3.3 Newton’s Law of Universal Gravitation From Kepler to Newton Newton used Kepler’s Laws to derive a law describing the nature of the gravitational force that causes planets to move in their orbits. Newton concluded that this force was a mutual force. If one objects pulls on another, then that objects pulls back on the first object with an equal but opposite force. New showed that the force between two objects varied directly with their individual masses, and inversely with the square of the distance between them. The universal gravitational constant, G = 6.67 x 10-11 Nm2/kg2 was not determined by Newton. Mm Fg = G R2

3.3 Newton’s Law of Universal Gravitation Cavendish’s Experiment to Measure G In 1797, Henry Cavendish performed a very sensitive experiment which both confirmed Newton’s Gravitation Law and gave a value for the Gravitational constant. A 2.0 m long rod was suspended from the ceiling by a wire. Two small lead spheres were fixed to the end of the rod. Two large masses were placed near the lead spheres. Cavendish was able to measure the period of the rotation of the rod due to the gravitation force between the lead spheres and the large masses. Through many difficult measurements Cavendish was able to calculate the value for the universal gravitation, G = 6.67 x 10-11 Nm2/kg2 (in modern units).

3.3 Newton’s Law of Universal Gravitation Gravitation Field Strength of the Earth Remember that force of gravity near the Earth’s surface is calculated by: Fg = m * g Mm Law of Universal Gravitation: Fg = G R2 Since both give the Force of gravity, let them equal each other: Mm m * g = G R2 Cancel out the small, m from each side results in an equation to determine the acceleration due to gravity: M g = G R2 (where M = mass of Earth)

3.3 Newton’s Law of Universal Gravitation Weightlessness According to Newton’s law of universal gravitation, the force of gravity varies inversely as the square of the distance between the center of masses. As this graph shows the gravitational force never reaches zero unless R = infinity. F  0 as R ∞ True weightlessness can only be achieved at infinite distances between two objects. Apparent weightless can be experienced when zero force is felt from supporting structure s like your chair seat, the floor, or the Earth’s surface.

3.3 Newton’s Law of Universal Gravitation Example of Apparent Weightlessness 1. A Falling Elevator A person standing on a scale in an elevator. When the elevator is stationary the person will experience the true weight of the person as measure by the scale. The normal force (which the scale actually measures) equals the force of gravity. (Figure 3.3.3 (a))

3.3 Newton’s Law of Universal Gravitation Example of Apparent Weightlessness 1. A Falling Elevator (con’t) When the elevator accelerates downwards the scale will measure a smaller apparent weight. The elevator and the scale are falling away from the person and hence will measure a smaller normal force. (Figure 3.3.3 (b)) If the elevator cable breaks, the elevator will accelerate downwards at 9.80 m/s2 (a = g) and be in free-fall. The elevator and the scale are falling downwards at the same rate as the acceleration of gravity, hence no force will be placed on the scale and the person will achieved zero apparent weightlessness. (Figure 3.3.3 (c))

3.3 Newton’s Law of Universal Gravitation Example of Apparent Weightlessness 2. Orbiting Weightlessness Astronauts in an orbiting space vehicle feel weightlessness for the same reason as a person in a free-falling elevator. The astronauts are free-falling towards the Earth but have enough horizontal velocity to actually travel in orbit around the Earth. 3. Momentary Weightlessness When a person jumps off the ground or dives off a diving board, or travels over a hill in car, that person may experience apparent weightlessness for brief periods of time while the person is no longer in contact with the Earth.

3.3 Newton’s Law of Universal Gravitation Satellites in Circular Orbits – Orbital Velocity The orbital (tangential) speed that an object has to achieve to maintain an orbit at a given altitude can be determined through Newton’s Gravitational Law and centripetal force. Fc = Fg v2 Mm m * = G R R2 M v2 = G R √ M Where M = mass of the Earth or other large body, and R = distance of orbit to center of Earth. v = G R

3.3 Newton’s Law of Universal Gravitation Gravitational Potential Energy To move a spaceship to another position above a planet requires work to be done on the spaceship against the gravitational field of the plant. Since the gravitational field varies as the distance changes this is not a simple W = F x d situation. The amount of work that must be done to escape a planet’s gravitational field is equal to the area underneath the force vs. distance graph, between R = Re and R  ∞. By using some calculus it can be shown that the change in gravitational potential energy is equal to: Mm ΔEp = G R

3.3 Newton’s Law of Universal Gravitation Gravitational Potential Energy (con’t) In situations involving space travel to distances away from a plant it is much more convenient to make the gravitational potential energy equal to zero when R  ∞. When this happens the gravitational potential energy equations changes to: Mm Ep (at R = Re) = - G R This would make the gravitational potential energy at some point above a planet a negative value. This makes sense if you understand that at R = ∞, there would be no force and therefore the Ep = 0. At any distance less than R = ∞, you would have to have a smaller Ep which would yield a negative value.

3.3 Newton’s Law of Universal Gravitation Escape Velocity Escape velocity is the speed that a spaceship has to achieve to completely escape a planets gravitational field. To determine this speed the kinetic energy of the spaceship plus the gravitational potential energy of the spaceship at some altitude above the planet will be equal to zero. EK + Ep = 0 Mm ½ mv2 + - G = 0 R Mm ½ mv2 = G R 2GM v2 = R Minimum speed for an object to completely escape the gravitational field of a planet. √ 2GM vescape = R

3.3 Newton’s Law of Universal Gravitation In this section, you should understand how to solve the following key questions. Page #173 Practice Problems 3.3.1 #2 – 3 Page# 175 Quick Check #1 – 4 Page# 176 Quick Check #1 – 2 Page# 178 Quick Check #1 - 2 Page #179 – 180 3.3 Review Questions #5, 6, 9, 10, 12 To be sure you understand the concepts presented in the entire chapter on Universal Gravitation: Page #181 – 184 Chapter 3 Review Questions #3, 5, 6, 10, 12, 13, 15, 18 and 19