Lecture 06: Velocity, acceleration; Kepler’s second law.
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Lec 6 | Mit 18.02 Multivariable Calculus, Fall 2007
03
Feb
Angiegirl2112
February 3, 2010 at 12:38 am
i understand all of this and im only 12
rgh1986aka199
February 3, 2010 at 1:34 am
Is anybody solving the assignment sets? I need help with Assignment 2> Problem 2….
studentLifeProject
February 3, 2010 at 2:19 am
Why isn’t the arc length from 0 to 2pi of the cycloid just the circumference of the wheel?
michalchik
February 3, 2010 at 2:31 am
Think of it this way. Let’s say the wheel did not turn but was simply dragged. The curve describing the path of a point on the wheel would be equal to the displacement of the wheel. Similarly, if the wheel did not move but simply turned in place the arc length would be 2pi for each revolution. The cycloid includes both the revolution and the translation as part of its length.
studentLifeProject
February 3, 2010 at 3:16 am
Thanks! That makes sense. Happy new year’s
rgh1986aka199
February 3, 2010 at 4:00 am
Points on the wheel do not have the same speed. Point of contact with the ground has 0 speed, the center has speed v, while the topmost point has speed 2v. By the time the centers moves some distance, the top point would have moved a greater distance. However if we fixate on a point on the rim, it will have variable speeds and would have covered various lengths in different time periods. The total length covered by one such point is arc length and it is more the circumference.
brijeshagarwal1975
February 3, 2010 at 4:28 am
for cycloid the proof was given in terms of angle,later it was told that at radius =1 and speed=unity, angle may be replaces with time.thus r vector of cycloid was arrived at in terms of t.also using this r in terms of t, speed was calculated as sqrt(2-2cost).my question is when relation in terms of t was arrived at by using speed =unity,why it is coming as sqrt(2-2cost) ?
tharinduuuu
February 3, 2010 at 5:12 am
@brijeshagarwal1975 in the first instance the speed at which the wheel moves to the right was set equal to 1(which was also set equal to the radius giving the magnitudes of angle and time equal to unity). The speed calculated afterwards was the velocity of the moving point P on the wheel which obviously varies with time. hope this helps.
michalchik
February 3, 2010 at 5:49 am
the angular speed of the wheel was set to 1. That is it completed on revolution in 2pi units of time. The speed we calculated was the linear speed of a point on the wheel.
rgh1986aka199
February 3, 2010 at 5:55 am
If the speed were non-unity say w, the relation would have come out to be root(2-2cos wt)…no biggie