What are the equations describing the position, velocity and acceleration of a simple pendulum which is has a length of 1.5m and an initial displacement of 12⁰ from vertical? Assume that the displacement at the start of the motion is positive. Would this change if the mass of the pendulum was doubled?

The equations for the displacement, velocity, and acceleration of a simple pendulum are derived from the principles of motion and involve the pendulum’s initial displacement, length, and the acceleration due to gravity. The mass of the pendulum does not affect these equations.

Explanation:

The equations of motion for a simple pendulum are derived from the principles of motion and acceleration. The displacement (θ), velocity (v), and acceleration (a) of a pendulum are given by the following equations:

Displacement: θ(t) = θ₀ * cos(√(g/L) * t)
Velocity: v(t) = - θ₀ * √(g/L) * sin(√(g/L) * t)
Acceleration: a(t) = - θ₀ * (g/L) * cos(√(g/L) * t)

Here, θ₀ is the maximum angle, L is the length of the pendulum, g is the acceleration due to gravity, and t is time. For the given problem, θ₀ is 12° and L is 1.5m. It's also important to note that the mass of the pendulum does not affect these equations – this is known as the independence of mass, a property of simple pendulums.

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