Students will learn to use Newton's Universal Law of Gravity equation to solve problems. They will also learn how the acceleration of gravity on Earth is calculated and a bit about gravitational fields in general.
FG=Gm1m2r2; the force of gravity between an object with mass m1and another object of mass m2and a distance between them of r.
G=6.67×10−11 Nm2/kg2; the universal constant of gravity
g=Gmr2; gravitational field strength or gravitational acceleration of a planet with mass mand radius r. Note that this is not really a separate equation but comes from Newton’s second law and the law of universal gravitation.
- Some data needed for the problems:
The radius of Earth is
The mass of Earth is about
The mass of Sun is about
The Earth-Sun distance is about
The Earth-Moon distance is about
- When using the Universal Law of Gravity formula and the constant
Gabove, make sure to use units of meters and kilograms.
- The direction of the force of gravity is in a straight line between two objects. It is always attractive.
- Newton invented calculus in order to prove that for a spherical object (like Earth) one can assume all of its mass is at the center of the sphere (thus in his formula, one can use the radius of Earth for the distance between a falling rock and Earth).
- Newton's Laws apply to all forces; but when he developed them only one was known: gravity. Newton's major insight --- and one of the greatest in the history of science --- was that the same force that causes objects to fall when released is also responsible for keeping the planets in orbit.
Any two objects in the universe, with masses
Here is an illustration of this law for two objects, for instance the earth and the sun:
Gravity on the Earth's Surface
On the surface of a planet --- such as earth --- the
For any object a height
We can do this because the quantity in braces only has constants; we can combine them and call their product
We call the quantity
Time for Practice
- Mo and Jo have been traveling through the galaxy for eons when they arrive at the planet Remulak. Wanting to measure the gravitational field strength of the planet they drop Mo’s lava lamp from the top deck of their spacecraft, collecting the velocity-time data shown below.
|velocity (m/s)||time (s)|
(a) Plot a velocity-time graph using the axes above. Put numbers, labels and units on your axes. Then draw a best-fit line (use a ruler) and use that line to find the gravitational field strength of Remulak. Explain below how you did that.
(b) Mo and Jo go exploring and drop a rock into a deep canyon – it hits the ground in 8.4 s. How deep is the canyon?
(c) If the rock has a mass of 25 g and makes a hole in the ground 1.3 cm deep, what force does the ground exert to bring it to a stop?
(d) Mo and Jo observe the shadows of their lava lamps at different positions on the planet and determine (a la Eratosthenes, the Greek astronomer, around 200 B.C.) that the radius of Remulak is 4500 km. Use that and your result for
Answer to 5