) with 3M1 (mass of planet). Another way of seeing the same thing is by realizing that weight (F) is directly related to mass of the planet (M1). If M1 triples, so does F.
Effects of Size
Now what about the fact that this planet has twice the diameter of the earth? This is where the inverse square law comes into play. Consider the equation if you replace d with 2d. Since this quantity is squared, we get (2d)2 = 4d2. This implies that the force drops by a factor of 4 (since it appears in the denominator). Another way of seeing this is to consider what would happen to your weight of you moved to a distance of 2 earth radii (from the center). The force drops to 1/4 the surface value (right?) ... well isn't that exactly what you are doing if you stand on a planet with twice the radius of the earth?
Now put the two effect together and you get a weight which is 3/4 the initial surface value. 200 pounds x 3/4 = 150 pounds.
Below is an excellent answer sent in by a fellow student exactly as it was received.
The weight of the astronaut
on the new planet will be 150 pounds.
The gravitational field experienced by the astronaut on the surface of the earth
is given by the equation:
(G*M)/(r*r)
Where M is the mass of the earth;
r is the radius of the earth;
and G is the universal gravitational constant
The mass of the new planet is 3M and its radius is 2r. So, the gravitational
field experienced by the astronaut on the surface of the new planet will be:
(3 * G*M)/(4 * r * r)
Or,
3/4 * (G*M)/(r*r)
Or, 3/4 times the gravitational field on the surface of the earth. So, the
weight of the astronaut on the new planet will be 3/4 times her weight on earth.
Weight on new planet = 3/4 * 200 lbs = 150 lbs
Part 2 - Use Newton's law of gravity to calculate the gravitational force (weight) of a 100 kg astronaut standing on the earth. You will need to look up several quantities and show all your work. Hint: Force in the metric system is measured in Newtons (not pounds). Please show all your work and convert your answer to pounds. This will require some knowledge of scientific notation and skills using them on a calculator.
You will need to look up the value of G, the mass of the earth in kilograms (M2), and the radius of the earth in meters (d).
F = G M1 M2 /d2
F= (6.67*10-11)*( 100*5.97*1024)/(6.371*106)2
F=981.3n
Convert this to pounds you will get 220 pounds. Hey, that is exactly what 100 kg weighs at the surface of the earth. It works!!!