Okay,
I have a question and i am completely stumped as to how to do it.
A wheel is rolling in a straight line along level ground. The coordinates of a point P(x(t), y(t)) on the perimeter are given by:
x(t) = 27t - 9sin(3t)
y(t) = 9- 9cos(3t)
How fast is the wheel moving?
What is the maximum speed of the the point P?
How far does the wheel move (along the straight line) during one revolution?
How far does the point P move during one revolution of the wheel?
Any help would be greatly appreciated
I have a question and i am completely stumped as to how to do it.
A wheel is rolling in a straight line along level ground. The coordinates of a point P(x(t), y(t)) on the perimeter are given by:
x(t) = 27t - 9sin(3t)
y(t) = 9- 9cos(3t)
How fast is the wheel moving?
What is the maximum speed of the the point P?
How far does the wheel move (along the straight line) during one revolution?
How far does the point P move during one revolution of the wheel?
Any help would be greatly appreciated
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(a) The speed of the wheel is the speed that it moves in the x-direction. The speed in the x-direction is |dx/dt|. By differentiating x, we see that:
dx/dt = -27cos(3t).
So the required speed is |-27cos(3t)| = 27|cos(3t)|,
(b) By the Pythagorean Theorem, the speed of point P is:
speed = √[(speed in the x direction)^2 + (speed in the y direction)^2]
= √[(dx/dt)^2 + (dy/dt)^2]
= √[729sin^2(3t) + 729cos^2(3t)], since dx/dt = -27cos(3t) and dy/dt = 27sin(3t)
= √{729[sin^2(3t) + cos^2(3t)]}, by factoring out 729
= √[729(1)], since sin^2(3t) + cos^2(3t) = 1
= √729
= 27 units/sec.Point P moves at a constant rate (as can be seen above), so the maximum speed is 27 units/sec.
(c) We want the horizontal distance between two cusps of this cycloid. The cusps occur when the y value is minimized, which occurs when:
9 - 9cos(3t) = 0, since 9 - 9cos(3t) ≥ 0
==> cos(3t) = -1
==> 3t = ±2πk, where k is an integer
==> t = ±2πk/3.
At this point, you can see that cusps occur every 2π/3 units, so this is the required distance.
(c) This is just the arc-length of the required path on an interval whose endpoints are the two successive cusps [for example, two cusps occur at 0 and 2π/3, so one interval is (0, 2π/3)].
I hope this helps!
dx/dt = -27cos(3t).
So the required speed is |-27cos(3t)| = 27|cos(3t)|,
(b) By the Pythagorean Theorem, the speed of point P is:
speed = √[(speed in the x direction)^2 + (speed in the y direction)^2]
= √[(dx/dt)^2 + (dy/dt)^2]
= √[729sin^2(3t) + 729cos^2(3t)], since dx/dt = -27cos(3t) and dy/dt = 27sin(3t)
= √{729[sin^2(3t) + cos^2(3t)]}, by factoring out 729
= √[729(1)], since sin^2(3t) + cos^2(3t) = 1
= √729
= 27 units/sec.Point P moves at a constant rate (as can be seen above), so the maximum speed is 27 units/sec.
(c) We want the horizontal distance between two cusps of this cycloid. The cusps occur when the y value is minimized, which occurs when:
9 - 9cos(3t) = 0, since 9 - 9cos(3t) ≥ 0
==> cos(3t) = -1
==> 3t = ±2πk, where k is an integer
==> t = ±2πk/3.
At this point, you can see that cusps occur every 2π/3 units, so this is the required distance.
(c) This is just the arc-length of the required path on an interval whose endpoints are the two successive cusps [for example, two cusps occur at 0 and 2π/3, so one interval is (0, 2π/3)].
I hope this helps!
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just go on wolframalpha and plug in the numbers