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 16.3 m with a 109-kg brass bob. It is set to swing with an amplitude of 3.55°.
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?
a) To find the period of the pendulum, we can use the formula for the period of a simple pendulum:
T = 2π√(L/g)
where T is the period, L is the length of the pendulum (16.3 m), and g is the acceleration due to gravity (9.81 m/s²). Plugging in the values:
T = 2π√(16.3 m / 9.81 m/s²) ≈ 2π√(1.662 s²) ≈ 2π * 1.288 s ≈ 8.087 s
So the period of the pendulum is approximately 8.087 seconds.
b) To find the maximum kinetic energy of the pendulum, we first need to find the potential energy when it is at its highest point (when the amplitude is 3.55°). We can use the following formula:
PE = mgh
where m is the mass of the bob (109 kg), g is the acceleration due to gravity (9.81 m/s²), and h is the height it swings.
We can find h by considering the triangle formed by the length of the pendulum and the arc it swings:
h = L - Lcosθ
where θ is the amplitude of the pendulum in radians. First, we need to convert 3.55° to radians:
(3.55°)*(π/180) ≈ 0.06194 radians
Now we can find h:
h = 16.3 m - 16.3 m*cos(0.06194) ≈ 0.0324 m
Now we can find the potential energy:
PE = (109 kg)*(9.81 m/s²)*(0.0324 m) ≈ 34.65 J
Since the pendulum's maximum kinetic energy matches its potential energy at the highest point, the maximum kinetic energy is 34.65 J.
c) To find the maximum speed of the pendulum, we can use the formula for kinetic energy:
KE = 0.5 * m * v²
where KE is the kinetic energy (34.65 J), m is the mass of the bob (109 kg), and v is the speed. Solving for v:
v = √(2 * KE / m) = √(2 * 34.65 J / 109 kg) ≈ 0.792 m/s
So the maximum speed of the pendulum is approximately 0.792 m/s.