Normal Forces and Ferris Wheels

Question

At the moment, I am reading an example problem regarding what was alluded to in the title. In this example problem, they say, "Based on experiences you may have had on a ferris wheel or driving over small hills on a roadway, you would expect to feel lighter at the top of the path. Similarly, you would expect to feel heavier at the bottom of the path."

Quite honestly, this is not what my intuition speaks to me: I would expect to have a greater normal force at the top of a hill, due to my inertia wanting to keep me at the bottom of the hill, thus the seat pushing up into me; for driving down a hill, I would expect my normal force to be less, because my inertia wants to keep me on top of the hill, therefore, the seat travels from beneath me.

I trust that the book is correct, but I would really like to know why I am wrong.

Thank you!

Edit:

Also, in the example problem, they analyze the forces acting on a person in a ferris wheel while at the bottom and top of it. For the bottom, $\Sigma F = N_{bottom} - mg = m \frac{v^2}{r}$; for the top, $\Sigma F = mg - N_{top} = m\frac{v^2}{r}$. Why does the normal force and weight change signs in each case?

Answer 1

I would expect to have a greater normal force at the top of a hill, due to my inertia wanting to keep me at the bottom of the hill

It seems like maybe the problem is that you're misinterpreting inertia. Remember the classic definition: a body at rest tends to stay at rest, and a body in motion tends to stay in motion. Your height from the ground is irrelevant to the question; all that matters is acceleration, speeding up or slowing down.

• At the top of the hill/ferris wheel, you're momentarily at rest (in the axis parallel to the ground anyway). When the ground moves away from under you, the perceived push-back is less, hence less normal force, resulting in the feeling of relative weightlessness.
• As you near the bottom, your body is in motion downwards, but now the ground is pushing back to slow that motion; in other words, the normal force increases, resulting in a heavy feeling.

I think a better example than the ferris wheel / hill would be an elevator. That's an experience that's more familiar to most people, and one where the effects of inertia are very easy to feel, and to recall.

Answer 2

Try riding a roller coaster to experience the effects mentioned here.

If you are near the surface of the earth, gravity 'wants' to accelerate you downward with an acceleration of 9.8 m/s². If you aren't accelerating downward it is because something is pushing (exerting a force) upward.

The effect here is not just because you are at the bottom or the top; if the Ferris wheel is not moving then you will have (to a very good approximation) the same force pushing up on you whether you are at the top or the bottom; it is motion along a curved path that produces the effect: when the Ferris wheel is turning you are accelerating downward when at the top, and accelerating upward when at the bottom. In both cases the magnitude of the acceleration is 1/2 mv². The difference in sign you ask about is because in one case the acceleration is downward (the same direction as force of gravity), in the other case it is upward (in the opposite direction to gravity).