Potential Energy and Conservative Forces?

Question

A skateboard track has the form of a circular arc with a 4.00 m radius, extending to an angle of 90 degrees relative to the vertical on either side of the lowest point (semicircle). A 57.0-kg skateboarder starts from rest at the top of the circular arc. What is the normal force exerted on the skateboarder at the bottom of the circular arc?

Thanks!

Answer 1

The normal force exerted on the skateboard at th bottom of the circular arc is:

Normal force = centripedal force - gravitational force

Usually, the normal force is equal to the weight of the person (due to gravity), but here the skater is accelerating around a circular arc that exerts a centripedal force upwards toward the center of the circle.

N = m v²/r - mg = m(v²/r-g)

Here v is the velocity of the skater when he reaches the bottom. This velocity is attained when he jumps off the top of the circular arc and rides down on the skateboard. This motion is called harmonic motion, like the swing of a pendulum,. Therefore, the velocity is given by

v = ωr = sqrt(g/r)r = sqrt(rg)

Plugging this back into the equation for the normal force:

N = m(v²/r-g) = m(rg/r -g) = 0

Hence, the skater experiences NO forces at all at the bottom of the circular arc!!! He'll feel weightness!