7. A pendulum of mass 5.0 kg hangs in equilibrium. A frustrated student walks up to it and kicks the bob with a horizontal force of 30.0 applied over 0.30 seconds. What is the length of the pendulum if it has a period of 5.0 seconds? What is the maximum angle of displacement of the swinging pendulum?

Explanation:

For pendulum , time period is

T = 2π √ l / g

T² = 4π² l/ g

l = / g T² / 4π²

= 9.8 x 5² / 4π²

= 6.21 m

Angular impulse = Torque x time

= 30 x 6.21 x .3 = increase in angular momentum

30 x 6.21 x .3 = I ω ; I is moment of inertia and ω is angular velocity .

I ω = 55.89

I = 1/3 m l²

= 1/3 x 5 x 6.21²

= 64.27

Angular kinetic energy = 1/2 I ω²

= 1/2(I ω)²/ I

= .5x 55.89²/ 64.27

= 24.30 J

Due to it , the centre of mass of pendulum increases by height h so that

h = l/2 ( 1 - cos θ )

1/2 I ω² = mg l/2 ( 1 - cos θ)

24.3 = 5 x 9.8 x 3.1 ( 1 - cos θ)

( 1 - cos θ) = .16

cos θ = .84

θ  = 33 degree.

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