PHY 3221 Classical Mechanics, Homework #9, due 8:30 am, 04/09/12
1 Consider the surface of revolution when a function, y = y(x), defined between x = x1 and x = x2 ,
is revolved around the x-axis.
a) Write down an (approximate) expression for the length of the curve, between x = x0 and x =
x0 + ∆x, that is valid when ∆x is sufficiently small, and any x0 such that x1 ≤ x0 ≤ x2 .
b) Use this to write down a corresponding expression for the area of the surface of revolution between
x = x0 and x = x0 + ∆x and so finally, for the area of the surface of revolution between x = x1
and x = x2 .
c) Show that the surface with extremal area arises when y is given by the cosh function (There should
be two undetermined constants which have to be solved for in terms of (x1 , y1 ) and (x2 , y2 )).
d) Show how one could solve for the two undetermined constants, and write out equations which
would allow you to solve for them separately (i.e., do not leave them as coupled equations, but
you may substitute from one equation into the other). You do not need to solve these equations.
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2 Two observers agree to discuss a number of Physics problems in relation to four particular events:
A, B, C, D. One observer is stationary, and he labels the (x, c ∗ t) coordinates of these events
as follows (in meters):
A (600, 0),
B (1000, 0),
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C (1666 , 0), and
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3
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D (1666 , 1333 ).
The other observer is moving with respect to the first with velocity vx = 0.8c and she sets her
clock to zero as she passes the first observer at O.
a) Draw a diagram, showing all these events, indicating the temporal and spatial relationship between
them.
b) What are the coordinates of the event D in a system moving with the second observer?
c) What is the velocity, measured in the moving coordinates, for a particle which moves from A to
D?
d) The line CD represents a particle not moving with respect to the x-axis. How far does the moving
observer say the particle is away from O at the instant that the particle passes through D?
e) A charged particle, of charge Q1 , moves along the straight line passing through A and D. Meanwhile, the moving observer carries another charged particle of charge Q2 . What is the electromagnetic force exerted on the second particle by the first when the second particle is at point
800m from the stationary observer?
f) An uncharged particle moves from B to D. Find its velocity with respect to the stationary
observer, and then use the appropriate velocity addition formula to find the velocity of this
particle in the moving system.
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3 A stationary observer observes two events, one, E1 , at (x1 , ct1 ) = (1000, 1000) and one, E2 ,
at (x2 , ct2 ) = (4000, 3000), while a moving observer observes a third event, E3 , at (x′3 , ct′3 ) =
(2460, −2040). All distances are in meters. The moving observer is moving at 12/13 the speed of
light.
a) Find the coordinates of events E1 and E2 in the moving frame and the coordinates of E3 in the
stationary frame.
b) The stationary observer uses light signals to determine when each event occurred. According to
him, which of the three events occurred earliest, and which occurred latest?
c) The moving observer also uses light signals to determine when each event occurred. According
to her, which of the three events occurred earliest, and which occurred latest?
d) The two observers have different impressions of the event ordering. Draw a diagram which clearly
shows what they each report, and why they can be correct.
e) Is event E3 in the future or the past of each of E1 and E2 . Explain your answer.
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