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course Mth 277
12 am 9/26/12
120917Text assignment:
Assignment 08: q_a_10.1, Text Section 10.1, Problem Set 10.1, query_10.1 *
Assignment 09: q_a_10.2, Text Section 10.2, Problem Set 10.2, query_10.2
Upcoming Ch 9 Test:
We will spend some time in the next class summarizing Chapter 9 and discussing testing procedures. You should plan to take the Chapter 9 test by the end of next week.
`q001. The vector function `R(t) = cos(t) `i + sin(t) `j defines a circle of radius 1. For this function:
What is the vector `R ' (t)? (If t is clock time, this is the velocity vector)
-sin(t)i + cos(t) j.
What is the unit tangent vector `T(t)?
T(t) = R'(t)/ || R'(t) ||.
In this case || R'(t) || is just 1, so T(t) is R'(t) = -sin(t)i + cos(t) j.
What is the derivative `T ' (t)?
-cos(t) i - sin(t)j.
The curvature of the path is defined to be the magnitude of the rate at which the unit tangent vector changes with respect to distance moved along the curve. Explain why this would be the magnitude of the rate at which the unit tangent vector changes with respect to clock time, divided by the rate at which distance along the curve changes with respect to clock time.
This is simply another rate, and it's defined as d T(t)/ ds. This is a scalar quantity, thus the magnitude.
It follows that the curvature is || `T ' (t) || / || `R ' (t) ||. What therefore is the curvature of this curve?
|| -cos(t) i - sin(t)j || / || -sin(t)i + cos(t) j ||
This simply becomes 1.
`q002. What is the curvature of the path of the vector function `R (t) = cos(t^2) `i + sin(t^2) `j? Note that the curvature had better come out the same as for the function of question `q001, since the path is identical.
R'(t) = -2 t sin(t^2)i + 2t cos(t^2)j
T(t) = R'(t) / || R'(t) || = [-2 t sin(t^2)i + 2t cos(t^2)j]/2t. = -sin (t^2 )i + cos(t^2) j.
T'(t) = - 2tcos(^2)i - 2t sin(t^2)j.
|| T' (t) ||/ ||R ' (t) ||. This reduces to 1 as well.
`q003. What is the curvature of the path of the vector function `R(t) = A ( cos(t) `i + sin(t) `j)?
How is the curvature related to the radius of the circular path?
Reduces to 1/A. Thus, the greater the radius, the smaller the curvature.
`q004. Consider again the vector function `R (t) = cos(t^2) `i + sin(t^2) `j. What is the vector function `R '' (t), which if t represents clock time is the acceleration vector?
[-4t^2 cos(t^2) - 2sin(t^2) ] i + [- 4t^2 sin(t^2) + 2cos(t^2) ] j.
As t increases, what happens to the magnitude of `R '' ( t)?
It increases.
As t increases, what happens to the direction of `R '' (t) relative to the direction of motion?
It goes inward.
Are the magnitudes of the centripetal and tangential accelerations in constant proportion?
AT = -2sin(t^2) i + 2 cos(t^2)j.
AN = -4t^2 cos(t^2) i - 4t^2 sin(t^2)
Yes, they will always be proportional.
How would it feel to travel this path?
You would definitely pull some Gs.
`q005. Suppose two points, which we will respectively refer to as point 1 and point 2, travel paths described by the vector functions
`R_1(t) = x_1(t) `i + y_1(t) `j + z_1(t) `k
and
`R_2(t) = x_2(t) `i + y_2(t) `j + z_2(t) `k
What is the function that describes each of the following quantities?
The distance at clock time t between the two points.
|| R_2(t) - R_1(t) ||
The rate at which the distance between the two points is changing.
V= ( R_2(t) - R_1(t) ) '
||V||= || ( R_2(t) - R_1(t) ) ' ||
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The expression would be
|| ( R_2(t) - R_1(t) ) || '
which is different from
|| ( R_2(t) - R_1(t) ) ' ||
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The direction from point 1 to point 2.
V/ ||V||
( R_2(t) - R_1(t) ) '/ || ( R_2(t) - R_1(t) ) ' ||
The rate at which the direction is changing.
( V / ||V|| )'
[ ( R_2(t) - R_1(t) ) '/ || ( R_2(t) - R_1(t) ) ' || ] '
The magnitude of the rate at which the direction is changing.
The magnitude of the above vector.
|| [ ( R_2(t) - R_1(t) ) '/ || ( R_2(t) - R_1(t) ) ' || ] ' ||
`q006. A Japanese mathematician recently published, without any fanfare, four paper totaling about 500 pages purporting to prove the abc conjecture. He has invented a new system connecting geometry and arithmetic in a completely novel way. So far nobody knows what he's talking about but the best mathematicians in the world hope to get it figured out within the next few years. The guy is really good with a history of being really innovative, and he's being taken very seriously. That event and the recent county fair were on my mind when I made this problem up. Everybody should be able to start the problem, and some will be able to sketch out the procedures for completely solving it (the complete solution has enough computational intensity to justify the use of a computer algebra system to actually implement the general solution).
Sketch out what you would do to answer these questions, up to the point where the solutions become calculationally intense (in the sense that they are either too time-consuming to do by hand, or you either suspect that closed-form solutions don't exist or don't know how to do them). Suppose point 1 describes a circular path of radius 10 meters centered at the origin, about which it moves with constant angular velocity 1 radian / second, while point 2 describes a circular path of radius 10 meters centered at the point (30 meters, 0, 0) about which it moves with constant angular velocity 1.1 radians / second.
At point 2 is an omnidirectional speaker and at point 1 an omnidirectional microphone which feeds its signal into a computer. The speaker emits a high-pitched sound at 4000 Hz.
The Doppler shift is a little more complex than the approximation we're going to use here, but when the relative velocity of source and receiver is small compared to the speed of sound the approximation is good. The approximation is simply this: the percent different between the emitted and the perceived frequency is equal to the rate of change of the distance of separation as a percent of the speed of sound (this calculation includes the sign of the rate of change, which when positive increases and when negative decreases the perceived frequency). Take the speed of sound to be 340 m/s.
What are the time intervals during which the frequency of the perceived sound exceeds 4100 Hz?
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Point 1 traced by:
x = 10 cos(t)
y = 10 sin(t).
Point 2 traced by:
x= 10cos(t) + 30.
y = 10sin(t).
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These would be good if the angular velocity of both was 1 radian / second, but the angular velocity of the second is 1.1 radian / second. Among other things, that and the irrationality of pi imply that the vector functoin won't repeat itself periodically.
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4100 is 102.5% of 4000.
Distance between the two points as vector= P2 - P1.
The rate of change of the distance of separation as a percent of the speed of sound:
|| (P2 - P1)' ||/ 340 * 100.
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|| (P2 - P1) || ' / 340 * 100
|| (P2 - P1)' || may well be different from || (P2 - P1) || '
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We are looking for the time intervals that the magnitude of the derivative of the vector between the two points (the rate of change of the distance between p2 and p1) exceeds 102.5 percent the speed of sound.
I'm a little stuck on the rest of these problems, but I may work on them and submit them again at a later date. Not sure though.
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If you were placed at position 1 and blindfolded, and if your orientation in the plane was held constant so that you were always facing in the direction of the `i vector, you would perceive not only increasing and decreasing in frequencies (which you would whether or not your orientation remained constant), and you would also perceive the sound as coming from a constantly changing direction.
At what clock times would this direction appear for an instant to be unchanging?
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What is the most rapid rate of change of direction that could be perceived?
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If the plane of the circle around point 2 is changed, without changing the center of the circle, so that its normal vector becomes `i + 2 `j - `k, what will be the answers to these questions? From this point on you just want to sketch out a strategy with as many specifics as practical. Ideally you will sketch out a strategy that can be easily implemented by computer.
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What if both paths are followed but with a constant acceleration of 0.01 radian / second^2?
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For any of the models above, the vector from point 1 to point 2 is a vector function of time. If this vector function itself describes a path in space, what would be the characteristics of this path?
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What would be the maximum curvature of the path?
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&#Good responses. See my notes and let me know if you have questions. &#