Equations

Write an equation for the centripetal force F in terms of reaction force R and weight W for someone rotating on the Earth.
Considering you need the radius to find the centripetal force, I do not really see how that can be achieved.

Reply 1

Stonebridge

Original post by Mlopez14

It looks like they don't want you to consider the radius of the Earth or its speed of rotation. Just those 2 forces, R and W. The question is, in this case, much simpler than you have imagined. :smile:

Just a simple resultant force question.

However , if there is more to this question than you have written here, please post again with the rest of it.

(edited 3 years ago)

Reply 2

Mlopez14

Original post by Stonebridge

Just a simple resultant force question.

However , if there is more to this question than you have written here, please post again with the rest of it.

Indeed the following is mentioned earlier:
Weighing scales don't measure your mass, just the reaction force R that the scales push back on you

I'd imagine that would help but I am still quite confused

Reply 3

Stonebridge

Original post by Mlopez14

Thanks. So it looks like it is as I thought.
When you stand on scales, there are 2 forces acting on you, W, your weight downwards and R, the force of the scales pushing up on you.

If the Earth was completely still - no rotation in this case - then you would be at rest on its surface.
If you are at rest then there is zero resultant force on you. (Newton's Law.)
This means that the resultant of those two forces is zero.
Taking down as positive direction it means W - R = 0
(or W = R)
In other words, the measurement on the scale (R) would be equal to your weight (W).

If the Earth is rotating, however, you are moving in a very large circle around with the Earth's surface at that point.
For you to move in a circle there must be an accelerating force towards the centre of that circle, the so called centripetal force.
As there are only 2 forces acting on you still, W and R, then these 2 forces combined must produce that centripetal force.
This force acts in the downwards direction towards the centre of the circle you are moving in.

In the previous paragraph, I wrote down the equation for the resultant downwards force when this force is zero. (When you were at rest)
This equation will always give me the resultant downwards force, but now it isn't zero. It's equal to the centripetal force (F) on you, because you are moving.

It is this that connects with the first bit of the question. As a result of the inclusion of the force F now, W is not equal to R.
In other words, the reaction of the scales on you pushing up, is not equal to your weight.
It is the reaction force of the scales on you that is actually measured by the scales as your 'weight'. (Through the spring system inside)
The conclusion is that now the scales are not (quite) measuring your weight. (Because R is not equal to W)
The difference is so small that it is insignificant. But it's a useful bit of physics. :smile:

BTW. At the north or south pole the scales would read correctly. Do you see why?

Reply 4

Mlopez14

Original post by Stonebridge

BTW. At the north or south pole the scales would read correctly. Do you see why?

Thanks for all of that. You are really nice
However, I am still confused about what you meant F not being equal to R
I would assume the scales would read correctly at poles since you are essentially at rest because of the poles being the axis of rotation of planet Earth?
Also if I understand correctly, the equation I am looking for is W - R = F (At rest), though I am still confused on what the missing force is when the person is moving?

(edited 3 years ago)

Reply 5

RogerOxon

Original post by Mlopez14

I would assume the scales would read correctly at poles since you are essentially at rest because of the poles being the axis of rotation of planet Earth?

Yes, for this level of analysis. The next level would consider the fact that the Earth is orbiting the Sun, so you're still accelerating in that plane. The axis of rotation of the Earth isn't perpendicular to the plane of its orbit.

Original post by Mlopez14

Also if I understand correctly, the equation I am looking for is W - R = F (At rest), though I am still confused on what the missing force is when the person is moving?

That's right. There is no missing force, as you are NOT at rest. You may think that you are relative to what you see, but it's all executing circular motion around the axis of rotation of the Earth - a net force is required for that (Newton's second law).

It's a poor question, IMO, because it only really makes sense at the equator (or on inclined land), as the reaction force is perpendicular to the ground, your weight is towards the centre of mass of the Earth, and your acceleration perpendicular to its axis of rotation. If we model the Earth as a perfect sphere, then, in most places, you'd need friction too. Consider standing at a point close to the North pole - your weight acts downwards, the reaction upwards, but you need friction to provide the force to accelerate you around the Earth's axis of rotation. Of course, the net forces are small, so almost always ignored.