# Week 14 (Apr. 15 -19)

**Read:**Planetary motion (ASGv2 Chap. 26), Universal Gravitation (ASGv2 Chap. 27), Hypothesis and Natural Theology (ASGv2 Chap. 28).

**Key topics:**Kepler's laws of motion, Newton's universal law of gravitation, natural theology, scientific apologetics, and Newton's argument from design.

**PHY 201 lecture:**gravitation of extended bodies; integral calculus approach to computing the gravitational attraction of a mass toward a curved bar

**Quiz:**Covering material from week 11, 12, and 13.

Homework:

Homework:

- Phases of the moon (Ex. 26.1)
- Phases of Venus (Ex. 26.2)
- Force of gravity and Modified Kepler's law: Suppose that it was discovered that the planets obey a modified form of Kepler's law: (T_1/T_2) = (a_1/a_2)^(2), instead of Kepler's law, that (T_1/T_2) = (a_1/a_2)^(3/2). What would you infer about the force holding the planets in orbit from this? Would it still be a 1/r^2 force? (Hint: think about Corollary 6 of Theorem 4 in Book 1 of Newton's
*Principia*.).**Answer:**No, it would not be an inverse square force law. It would be an 1/r force law. How? By plugging T_1/T_2 = (R_1 / R_2)^2 into the formula (F_1/F_2) = (R_1/R_2)/(T_2/T_1)^2, one gets (F_1/F_2) = (R_2 / R_1). - PHY 201 Pulley problem: A 10 kg mass is suspended from a thin string. The string is wrapped a bunch of times around a solid disk that acts as a pulley. The disk/pulley has a mass of 2 kg and a diameter of 5 cm. The mass is released from rest and so it begins to descend due to its weight. As it does so, the string, which is wrapped around the disk, causes the disk to rotate (the string does not slip). (a) How much torque does the 10 kg mass exert on the disk? Be careful: it is only equal to the weight of the mass
*if*the 10 kg mass is*not*accelerating!) (b) What is the angular acceleration of the pulley as the 10 kg mass falls? Is it constant? (c) What is the angular velocity of the pulley? Is it constant, or time dependent? (d) The floor is 3 meters beneath the release point of the 10 kg mass. How long does it take to hit the floor? (e) What is the speed of the mass just before it hits the floor? Video Solution

**Let's put the following problems off until next week….**

- Acceleration and the force of gravity (Ex. 27.1)
- Meteor problem:
- Mass of Jupiter: Io, a small moon of Jupiter, has an orbital period of 1.77 days and an orbital radius of 4.22 x 10^5 km. From this data, determine the mass of Jupiter. (Video solution).
- PHY 201: Steel, silk and gravity (Ex. 27.3)
- PHY 201 challenge problem: A 10 kg thin metal bar of length 1 meter is curved into a semicircle having a radius of curvature of 3 meters. A small 5 kg mass is placed at the center of curvature. What is the gravitational force exerted by the curved bar on the small mass? (Hint: we worked out a similar problem in class on Thursday. You can use the solution from class; no need to go through the whole derivation. You just need to change the mass per length, lambda, and the angles involved in the integration.
**Answer**: 3.7 x 10^(-10) Newtons)

**Lab:**Centripetal Force Laboratory (Ex. 24.4). See the videos below!

**Four videos on planetary motion**

Chapter 26:

Chapter 26:

**Chapter 27:**Six videos explaining how Newton arrived at his universal law of gravitation.

**Chapter 28:**One video describing Newton's

*natural theology.*Newton believes that looking at nature points clearly to God and his attributes.