A pendulum has length 0.20 meter. A bob is released from rest from a starting angle . Find an equation for the position at any time t and find the amplitude and period of the motion. [The acceleration due to gravity is 9.8 m s^-2.] A) amplitude: 0.15 m; period: 0.898 rad B) amplitude: 0.15 m; period: 0.449 rad C) amplitude: 0.3 m; period: 0.449 rad D) amplitude: 0.3 m; period: 0.898 rad

Question

Quizplus · Accepted Answer

The equation for the position of a simple pendulum is given by:

θ(t) = θ0cos(ωt)

where θ0 is the starting angle, ω is the angular frequency, and t is the time.

To find the amplitude of the motion, we need to find the maximum displacement of the pendulum from its equilibrium position. The maximum displacement occurs when the angle is equal to the amplitude, so we have:

θ0 = A

where A is the amplitude of the motion.

Using the small angle approximation sin(θ) ≈ θ and T = 2π√(L/g), the angular frequency and period of the pendulum are given by:

ω = √(g/L)
T = 2π√(L/g)

where g is the acceleration due to gravity and L is the length of the pendulum.

Substituting the given values, we have:

L = 0.20 m
g = 9.8 m/s^2
θ0 = 11° = 0.192 radians

So the amplitude is:

A = θ0 = 0.192 radians ≈ 0.3 m

And the angular frequency and period are:

ω = √(g/L) = √(9.8/0.20) ≈ 2.21 rad/s

T = 2π√(L/g) = 2π√(0.20/9.8) ≈ 0.898 s

Therefore, the best choice is D, with an amplitude of 0.3 m and a period of 0.898 rad.

A Pendulum Has Length 0