A Foucault pendulum is designed to demonstrate the effect of the Earth's rotation. A Foucault pendulum displayed in a museum is typically quite long, making the effect easier to see. Consider a Foucault pendulum of length 16 m with a 109-kg brass bob. It is set to swing with an amplitude of 3.1°.

(a) What is the period of the pendulum?


(b) What is the maximum kinetic energy of the pendulum?


(c) What is the maximum speed of the pendulum?

To answer these questions, we need to use some formulas related to pendulum motion.

(a) The period of a simple pendulum is given by the formula:

T = 2π * √(L / g)

Where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Using the given values, substitute them into the formula:

T = 2π * √(16 / 9.8) ≈ 6.28 * √(1.63) ≈ 6.28 * 1.28 ≈ 8.05 seconds

So, the period of the given pendulum is approximately 8.05 seconds.

(b) The maximum kinetic energy of a pendulum is given by the formula:

K = 0.5 * m * v^2

Where K is the kinetic energy, m is the mass of the pendulum bob, and v is the maximum speed of the pendulum.

To find the maximum kinetic energy, we need to find the maximum speed of the pendulum first.

(c) The maximum speed of a pendulum can be calculated using the formula:

v = √(2 * g * L * (1 - cosθ))

Where v is the maximum speed, g is the acceleration due to gravity, L is the length of the pendulum, and θ is the amplitude of the swing in radians.

First, convert the amplitude from degrees to radians:

θ = 3.1° * (π / 180°) ≈ 0.054 radians

Now, substitute the given values into the formula:

v = √(2 * 9.8 * 16 * (1 - cos(0.054))) ≈ √(313 * (1 - 0.998)) ≈ √(313 * 0.002) ≈ √0.626 ≈ 0.79 m/s

So, the maximum speed of the pendulum is approximately 0.79 m/s.

Now, substitute the values of m and v into the kinetic energy formula:

K = 0.5 * 109 * (0.79)^2 ≈ 0.5 * 109 * 0.6241 ≈ 34.06 J

So, the maximum kinetic energy of the pendulum is approximately 34.06 Joules.