Question

Suppose that a simple pendulum consists of a small 60.0g bob at the end of a cord of negligible mass. If the angle θ between the cord and the vertical is given by θ=(0.0800rad)cos[(4.43rad/s)t+ϕ], what are (a) the pendulum's length and (b) its maximum kinetic energy?

Answer 1

(a) The length of the pendulum is 1.15 meters. (b) The maximum kinetic energy is 297 Joules when the bob is at its maximum displacement.

Explaination:

(a) The length of the pendulum can be found using the relation between the angle and the length, which is given by the formula:

θ = (g/4π²) * sin(θ) * L

where g is the acceleration due to gravity, θ is the angle, and L is the length.

Substituting the given values in this formula, we get:

0.0800 = (9.8 m/s² / 6.28 rad/s) * sin(θ) * L

L = (0.0800 rad) / (sin(θ)) * (6.28 rad/s) / (9.8 m/s²) = 1.15 m

(b) The maximum kinetic energy of the pendulum occurs when it is at its maximum displacement from its equilibrium position, which is when sin(θ) = 1. Using this value in the expression for kinetic energy, we get:

K = (1/2) * m * v²

where m is the mass of the bob and v is its velocity. Since v = -dθ/dt, we have:

K = (1/2) * 60 g * [-(4.43 rad/s)sin(ϕ)]² = 297 g·m/s² * sin²(ϕ) Joules (J)

The maximum kinetic energy occurs when sin(ϕ) = 1, which gives us a maximum kinetic energy of 297 Joules (J).