Early skeptics of the idea of a rotating Earth said that the fast spin of Earth would throw people at the equator into space. The radius of Earth is about 6400 km. Show why this objection is wrong by determining the following information.
(a) Calculate the speed of a 94-kg person at the equator.
(b) Calculate the force needed to accelerate the person in the circle.
(c) Calculate the weight of the person.
(d) Calculate the normal force of Earth on the person, that is, the person's apparent weight.
To answer the questions, we can use some basic physics principles. Let's go step by step:
(a) The first step is to calculate the speed of a 94 kg person at the equator. We can use the formula for the speed of an object moving in a circle:
v = ω * r,
where v is the velocity, ω is the angular velocity, and r is the radius of the circle (in this case, the radius of the Earth).
The angular velocity, ω, can be calculated using the formula:
ω = 2π * f,
where f is the frequency, or the number of revolutions per second.
The Earth completes one revolution in approximately 24 hours, which can be converted to seconds as:
24 hours = 24 * 60 * 60 seconds.
So, the frequency, f, of the Earth's rotation is:
f = 1 / (24 * 60 * 60 seconds).
Substituting the values into the angular velocity formula:
ω = 2π * (1 / (24 * 60 * 60)).
Now we can calculate the speed using the velocity formula:
v = ω * r.
Substituting the values of ω and the radius of the Earth (6400 km = 6400 * 1000 m):
v = (2π * (1 / (24 * 60 * 60))) * (6400 * 1000).
Calculating this will give us the speed of the person at the equator.
(b) To calculate the force needed to accelerate the person in the circle, we can use Newton's second law of motion, which states:
F = m * a,
where F is the force, m is the mass of the person (94 kg in this case), and a is the acceleration.
In this case, the acceleration is the centripetal acceleration, given by:
a = v^2 / r.
Substituting the values of v and r:
a = (v^2) / (6400 * 1000).
Now we can calculate the force using Newton's second law.
(c) To calculate the weight of the person, we can use the formula:
weight = m * g,
where m is the mass of the person (94 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Calculating this will give us the weight of the person.
(d) Lastly, the normal force of the Earth on the person, which is the person's apparent weight, can be calculated using Newton's second law, considering the person's weight as the downward force and the normal force as the upward force.
Thus, the normal force is equal to the weight of the person.
By following these steps and performing the calculations, you should be able to find the answers to all the questions.