Section 7.2 Using the Law of Universal Gravitation

1. Section 7.2 Using the Law of Universal Gravitation Objectives Solve orbital motion problems. Relate weightlessness to objects in free fall. Describe gravitational fields. Compare views on gravitation.

2. ORBITS OF PLANETS AND SATELLITES • Go over Cannonball example p. 179 • The curvature of the projectile would continue to just match the curvature of Earth, so that the cannonball would never get any closer or farther away from Earth’s curved surface. The cannonball therefore would be in orbit. This is how a satellite works. • Thus a cannonball or any object or satellite at or above this altitude could orbit Earth for a long time. • A satellite in an orbit that is always the same height above Earth move in uniform circular motion.

3. ORBITS OF PLANETS AND SATELLITES • ac = v2 / r • Fc = mac = mv2 / r • Combining the above equation with Newton’s Law of Universal Gravitation yields the following equation GmEm = mv2 r2 r

4. ORBITS OF PLANETS AND SATELLITES • Speed of a Satellite Orbiting Earth – is equal to the square root of the universal gravitation constant times the mass of Earth divided by the radius of the orbit. • v = √ (GmE) / r • Period of a Satellite Orbiting Earth – is equal to 2Π times the square root of the radius or the orbit cubed divided by the product of the universal gravitation constant and the mass of Earth. • T = 2Π √ r3 / (GmE)

5. ORBITS OF PLANETS AND SATELLITES • The equations for the speed and period of a satellite can be used for any object in orbit about another object. • The mass of the central body will replace mE in the equations and r will be the distance between the centers of the orbiting body and central body. • Since the acceleration of any mass must follow Newton’s 2nd Law (F = ma) more force is needed to launch a more massive satellite into orbit. Thus the mass of a satellite is limited to the capability of the rockets used to launch it.

6. ORBITS OF PLANETS AND SATELLITES • Do Example Problem 2 p. 181 • v = √ (GmE) / r • v = √ (6.67 * 10-11)(5.97 * 1024) / (6.38 * 106 + .225 * 106) • v = √ (3.98199 * 1014) / (6.605 * 106) • v = √ (6.029 * 107) • v = 7.765 * 103 m/s • T = 2Π √ r3 / (GmE) • T = 2Π √ (6.605 * 106)3 / (6.67 * 10-11)(5.97 * 1024) • T = 2(3.14) √ (2.8815 * 1020) / (3.98199 * 1014) • T = 6.28 √ (7.236 * 105) • T = 6.28 (850.647) • T = 5342.062 s (If convert = 89 min or 1.5 hours • Do Practice Problems p. 181 # 12-14

7. ACCELERATION DUE TO GRAVITY • F = GmEm / r2 = ma • a = GmE / r2 • a = g and r = rE so • g = GmE / rE2 • mE = grE2 / G • a =g (rE / r)2 • This last equation shows that as you move farther from Earth’s center (r becomes larger) the acceleration due to gravity is reduced according to the inverse square relationship.

8. ACCELERATION DUE TO GRAVITY • Weightlessness – an object’s apparent weight of zero that results when there are no contact forces pushing on the object. This is also called Zero g. • There is gravity in space. Gravity is what causes the shuttle and satellites to orbit Earth.

9. THE GRAVITATIONAL FIELD • Gravity acts over a distance. It acts on objects that are not touching. • Michael Faraday – invented the concept of the field to explain how a magnet attracts objects. Later the field concept was applied to gravity. • Gravitational Field – is equal to the universal gravitational constant times the object’s mass divided by the square of the distance from the object’s center. • g = Gm / r2

10. THE GRAVITATIONAL FIELD • The Gravitational Field can be measured by placing an object with a small mass in the gravitational field and measuring the force on it. Then the gravitational field is the force divided by a mass. It is measured in Newtons per kilogram (N/kg) which = m/s2. Thus we have • g = F / m • The strength of the field varies inversely with the square of the distance from the center of Earth. The gravitational field depends on Earth’s mass but not on the mass of the object experiencing it.

11. TWO KINDS OF MASS • Inertial Mass – is equal to the net force exerted on the object divided by the acceleration of the object. It is a measure of the object’s resistance to any type of force. • mInertial = FNet / a • Gravitational Mass – is equal to the distance between the objects squared times the gravitational force divided by the product of the universal gravitational constant times the mass of the other object. • mGravitational = r2FGravitational / Gm • Newton claimed that Inertial and Gravitational Mass are equal in magnitude. All experiments done since then show this is the case.

12. EINSTEIN’S THEORY OF GRAVITY • Einstein proposed that Gravity is not a force but rather an effect of space itself. According to Einstein mass changes the space around it. Mass causes space to be curved and other bodies are accelerated because of the way they follow this curved space. • Einstein’s General Theory of Relativity – makes many predictions about how massive objects affect one another. It predicts the deflection or bending of light by massive objects. In 1919, an eclipse of the sun proved Einstein’s theory. • Do 7.2 Section Review p. 185 # 15-21