Rotation Problem: Earth lengths/Apparent Weights

[a Vortex]

Homework Statement

A friendly Brazilian has a mass of 150 kg. Being in Brazil, he rotates in a circle around the center of the Earth once per day, The radius of this circle (which is essentially the radius of the Earth) is 6.40 x 10^6 m.

I have found that the normal force is 1.4649 x 10^3.

Homework Equations

a_c = v²/r , centripetal acceleration
$\Sigma$F=ma
v = (2$\pi$r)/T , T is period of one revolution

The Attempt at a Solution

I only got to mg - F_n = ma_c
I know what the apparent weight is but I'm not sure about the normal force. If I can find this, then I can then find a_c, v, and then T to find about the length of day.

 

[ehild]

What is the question?

ehild

 

[a Vortex]

Sorry forgive me. The question is: How long would a day have to be for the Brazilian's apparent weight to be 1.46 x 10^3 N?

Btw, the answer should be about 6.16 x 10^4 sec or 17.1 hours.

 

[ehild]

The weight is equal to the force the man presses a horizontal support - scales, ground, chair ... So its is the same as the normal force.

ehild

 

[asteriatic]

Your calculations so far are correct. The normal force, Fn, is the force that the ground exerts on the person to balance out the force of gravity, mg. In this case, the centripetal force (mac) is also acting on the person, causing them to rotate around the Earth. Therefore, the normal force must also equal the sum of the person's weight (mg) and the centripetal force (mac).

To find the length of a day, you can use the equation T = 2πr/v, where T is the period of one revolution, r is the radius of the circle (in this case, the radius of the Earth), and v is the velocity. Since the person is rotating once per day, the period is 24 hours or 86400 seconds. You can use this value for T and solve for v, which will give you the velocity at which the person is rotating around the Earth.

Once you have the velocity, you can use the equation ac = v^2/r to find the centripetal acceleration. This will also give you the value for the normal force, which you have already calculated. From there, you can continue with your calculations to find the length of a day.

It is important to note that this calculation is assuming that the person is rotating on the surface of the Earth, which is not exactly true. The Earth's rotation is not a perfect circle and the person's rotation would also be affected by the Earth's tilt and other factors. However, this calculation provides a good estimate for the length of a day on Earth.