The figure shows a pendulum of length L = 2.7 m. Its bob (which effectively has

Question

The figure shows a pendulum of length L = 2.7 m. Its bob (which effectively has all the mass) has speed vo when the cord makes an angle theta 0 = 390 with the vertical. (a) What is the speed of the bob when it is in its lowest position if v0 = 7.1 m/s? What is the least value that v0 can have if the pendulum Is to swing down and then up (b) to a horizontal position, and (c) to a vertical position with the cord remaining straight? (d) Do the answers to (b) and (c) increase, decrease, or remain the same if theta 0 is increased by a few degrees?

Explanation / Answer

apply KE i + PE i = KEf + PEf

0.5 mv1^2 + mgL(1-cos theta) = 0.5 mv2^2 + 0

so

v2^2 = sqrt(v1^2 + 2gL (1-cos 39)

v2^2 = sqrt(7.1^2 + (2*9.81 * 2.7*(1-cos 39)

v2^2 = 62.21

v2 = 7.88 m/s

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b. again

K+U)i = U2

0.5 mv1^2 + mgL (1-cos theta) = = mgL

v1^2 = sqrt( 2*(gL(1-c0s39)

v1^2 = sqrt(2*9.8 *2.7 * (1-cos 39)

v1 = 3.42 m/s

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C. Ki +Ui = Kf + Uf

T + mg = mv^2/L  

if T = 0, then v^2 = gL

so

0.5 mv1^2 + mgL(1-cos 39) = mg2L + 0.5 mgL

v1^2 = 4gL *(1-cos 39) = 2mgL + 0.5 mgL

v1^2 = 4gL-2gL(1-cos theta) + gL

v1^2 = 4*9.8* 2.7*(1-cos 39) + 9.8*2.7

V1 = 5.07 m/s

if theta slightly increases, all other factors decreaases

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