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Figure 6.11 Problem 6.25

Figure 6.11 Problem 6.25

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6.25 classic mechanics 

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.]
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.]
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
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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