Understanding centripetal force

Doc Al

This is correct.

This is not correct.

Take the correct equation and solve for R.

jsmith613

sorry
R = mg - mv^2/r

Doc Al

Good. Now how would you answer the question?

jsmith613

as speed increases, Reaction force decreases so apparent weight decreases

jsmith613

This matches the answer
could we now go through Q2 (with the change of "space habitat" to "satellite")

Doc Al

Good.
Please state the problem you are asked to solve.

jsmith613

Q:As the rotational speed of a satellite increases what happens to the apparent weight of the people inside

The force, mg, of the space station acts towards the earth

F = ma
F = mv2/r
BUT v = omega (w) * r
SO

F = m*w2*r

F = (GMm)/r2

(GMm)/r2 = m*w2*r

If w (omega) increases then Force increases
this seems wrong because GMm/r2 should always be constant and therefore not affected by speed :S

this is where i get stuck

Doc Al

I assume they mean orbital speed, not that the satellite is rotating about it's center of mass.

Note that the satellite is in free fall as it orbits. So what's the apparent weight of the people inside? No calculation needed.

PeterO

I don't like the presenters explanation very much. I look at the problem this way:

When the bucket is overhead, the water will tend to accelerate down with acceleration g, just as it would if there was no bucket there at all.

In order to not have the water coming out of the bucket, you must accelerate the bucket down at g or higher.

You could pull the bucket down with great acceleration, but it will soon hit the ground.

If you rotate the bucket fast enough, the bucket will have acceleration [centripetal acceleration towards the centre] given by V² / R. Provided that is at least equal to g, we have the bucket accelerating down with a sufficiently large acceleration for the water not to fall out of the bucket.

jsmith613

I copied the question exactly from a website. The website has the answer as an increase.
The only force acting on the satellite is mg (towards the earth) and we would use the F = GmM/r^2 to find this force (I presume)

I don't see why no calculation is needed for your interpretation of the Q

jsmith613

i would presume the logic is
if the bucket accelerates fast than the water downwards then the bucket will move before the water has a chance to get out.
My next question is why won't the bucket and the water accelerate at the same speed downward as they are effectively one object

Doc Al

So they are talking about the rotation of the satellite about its center of mass, not its orbiting around the earth.
What you need to consider is the force exerted by the rotating satellite (a space 'habitat') on the passenger. That's his apparent weight. The force of the earth on the satellite is not relevant to this question.

Imagine the 'satellite' as a rotating cylinder and a person is standing on the inner surface of that cylinder. As the rotational speed increases, what happens to the force with which he is pressed against the 'floor'?

FYI: 'mg' is only valid near the earth's surface.

Think of astronauts in the space shuttle once they are in orbit. What's their apparent weight?

jsmith613

ok so with a free body diagram,
mg = down
Reaction force = towards centre
he is accelerating towards the centre of the satellite (i presume this is up)
F = ma
ma = R - mg
mv^2/r = R - mg
mg = R + mv^2/r

so as the satellite rotates faster, his weight increases :)

I am confused though as to why on earth the effect of a faster rotation is completley different than on a satellite

well weight = mass * acceleration
I would presume their weight is dependant on how fast the rocket is accelerating
Once in orbit, they must have a net acceleration towards the earth (v^2/r) so the guys weight = mass * (v^2/r)

Doc Al

The only force on the person that you need to consider is the 'reaction force' from the 'floor' of the satellite.

On the earth you are standing on the outside of the rotating body; in the satellite, you're standing on the inside. (Look up 'artificial gravity'.)

So I guess you've never see pictures of the astronauts in orbit floating around in their ship?

jsmith613

of course, but they must weight something
Although, if your weight is given by the reaction force then floating astronouts have no weight

jsmith613

so was my answer wrong?

(please also see last post for the previous question)

PeterO

I don't like the way you used speed [I underlined it]!!!!
They do accelerate at the same rate [speed is not a term that relates well to acceleration]. The fact is that acceleration is greater than g, so the only way the water can accelerate at a rate greater than g, is if something applies an extra force to it [gravity is not enough] The thing that applies the extra force is the bucket - but that can only happen if the water remains in contact with the bucket - it doesn't come out.