Figure 6.11 Problem 6.25
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6.25
please provide a solution for the following
![Po
P
Figure 6.11
Problem 6.25
is, P, has parameter 0 < 0, < ). Show that the time for the cart to roll from P, to P is given by the
integral
cos e
de
а
1
time(P, → P) =,
Ele V cos 6, – cos®
and prove that this time is equal to a Ja/g. Since this is independent of the position of Po, the cart
takes the same time to roll from P, to P, whether P, is at 0, or anywhere between O and P, even
infinitesimally close to P. Explain qualitatively how this surprising result can possibly be true. [Hint:
To do the mathematics, you have to make some cunning changes of variables. One route is this: Write
0 = 1 - 2a and then use the relevant trig identities to replace the cosines of 0 by sines of a. Now
substitute sin a = u and do the remaining integral.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d945e56-c5e1-480a-9c03-390d785d9d79%2F83ee4524-c1c2-49c2-bf6d-b8ecc6e280b7%2Fi13uh2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Po
P
Figure 6.11
Problem 6.25
is, P, has parameter 0 < 0, < ). Show that the time for the cart to roll from P, to P is given by the
integral
cos e
de
а
1
time(P, → P) =,
Ele V cos 6, – cos®
and prove that this time is equal to a Ja/g. Since this is independent of the position of Po, the cart
takes the same time to roll from P, to P, whether P, is at 0, or anywhere between O and P, even
infinitesimally close to P. Explain qualitatively how this surprising result can possibly be true. [Hint:
To do the mathematics, you have to make some cunning changes of variables. One route is this: Write
0 = 1 - 2a and then use the relevant trig identities to replace the cosines of 0 by sines of a. Now
substitute sin a = u and do the remaining integral.]
Transcribed Image Text:6.25 *** Consider a single loop of the cycloid (6.26) with a fixed value of a, as shown in Figure 6.11.
A car is released from rest at a point P, anywhere on the track between O and the lowest point P (that
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