Investigation 6

Gravitation

These are the units for weight and mass in the three common systems of units.

Unit in the SI System of Units / Unit in the British Engineering System of Units / Unit in the cgs System of Units
Weight / Newtons (N) / pounds (lb) / dynes
Mass / kilograms (kg) / slugs / grams (g)

1.What weighs more, one kilogram of lead or one kilogram of feathers? (Do you know why?)

Same weight – both 10 Newtons

2.a.How much do you weigh? Express your weight in pounds.

160 pounds

b.Write your weight in Newtons. In order to do this, you must multiply your weight in pounds by 4.45.

160 x 4.45 = 712 Newtons

c.Now calculate your mass in kilograms. Remember that weight = mass x g, and that mass = (weight/g).

712/10 =71.2 kg

d.The force the earth exerts on you as calculated by using Newton’s Universal Law of Gravitation is , where G = 6.67 x 10-11 N m2/kg2, m1 = mass of earth = 5.98 x 1024 kg, m2 = your mass in kg, and d = radius of the earth = 6.378 x 106 m. Should the value of this force be the same as the value of your weight found in part b? Explain – do not calculate.

Yes – they represent the same thing – the force the earth exerts on you.

e.The gravitational force on objects on the surface of the moon is approximately 1/6 of the gravitational force the earth exerts on the same objects.

1)What is the value of your mass when you are on the moon? Express your result in kilograms.

71.2 kg – same no matter where I am.

2)How much do you weigh on the moon? Express your result in pounds and Newtons.

160/6 = 26.7 pounds

712/6 = 119 Newtons

3.A physics student weighs 600 N at the surface of the earth. Find her weight at the following positions:

a.a distance above the surface of the earth equal to the radius of the earth.

Distance from center of the earth is doubled, so the force decreases by 2 squared = 4, so the student’s weight is 600/4 = 150 Newtons.

b.a distance above the surface of the earth equal to four times the radius of the earth.

Distance from center of earth is 5 times greater so the force is 600/(5 squared) = 600/25 = 24 Newtons.

c.at the center of the earth.

Zero

4.Let’s take a look at what happens to the gravitational force on a student who weighs 600 Nas the radius of the earth is reduced, but the massremains the same (the density of the earth will increase). This may give you some idea of how a black hole results from the gravitational collapse of a star.

The gravitational force on a person at the surface of the earthis , where mperson is the person’s mass. This is called an “inverse square” force. This means that when the distance is doubled, the force is reduced by a factor of 2 squared, or four. And when the distance is halved, the force is increased by a factor of 2 squared, or four. If the distance is tripled, then the force is decreasedby a factor of 9 (3 squared). And when the distance is reduced to a third, then the force is increased by a factor of 9. And so on.

So, if the radius of the earth is reduced to half of its value , then the student will increase in weight to 4 x 600 = 2400 N. If the radius of the earth is reduced to one-third of its original value, , then the student will weigh 9 x 600 = 5400 N, and so on.

a.Suppose the radius of the earth is reduced to one-tenth (1/10) of its original value, then how much will the student weigh at the surface of the earth? (Remember that the mass of the earth isn’t changing.)

100 squared = 100, so her weight is 100 x 600 = 600000 Newtons ~ 13,500 pounds

b.When the radius of the earth is reduced to about one millimeter (remember, all of the mass of the earth is contained within that 1 mm radius sphere), the earth theoretically becomes a black hole. In this case, the radius of the earth is. How much does the student now weigh? (Obviously, this can’t happen, but it is fun to speculate.)

10,000,000,000 squared = 100,000,000,000,000,000,000 so her weight is

100,000,000,000,000,000,000 x 600 = 60,000,000,000,000,000,000 Newtons