Due Tuesday, November 21, 2:00 p. m.
1. Scaling. For each of the following questions, write a few lines of explanation, and, where appropriate, give a page reference to Two New Sciences.
(a) A big ship, which is a scaled-up version of a little ship, is less safe. Why?
(b) King Kong would never make it to the Empire State Building. Why?
(c) Why does fine sand take longer to settle out of water than coarse sand, even though they are the same types of rock?
(d) Why are there no giant insects (our size, for example)?
(e) Why are there no wild mice in the Arctic?
(f) How far, approximately, could you safely drop a horse? A cat? A grasshopper?
Explain what physical factors come into play in each case.
2. Newton succeeded where both Galileo and Kepler fell short, in formulating a theory that showed the heavenly motions and those here on earth obeyed the same laws.
(a) Why do you think Kepler didn't manage to do this? In other words, what extra insight, on top of what he already understood, would have given him the essential clue?
(b) Same as (a), but for Galileo.
3. (a) Taking the natural acceleration due to gravity to be 32 feet per second per second, how far does a dropped ball fall in the first second? (Find its average speed and multiply by time - no other formulas!) How far does it fall in the first half second? (Same routine).
(b) A California earthquake opens a forty-foot wide crack across a freeway, so that the road is level on both sides of the crack, but four feet lower on one side than the other. What speed would a car approaching from the higher side have to be moving to make it across the crack? Is 55 m.p.h. enough?
4. (a) Redraw for yourself the diagram on acceleration in uniform circular motion from the Newton's Laws lecture, and write down Newton's Second Law for the cannonball, taking the force to be the gravitational attraction of the Earth on the cannonball, to get a connection between the radius of its circular orbit and its speed v.
(Note: to find the Earth's gravitational pull on the cannonball, don't assume that the cannonball is necessarily very close to the Earth's surface.)
(b) Find the time T for the cannonball to make a complete orbit of its circular path of radius r at steady speed v.
(c) Prove Kepler's Third Law: that for several cannonballs in circular orbits around the earth with different radii r(1), r(2), etc., and consequently different times for one revolution T(1), T(2), etc., the ratio r3/T2 is the same for all of them.
5. Imagine a souped up car that can get from 0 to 60 m.p.h. in 6 seconds.
(a) Approximately, how does that acceleration compare with that for a falling ball? (Translate m.p.h. into meters per second, and round off your answer. I don't want to see more than two significant figures - don't copy down all your calculator produces!)
(b) From Newton's Second Law, when the car is accelerating, there is an external force acting on it. State clearly what that force is.
(c) The driver is accelerated by the push from the back of the seat. How does that force compare with the driver's weight?
(d) Suppose the road curves around with a 200 meter radius, but the road isn't banked - meaning that it stays flat. The car is still going at 60 m.p.h. How does the sideways force of the seatbelt on the driver compare with the driver's weight?
(e) All new cars now have accelerometers built in, which trigger the airbag deployment if deceleration passes a certain point. Explain, in terms of forces felt by the driver during too rapid deceleration, with and without the bag, why the airbag lessens damage. Can you guesstimate the deceleration that would cause the bag to deploy? (Hint: stopping distance from 60 m.p.h. for a good car is about 125 feet - and you don't want the airbag to deploy! Take this as 40 meters from 30 meters per sec, assume uniform deceleration, find the average speed and hence time taken to stop, etc.)