Chapters 9 & 10: Gravity & Satellite Motion

Gravity

• Gravitational forces are exerted between two objects because of their masses.

• This force is always attractive and the amount of force is proportional to the product of their masses.

• The gravitational force decreases as the masses move farther apart and increases as they move closer together. It is inversely proportional to the square of the distance.

The mathematical expression is     

where F is the magnitude of the force, G is the universal gravitation constant, M₁ and M₂ are the masses of the two objects and d is the distance between them. Note that G is measured by measuring the force between two known masses separated by a known distance.

• Note that if I double one of the masses, the force doubles. If the distance between the masses is doubled, the force decreases by a factor of 4, i.e. it becomes ¼ of what it was. For instance, if it was 100N it would become 25N.

• For uniform spherical masses the force can be calculated assuming all the mass is located at the center of the sphere.

Satellite Motion

• Objects moving in a circle with a constant speed are said to be in uniform circular motion. However, note that the velocity is changing direction as it moves in a circle, so the object is accelerated and the acceleration is toward the center of the circle. Since it requires a force to accelerate something, there must be a force directed toward the center of the circle.

• Many of the planets move in orbits about the sun that are approximately circular and the motion of the planets is close to uniform circular motion. Many of the satellites moving about the earth are also in approximately circular orbits and their motion is close to uniform circular motion. In these cases the force that produces the inward acceleration is gravity.

• For a circular orbit, the velocity is tangential to the circle, and is called tangential velocity. (The tangent to the circle is perpendicular to the radius of the circle.) Since the force is perpendicular to the motion, i.e. the velocity, it doesn’t do any work on the object and its kinetic energy does not change, i.e. its speed is constant.

• The tangential velocity keeps the planet from "falling" into the sun. Even through gravity accelerates it inward, the acceleration is just enough to "bend" the motion around into a circle, not enough to pull it closer. Another way of thinking of it is that the planets are in free fall toward the sun, but just keep missing it because of their large tangential velocities. Similarly satellites in orbit about the earth are in free fall toward the earth, but just keep missing because of their tangential velocity.

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