Warm-Up Hints: Proportional Reasoning Gravitational force is directly proportional to the mass of one object—so any change to the mass and the gravitational force changes by the same factor. If the mass of one object increases by a factor of three, the gravitational force also increases by a factor of three. Gravitational force is inversely proportional to the square of the distance (inverse square)—so any change to the distance and the gravitational force changes by the inverse of the square of the factor. If the distance increases by a factor of three, the gravitational force decreases (inverse) by a factor of nine (square). If multiple changes occur at the same time, simply multiply the resulting changes to the gravitational force together. If the mass of one object triples and the distance triples, the gravitational force would change by a factor of (3) x (1/9) = 1 / 3.

Universal Gravitation Mathematics

Target #1: How is the mass of one object related to the gravitational force between two objects?

Target #1: How is the mass of one object related to the gravitational force between two objects?

Target #1: How is the mass of one object related to the gravitational force between two objects?

Target #1: How is the mass of one object related to the gravitational force between two objects? The mass of one object is directly proportional to the magnitude of the gravitational force.

Target #1: How is the mass of one object related to the gravitational force between two objects? The mass of one object is directly proportional to the magnitude of the gravitational force.

Target #2: How is the distance between two objects related to the gravitational force between two objects?

Target #2: How is the distance between two objects related to the gravitational force between two objects?

Target #2: How is the distance between two objects related to the gravitational force between two objects?

Target #2: How is the distance between two objects related to the gravitational force between two objects? The distance between two objects is related by the inverse square to the magnitude of the gravitational force.

Target #2: How is the distance between two objects related to the gravitational force between two objects? The distance between two objects is related by the inverse square to the magnitude of the gravitational force.

Universal Gravitation Equation

“Universal Gravitation” Isaac Newton reflected on the attraction between any two objects in nature. He concluded that objects exert an attractive force (gravity) on each other that is: directly proportional to the mass of each object inversely proportional to the square of the distance between the objects

“Universal Gravitation” The force of gravity between objects depends on the distance between their centers of mass.

Would you weigh more or less at the top of a mountain? Why?

Your weight is less at the top of a mountain because you are farther from the center of Earth.

This equation can describe the relative strength of gravity between any two objects: m1 m2

This equation can describe the relative strength of gravity between any two objects: m1 m2 To write this proportional relationship as an actual equation, we have to make sure the right side of the equation gives us the correct number in units of Newtons (since gravity is a force).

If we use a proportionality constant—let’s call it “G” for gravity—then the force of gravity can be expressed as an exact equation. m1 m2

Universal Gravitational Constant This equation tells us the strength of gravity between any two objects in the universe—universal gravitation. To find the force of gravity between two objects, multiply their masses, divide by the square of the distance between their centers, and then multiply by G. The magnitude of G is defined as the force between two masses of 1 kilogram each at 1 meter apart: 0.0000000000667 N. G = 6.67 x 10-11 N m2/kg2

Philipp von Jolly (1809-1864) developed a method of measuring the attraction between two masses, and could calculate the universal gravitational constant, G.

The low value of G tells us that gravity is a very weak force. G = 6.67 x 10-11 N m2/kg2

Example 1: What is the force of gravity between a physics student (m1 = 60.0 kg) and his date (m2 = 55.0 kg) as they stand 5.0 meters apart on the dance floor?

Example 2: What is the force of gravity between a physics student (m1 = 60.0 kg) and the Earth? Mass of Earth: 6.0 x 1024 kg Radius of Earth (distance to center from edge): 6.4 x 106 meters

Example 3: What is the distance between the moon and the Earth, if the force of gravity between them is 2.0 x 1024 Newtons? Mass of Earth: 6.0 x 1024 kg Mass of Moon: 7.3 x 1022 kg

Gravity “Math” Practice