Using conservation of energy with pendulums

[Mr Davis 97]

I am confused about how the principle of conservation of energy can be used to predict the velocity of a pendulums for any given height and angle. For example, say I use the equation U = -K (potential is equal to kinetic) to solve for the velocity at 15 degrees of a pendulum bob whose wire length is 1.2 m. The answer turns out to be 1.5 m/s. But this confuses me. The bob is not in free fall because there is the tension of the wire pulling on the pendulum bob. So what does this 1.5 m/s refer to? How does the equation "know" that the bob is swinging in an arc and not vertically falling?

 

[HallsofIvy]

The equation doesn't "know" and doesn't care! The kinetic energy is a scalar and has no direction. YOU have to use the properties of the pendulum to determine the direction of motion. IF the pendulum is "rigid", so has a constant length, then pendulum bob must move along a circle with radius equal to the constant length of the pendulum so the velocity vector must be tangent to that circle.

 

[robphy]

Be careful with the "reference level for potential energy". It is better to say U+K=constant, which could be zero, if the reference level is chosen correctly.

The tension in the rope does zero work (because the that force is always perpendicular to the displacement)... that's why you can use conservation of total energy.

 

[Mr Davis 97]

The equation doesn't "know" and doesn't care! The kinetic energy is a scalar and has no direction. YOU have to use the properties of the pendulum to determine the direction of motion. IF the pendulum is "rigid", so has a constant length, then pendulum bob must move along a circle with radius equal to the constant length of the pendulum so the velocity vector must be tangent to that circle.

So if I had a free falling object, would the velocity gained in the distance "fallen," .12 meters , be the same as the velocity of the pendulum, given that for the free falling object the velocity vector is pointed downwards while for the pendulum it is tangent to the arc? If the vectors are in completely different directions, how are the magnitudes of the velocity equal?

 

[Doc Al]

So if I had a free falling object, would the velocity gained in the distance "fallen," .12 meters , be the same as the velocity of the pendulum, given that for the free falling object the velocity vector is pointed downwards while for the pendulum it is tangent to the arc?

Yes. (At least for a simple pendulum.)

If the vectors are in completely different directions, how are the magnitudes of the velocity equal?

The wire changes the direction of the velocity. Since the tension is perpendicular to the velocity, it changes the direction but not the magnitude. (The tension force does no work on the pendulum bob, as robphy noted.)