To find the maximum velocity of a pendulum, how do you derive the formula for the height?

Question

mgh = .5mv^2
if L = .66 m, and the pendulum is left at an angle of 12 degrees to the vertical, how do I DERIVE the height? What's the physics behind it?

Answer 1

Draw an arc of circle of radius 0.66 m, with one radius in vertical direction and the other at the left at angle of 12 degree from the vertical.

In the vertical radius mark a point which is equal to 0.66 cos 12 from the center of the arc of the circle.

( A horizontal line drawn from the highest point passes through this point)

The height through which the pendulum has raised is (0.66 - 066 cos 12.)

h = 0.66 (1-cos 12) = 0.014m.

Use this height to find the medium speed using, mgh = 0.5 m v^2.

Or simply v^2 = 2gh.

The physics behind this formula is the pendulum is raised to height of h from the equilibrium position.
At this point it has a potential energy of mgh.
As it falls it looses its P.E and gains K.E
At the equilibrium position all its P.E is converted to K.E and hence it has maximum speed at this position.

Also mgh = ½ mv^2.

Answer 2

Actually, deriving the height in this situation is more of a geometry problem. Imagine that you have a triangle - one side is the pendulum string and one side is the "vertical." (The other side is the ground, but that's unimportant right now.) The pendulum string forms the hypotenuse of the triangle, and the angle you have - 12 degrees - the between the hypotenuse the vertical, or adjacent, side. From our favorite politically incorrect acronym, SOH-CAH-TOA, we should be using the cosine function. The cosine of 12 degrees is equal to the length of the adjacent side over the hypotenuse. Solve for the length of the adjacent side.

Why do you want the adjacent length? Because this is the vertical height of the pendulum, as measured from the pivot point. If you are defining potential energy from some other point, this will at least give you a start.

Answer 3

Why not. Conservation of energy relates the height to the speed of the pendulum. The height can be related to the angular deviation and thus to the position along the arc of movement (a circle) Thus, you know the velocity at every point along the arc without having to consider acceleration (or force). Integrating the velocity function will give position against time, and thus, you may determine the period.

Answer 4

What Physics?? It's simple trig. If the length of the pendulum is .66m and it's at an angle of 12° to the verticle, then the end of the pendulum is 14 mm (.014 m) above its height when the pendulum is vertical.

Dewd.... If you can't figure out the geometry and use simple trig to calculate that, you are in *way* over your head and you need to drop back a bit and really *learn* the previous math subjects that you've taken. Not just learn how to get through the courses.


Doug

Answer 5

Split the velocity into components, and apply energy conservation to the vertical component
(vsin(12)^2)=gh