Derive a formula for the maximum speed vmax of a simple pendulum bob in terms of g, the length ℓ, and the maximum angle of swing θmax.

Express your answer in terms of the variables g, l, and θmax.

Elena's answer seems to be incorrect

v max = sqrt( 2gL( 1−cos(θmax) ) )

Well, neither do I 🤔

To derive the formula for the maximum speed of a simple pendulum bob, we can use the principles of conservation of energy.

The total mechanical energy of the pendulum bob is the sum of its potential energy and kinetic energy. At the highest point of its swing, the bob is momentarily at rest, which means the entire mechanical energy is potential energy. At the lowest point of its swing, the bob has reached its maximum speed, and thus the entire mechanical energy is kinetic energy.

Let's go through the steps to derive the formula:

1. Potential Energy (PE):
The potential energy of the bob at its highest point (θ = θmax) is given by:
PE = m * g * h
where m is the mass of the bob, g is the acceleration due to gravity, and h is the vertical height above the lowest point. In this case, the height can be approximated as ℓ - ℓ * cosθmax = ℓ(1 - cosθmax) since the pendulum's length (ℓ) remains constant throughout. Therefore:
PE = m * g * ℓ(1 - cosθmax)

2. Kinetic Energy (KE):
The kinetic energy of the bob at its lowest point (θ = 0) is given by:
KE = (1/2) * m * v^2
where v is the velocity of the bob at its lowest point.

Using the principle of conservation of energy, the potential energy at the highest point is equal to the kinetic energy at the lowest point:
PE = KE
m * g * ℓ(1 - cosθmax) = (1/2) * m * v^2

Simplifying the equation:
2 * g * ℓ(1 - cosθmax) = v^2

Taking the square root of both sides:
v = √(2 * g * ℓ(1 - cosθmax))

Therefore, the formula for the maximum speed (vmax) of a simple pendulum bob in terms of g, ℓ, and θmax is:
vmax = √(2 * g * ℓ(1 - cosθmax))

This equation relates the maximum speed of the pendulum bob to the acceleration due to gravity (g), the length of the pendulum (ℓ), and the maximum angle of swing (θmax).

how to dirive a furmua to find the perimiter of following figures

θ=θ(max) •sin(ωt+α)

dθ/dt =θ(max) •ω•sin(ωt+α)
(dθ/dt)(max) = θ(max)•ω= θ(max)√(g/L)
v(max) =(dθ/dt)(max) •L = L•θ(max)√(g/L)