Suppose that a simple pendulum consists of a small 55.0 g bob at the end of a cord of negligible mass. Suppose that the angle between the cord and the vertical is given by

θ = (0.0800 rad) cos[(4.91 rad/s)t + ϕ].
(a) What is the pendulum's length?
units in m

(b) What is its maximum kinetic energy?
units in J

I got .41 m for L in part a and trying to figure out part B!!!

To find the length of the pendulum, we need to analyze the equation given for the angle θ. The general equation for the displacement of a simple harmonic motion (SHM) is given by:

θ = A * cos(ωt + ϕ)

Where:
θ = angle of displacement
A = amplitude of the oscillation
ω = angular frequency
t = time
ϕ = phase constant

Comparing this equation to the one given, we can determine the values of A and ω. In this case, A = 0.0800 rad and ω = 4.91 rad/s.

For a simple pendulum, the angular frequency can be related to the length (L) of the pendulum using the equation:

ω = √(g/L)

Where:
g = acceleration due to gravity (approximately 9.8 m/s²)

By substituting the values of ω and g into the equation, we can solve for L:

4.91 = √(9.8/L)

Squaring both sides of the equation, we get:

24.0841 = 9.8/L

Rearranging the equation and solving for L, we find:

L = 9.8 / 24.0841 ≈ 0.407 m

Therefore, the length of the pendulum is approximately 0.407 m.

Now, to find the maximum kinetic energy of the pendulum, we need to consider the relationship between kinetic energy (K) and angular displacement (θ) for a simple pendulum. The kinetic energy of a simple pendulum is given by the equation:

K = (1/2) * m * L² * ω² * sin²(θ)

Where:
m = mass of the bob (55.0 g or 0.055 kg)
L = length of the pendulum (0.407 m)
ω = angular frequency (4.91 rad/s)
θ = maximum angular displacement (A = 0.0800 rad)

Substituting the values into the equation, we have:

K = (1/2) * 0.055 * (0.407)² * (4.91)² * sin²(0.0800)

Evaluating this equation, we find:

K ≈ 0.089 J

Therefore, the maximum kinetic energy of the pendulum is approximately 0.089 J.