Physics, Energy?

Question

A simple pendulum, 0.855 m long, is held at rest so that its amplitude will be 0.245 m. Neglecting friction and air resistance, use energy concepts to determine the maximum speed of the pendulum bob after release.
Energy concept==> Etotal = Epotential + Ekinetic
Et = mgh + 1/2mv^2

Answer 1

From the conservation of energy, TE = KE + PE + WE anywhere in the pendulum's swing. WE, work energy, = 0. This results because you've assumed no work done on or by the pendulum (e.g., air friction would cause work done by the pendulum to overcome that drag). So we are left with TE, total energy, = KE, kinetic, + PE, potential.

At the release point, TE = PE only because the pendulum is not moving and KE = 0. At the bottom of the swing, TE = KE because the height above ground state is h = 0 and we have PE = mgh = mg0 = 0.

Since TE = constant anywhere in that swing, we have TE(h) = mgh = 1/2 mv^2 = TE(0); so that v = sqrt(2gh); where h = .245 m if, by "amplitude" you mean vertical height of the tip of the pendulum above ground zero (h = 0), which is the bottom of the swing. If you mean x distance horizontally as your amplitude, you can solve for h = L (1 - sin(theta)); where L = .855 m long and theta = arccos(x/L). You can do the math.

[By convention, the amplitude of a pendulum, when drawn out as a sinusoidal curve on a graph is usually the x distance from the neutral point of the swing; not h, which is the vertical distance from the neutral point. An earlier answer assumed you meant h, I'm not that clear on that point; so I gave you solutions for both assumptions of what "amplitude" means.]

But the physics is the important thing here. Conservation of energy states energy is only converted from one form to another. It is not destroyed. That's why TE(h = .245) = TE(h = 0) is true and we can solve for v, the velocity at the bottom of the swing.

Had you added axle and/or air drag friction, there would have been work done against that friction. So KE = PE - WE and the v would have been less at the bottom of the swing because some of that PE was converted to work, not kinetic energy. Even so, the total energies at start and at the bottom would still have been equal.

One more point...the resulting velocity in your frictionless problem was independent of the mass of the pendulum m. And v^2 = 2 gh resulted. In case you didn't recognize this...it's the gravity version of the SUVAT equation v^2 = u^2 + 2 aS, with u = 0. Cool stuff, the conservation of energy.

Answer 2

sqrt(0.855^2 - 0.245^2) is height in metres that it is raised.