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A wagon descending an inclined plane AB and proceeding on a circular rail

To my understanding, normal reaction is a force that pushes in the opposite direction of gravity, in order to keep the object from falling, or phasing through the surface. Though, what happens when we enter a circular rail? In this case, the guide book is saying that the normal reaction is pushing alongside gravity. Then, shouldn't it fall down? I thought the normal reaction was supposed to be pushing upwards in this case. If not, then is the normal reaction always bound to the center of the circle?

I can provide the exercise with the answers, in case there's not much details in there.

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2 Answers 2

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To my understanding, normal reaction is a force that pushes in the opposite direction of gravity

Nooooo...

or phasing through the surface

This, and only this.

If not, then is the normal reaction always bound to the center of the circle?

No. Normal reaction force should only ever prevent penetration into the surface.

In this case, the guide book is saying that the normal reaction is pushing alongside gravity. Then, shouldn't it fall down?

It seems like you have not yet understood what Newton's Laws of Motion are meaning. The book is correct. You need to understand what forces are really doing, then you can understand what is happening here.

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  • $\begingroup$ I see. I think I got it now. "A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force," shows the presence of centripetal force. If centripetal f + weight + normal f = m*a, then the wagon won't fall off even if it reached the top of the rail. Then, the reason it didn't fall is because the acceleration was greatly balanced with the net force acting on it, am I right? $\endgroup$
    – Vocelia
    Commented May 26, 2023 at 8:15
  • $\begingroup$ Newton's $2^\text{nd}$ Law, usually written as $\sum\vec F=m\vec a$ states a relationship between cause and effect. The left side is causes, and the right side is effect. The causes must be real interactions, real forces. Weight and normal forces belong on the left. Centripetal acceleration is an effect that you see. It belongs on the right. You cannot have centripetal acceleration on the left, added to the real forces. $\endgroup$ Commented May 26, 2023 at 8:34
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Then, shouldn't it fall down?

It does "fall down" and in doing so continues on its circular trajectory.

The equation of motion at the top of its trajectory is $mg+N = m\frac{v^2}{r}$.

The gravitational force $mg$ alone is not able to provide enough force to cause a centripetal acceleration of $\frac{v^2}{r}$ and so it gets "help" from the normal force.

Note that as $v$ increases so must the normal force increase and as $v$ decreases so must the normal force decrease.
There is a speed $v'$ such that no normal force is needed, $mg = m\frac{v'^2}{r}$, and the gravitational force alone produces the centripetal acceleration.

If the body is travelling slower than speed $v'$ then either the body loses contact with the track or with a suitable arrangement of the rails the track provide a normal force $N'$ which is upwards so the equation of motion is now $mg-N = m\frac{v^2}{r}$.
Just imagine the situation with $v=0$ and then $mg-N=0 \Rightarrow N=mg$.

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