SPRING 2010
ME 107 MECHANICS & OPTICS IN ANTIQUITY
Homework 2, Due Mon Feb 15, 2010
Reading assignment From the class notes on Ancient Statics, read the portions of Aristotle’s Mechanical Problems as you need for the problems below. Also read the excerpts on The Five Simple Machines from the notes.
1. For the questions below, limit your answers to not more than 3 sentences for each question. Make sure that your sentences are complete, and not fragments. It is required that you attach a diagram for each of these questions.
1.1 Summarize Aristotle’s Mechanical Problem 13. What is the mechanics principle underlying Aristotle’s explanation? Draw a capstan. Draw a winch.
1.2 Summarize Aristotle’s Mechanical Problem 14. What is the mechanics principle underlying Aristotle’s explanation?
1.3 Summarize Aristotle’s Mechanical Problem 16. What is the mechanics principle underlying Aristotle’s explanation?
2. Summarize the principle of operation for each of the five simple machines described in Pappus’ work (but attributed to Hero.) Make sure that you attach a diagram to demonstrate each machine.
3. Briefly discuss the similarities and differences between the Hero and Pappus approaches to solving the inclined plane problem.
4. In this problem we wish to explore the implications of the Hero and Pappus calculations of the inclined plane. We will need the following information. One talent of weight is equivalent to 26 kg. One worker can exert a pushing force of 40 kg. The angle of the inclined plane is 30º or π/6.
It is observed that a force of 3 workers is needed to push a 19-talent body on a horizontal plane, but a force of 9-10 workers to push the same body on the inclined plane. You do not need to calculate these numbers.
According to Hero, what force is required to push slowly the body up the plane? According to Pappus?
5. In this problem, we wish to determine the extent to which it is possible to determine the difference in the amount of water displaced when an adulterated crown of gold/silver is compared to an equal weight of pure gold being immersed separately.
The basic idea is that a body immersed in a liquid displaces an amount of liquid equal to the body’s volume.
Let’s say we have one crown weighing exactly 1 kg. It is made of gold and silver in equal weights, i.e 0.5 kg gold and 0.5 kg silver. We also have a chunk of pure gold, also weighing 1 kg.
The density of gold is 19.3 g/cm3. For silver it is 10.5 g/cm3.
The water container is a cylindrical tub, which must be wide enough for the crown to be immersed in it. Since the crown should fit a human head, it is about 18 cm in diameter. Therefore, let’s take the tub radius as being 9 cm.
How much does the surface of the water rise when the 1-kg crown is fully immersed in it?
How much does the surface rise when the 1-kg chunk of pure gold is immersed in it?
Do you think that this is a likely method that Archimedes may have used to determine whether the king’s crown was all gold or adulterated? Discuss.
3. According to Aristotle, “If a force moves a body in a given time over a given distance, the same force will move half this mass through the same distance in half the time.”
First, use the Aristotelian equation of motion (for unnatural motion),
F = m V, with velocity V = S / T, i.e.
for a body of mass m moved under the action of a force F and acquiring a velocity V. In this way, show that the quote above is a direct prediction of the Aristotelian mechanics.
Next, use the modern view of mechanics (based on Newton’s 2nd law F = m a)
a = F / m, and that S = (1/2) a T2, i.e.
and compare with the Aristotelian predictions.
4. This exercise was prepared by Isaac Newton for his students in Cambridge university in 1673-1675. At that time Newton had already completed his seminal work on colors, but had not quite finished his theory of gravitation and motion. These problems were assembled into a book around 1683, and published in 1707.
The reason we discuss this problem is to show how vague was the idea of “effort”, “work”, “force” or “effect” at that time.
Solve the following problem (Problem 6 in Newton’s book):
If a scribe in 8 days can write 15 folio pages, how many scribes of the same sort will be necessary to write 405 folio pages in 9 days?
5. This problem is also from the same collection of exercise by Newton (problem 7 in Newton’s book):
Three workers can accomplish a certain job in a certain time, namely worker A once in three weeks, worker B three times in eight weeks, and worker C five times in twelve weeks.
In how much time can they complete the job by working together?
[Hint: This is my hint, not Newton’s: If worker B accomplishes the job three times in eight weeks, what fraction of the job will B accomplish in time x?]
6. A lever has length 6 m. The fulcrum point is 1.5 m from the left end. If a weight of 1 stone is suspended from the left end, what weight is required at the right end for equilibrium?
Neglect the weight of the lever.
7. Solve the same problem, but now the weight of the lever is ½ stone. What other assumption(s) do you need to make in order to solve the problem?