Question

Part 2 of 2 (b) Write a model for the angular displacement of the pendulum after ( t ) seconds. (Hint: Be sure to convert the initial position to radians.) The model for the angular displacement of the pendulum is __________.

a) ( θ(t) = θ_0 sin(t) )
b) ( θ(t) = θ_0 cos(t) )
c) ( θ(t) = θ_0 tan(t) )
d) ( θ(t) = θ_0 sinleft(t2right) )

Answer 1

The model for the angular displacement of a pendulum in simple harmonic motion is given by θ(t) = θ_0 cos(ωt + φ), assuming the pendulum has a small angular displacement and starts from rest.

Step-by-step explanation:

The question seeks a model for the angular displacement of a pendulum after t seconds. In simple harmonic motion (SHM), assuming a small angular displacement, the motion of a pendulum can be modeled using the function of sine or cosine because these functions represent periodic motion. Given that the motion starts with the initial condition converted to radians, the motion can be modeled using a cosine function if the initial angular displacement is at its maximum and starts decreasing, or a sine function if it starts at zero and increases.

Therefore, the correct model for the angular displacement of the pendulum would be:
θ(t) = θ0 cos(ωt + φ)
where θ0 is the initial angular displacement in radians, ω is the angular frequency, and φ is the phase shift (which is zero if the pendulum starts at its maximum displacement).