Einstein references

John Norton's "Einstein for Everyone" e-book:

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/index.html

Supplementary readings:

J. Schwartz and M. McGuinness, Einstein for Beginners. New York: Pantheon.

J. P. McEvoy and O. Zarate, Introducing Stephen Hawking. Totem.

J. P. McEvoy, Introducing Quantum Theory. Totem.

Pre-Final Problems

1. Consider a 7-m long pendulum (a Foucault pendulum, typically used to demonstrate the revolution of the Earth). Find its period on Earth, and on the Moon.

2. How long must a pendulum be such that its period is 0.5 seconds?

3. A spring-mass oscillator has a period of 0.8 seconds. What are the first 3 times where the oscillator will have its maximum speed?

4. What is the acceleration due to gravity at a point above the surface of the Earth equal to the radius of the Earth?

5. Mercury orbits the Sun once every 88 days. What is the size of its orbit (semi-major axis) and what is its average speed around the Sun (in km/hr)?

6. What is the angular velocity of a 33 1/3 album? If it takes 0.5 seconds to accelerate up to this speed, what is the angular acceleration and how many turns does it take to get up to this speed?

*7. Two masses (2 kg and 8 kg) are 5-m apart. Where is the center of mass located (as measured from the 2 kg mass)? Hint: consider one distance as x and the other distance as (5-x), then solve for x.

8. A meter stick is set up such that the fulcrum is located at the 25-cm mark. If a 100-g mass is at the 15-cm mark, what is the mass of the meter stick (assuming that it is at the 50-cm mark).

Simple Harmonic Motion

Practice Problems

Gravitation

1. Consider Jupiter, which has an orbital size (a) of 5 AU.

- How long does it take to orbit the Sun once?

- What exactly is 5 AU, in this problems?

2. If an asteroid were discovered that took 2.5 years to orbit the Sun once, how far away from the Sun must it be (on average)?

3. Consider the planet Mars, with mass 1/10 that of Earth and a radius 1/2 as much. What is its acceleration due to gravity? Also, if it is 1.8 AU from the Sun, how long does it take to orbit the Sun? Finally, what is its average speed (in km/sec) around the Sun? To do this, you'll need to convert AU to km first.

Torque and Center of Mass

4. On a see-saw, a 40-kg child is located 1.5-m away from the fulcrum. Where must a 75-kg adult be located, to balance with the child?

5. In the above problem, the 40-kg child now moves twice as far away from the fulcrum as she originally was. A third child (25-kg) wanders in. If the adult remains in the same location as above, where can the third child sit to balance the see-saw?

Rotation

6. If a cd can go from rest to 400 revolutions per minute in 4 seconds, find the following:

a. the final angular velocity (in radians/sec) - this is a conversion

b. the angular acceleration required to get to this angular velocity

c. the linear speed of a point at the edge of the cd (radius = 0.06 m)

13 - Angular Motion / Rotation

12 - Center of Mass and Torque

11 - Gravitation -- Kepler and Newton

As discussed in class, Kepler's laws were based on Tycho Brahe's massive amount of data. The laws can be summarized as follows:

1. Planetary orbits are elliptical with the Sun at one focus.

2. Planets sweep out equal areas in equal amounts of time.

3. The square of the period of orbit is proportional to the cube of the semi-major axis. If the units are years and AUs, this is an equality:

T^2 = a^3

e.g, Consider an asteroid with a 4 AU semi-major axis of orbit. How long does it take to orbit once?

Answer: 8 years

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Several decades after Kepler, Newton derived (from geometry) his law of universal gravitation. The derivation is prohibitive to discuss here, but it can be found in Principia (prop. 71). I don't recommend that you read it - many key points are omitted by Newton. In modern language:

F = G m1 m2 / r^2

That is, the force of gravitational attraction is equal to a constant (6.67 x 10^-11 Nm^2/kg^2) times the product of the masses, divided by the distance between the masses squared.

Setting this equal to the local force of gravity (weight) yields a simple expression for local gravitation:

g = G m(planet) / r^2

Finally, we saw in class how Newton's law of gravitation, along with the expression for centripetal acceleration (v^2 / r) can yield Kepler's third law. That is, Newton's law was powerful enough to predict anything known before it, as well as make predictions about the future.