The figure shows a pendulum of length L = 1.6 m. Its bob (which effectively has

Question

The figure shows a pendulum of length L = 1.6 m. Its bob (which effectively has all the mass) has speed v0 when the cord makes an angle ?0 = 35 with the vertical. (a) What is the speed of the bob when it is in its lowest position if v0 = 7.4 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 ?0 is increased by a few degrees?

The figure shows a pendulum of length L = 1.6 m. Its bob (which effectively has all the mass) has speed v0 when the cord makes an angle ?0 = 35½ with the vertical. (a) What is the speed of the bob when it is in its lowest position if v0 = 7.4 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 ?0 is increased by a few degrees?

Explanation / Answer

a) we have (K + U) 1 = K2

So 1/2*m*v1^2 + m*g*L*(1-cos(35)) = 1/2*m*v2^2

So v2 = sqrt(v1^2 + 2*g*L*(1 - cos(35)) = sqrt(7.4^2 + 2*9.8*1.6*(1 - cos(35)) = 6.478m/s

b)Now (K + U)1 = U2

so 1/2*m*v1^2 + m*g*L*(1-cos(35)) = m*g*L

so v1 = sqrt(2*(g*L*(1 - (1-cos(35))) = sqrt(2*(9.8*1.6*cos(35))) = 5.068m/s

c) Now (K + U)1 = K2 + U2

There is a K2 if the cord is to remain straight

We have T + m*g = m*v^2/L f T = 0 then v^2 = g*L

so 1/2*m*v1^2 + m*g*L*(1-cos(35)) = m*g*2*L + 1/2*m*g*L

so v1^2 = 4*g*L - 2*g*L*(1 - cos(35)) + g*L = 4*9.8*1.6 - 2*9.8*1.6*(1- cos(35)) + 9.8*1.6 = 72.72

So v1 = sqrt(72.72) = 8.52m/s

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