Help with circular motion of zero gravity

[koopa347]

Homework Statement

How long would the day have to be in order that for a person standing at the equator of the Earth would see a bathroom scale that she is standing on read zero?
This is all the info that is given.

Homework Equations

Fc=m(v^2)/r

The Attempt at a Solution

I setup my force equations so the sum of f=Fc=Fn-Fg. I then found out the radius of the earth, and this is where i got stuck , i know that the radius of the Earth is 6377.34km, but how do i figure out v and m; if v has to be greater than Earth's current centripetal force?

 

[cepheid]

Welcome to PF

You're spinning so fast that gravity is just barely providing enough force to keep you in your circular "orbit." In other words, the centripetal force required for this orbit is equal to your weight (which is why you feel no normal force, which as you've correctly stated, is the difference between the two).

If the centripetal force is equal to the weight, and both are proportional to m, then m cancels from both sides of the equation (the mass of the object in the orbit doesn't matter). Hence you're finding the speed at which the centripetal acceleration equals g. You know r. You know g. You just have to solve for v.

 

[koopa347]

What do you mean by both are proportional to m? Could you type out an equation that shows this step? Do you just mean
(m)(9.8)=mv^2/r
so those m's cancel, so we know now that we're left with 9.8=v^2/r?

 

[cepheid]

What do you mean by both are proportional to m? Could you type out an equation that shows this step? Do you just mean
(m)(9.8)=mv^2/r
so those m's cancel, so we know now that we're left with 9.8=v^2/r?

Yes, that's exactly what I meant. To be proportional to "x" means to be equal to "x" multiplied by some factor.

 

[rwisz]

To solve this problem, we can use the equation for centripetal force, Fc=m(v^2)/r, where Fc is the centripetal force, m is the mass of the person, v is the velocity, and r is the radius of the Earth.

Since the person is standing at the equator, the radius of the circle they are moving in is equal to the radius of the Earth (6377.34km). We can also assume that the person's mass remains constant.

To find the velocity, we can use the fact that the person is in zero gravity. In zero gravity, the only force acting on the person is the centripetal force. This means that Fc=0, and we can rearrange the equation to solve for v.

0=m(v^2)/r
v^2=0
v=0

This means that the person is not moving at all, and therefore the day would have to be infinite in order for the person to see a bathroom scale reading of zero. This is because the Earth would have to stop rotating for the person to not experience any centripetal force.

However, if we assume that the person is still moving with the rotation of the Earth, we can calculate the velocity needed for the bathroom scale reading to be zero. This would require the centripetal force to be equal to the force of gravity, which is given by Fg=mg, where g is the acceleration due to gravity (9.8 m/s^2).

Setting Fc=Fg, we get:

m(v^2)/r=mg
v^2=gr
v=sqrt(gr)

Plugging in the values, we get:

v=sqrt(9.8*6377.34km)
v=7904.97 m/s

This means that the person would have to be moving at a speed of 7904.97 m/s in order for the bathroom scale reading to be zero. The day would then have to be approximately 86 minutes long (assuming the Earth's radius is 6377.34 km and the person is at the equator).