The figure shows a pendulum of length L = 1.4 m. Its bob (which effectively has

Question

The figure shows a pendulum of length L = 1.4 m. Its bob (which effectively has all the mass) has speed v_0 when the cord makes an angle thega_0 = 46 degree with the vertical, (a) What is the speed of the bob when it is in its lowest position if v_0 = 8.3 m/s? What is the least value that v_0 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

a) Using work energy theorem,

work done by gravity = change in KE

mg(L - Lcos@) = mv^2 /2 - mv0^2 /2

9.81 x 1.4 ( 1- cos46) = v^2 /2 - 8.3^2 /2

v = 8.79 m/s

b) again,

work done by gravity = change in KE

- mgLcos@ = 0 - mv0^2/2

v0 = sqrt(2gLcos@) = 4.37 m/s

c) at highest point,

T + mg = mv^2 /L

T = mv^2/L - mg = 0

for least v

v = sqrt(Lg) at highest point,

using W-E theorem,

- mg(L + Lcos@) = m(sqrt(Lg))^2 /2 - mv0^2 /2

v0 = sqrt(2gL(1 + cos@) + Lg )

v0 = 7.76 m/s

d) as theta increase, v0 decreases.

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