7. An early objection to the idea that the earth is spinning on its axis was that the earth would turn so fast at the equator that people would be thrown into space.

Show the error in this logic by calculating the centripetal force needed to hold a 100. kg person in place at the equator. The radius of the earth is about 6400 km. Compare this force with the force of gravity of the 100. kg person.

So I know r=6,400,000 m and m=100. kg. I think F(centripetal)=(m*v)/r. How do I solve this?

3.4N

To solve for the centripetal force, you will need to find the velocity of the person at the equator. The velocity can be calculated using the formula: v = rω, where r is the radius of the Earth and ω is the angular velocity.

1. Convert the radius of the Earth from kilometers to meters:
r = 6,400,000 m

2. Determine the angular velocity using the formula ω = 2πf, where f is the rotational frequency of the Earth. The rotational frequency of the Earth is 1 rotation per day, which corresponds to approximately 24 hours or 86,400 seconds.
ω = 2π(1/86,400 s) ≈ 7.27 × 10^(-5) rad/s

3. Calculate the velocity at the equator:
v = rω = (6,400,000 m)(7.27 × 10^(-5) rad/s) ≈ 464 m/s

Now that you have the velocity, you can calculate the centripetal force using the formula:

F(centripetal) = (m * v^2) / r

4. Plug in the values:
F(centripetal) = (100 kg)(464 m/s)^2 / 6,400,000 m ≈ 339.5 N

So the centripetal force required to hold a 100 kg person in place at the equator is approximately 339.5 N.

To compare this force with the force of gravity, you can calculate the gravitational force using the formula:

F(gravity) = m * g

where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.

5. Plug in the values:
F(gravity) = (100 kg)(9.8 m/s^2) = 980 N

Comparing the centripetal force and the force of gravity, we can see that the centripetal force required to hold a person in place at the equator is significantly smaller than the force of gravity. Therefore, the notion that people would be thrown into space due to Earth's rotation is incorrect.

To solve this problem, you will need to use the following formula for centripetal force:

F(centripetal) = (m * v^2) / r

where:
F(centripetal) is the centripetal force
m is the mass of the object
v is the velocity of the object
r is the radius of the circular path

In this case, we are trying to find the centripetal force needed to hold a 100 kg person in place at the equator. The radius of the Earth (r) is approximately 6,400,000 m.

To calculate the velocity (v), we need to know the rotational speed of the Earth. The Earth completes one full rotation in approximately 24 hours. To convert this to seconds, we multiply by 60 (minutes) and 60 (seconds):

1 rotation = 24 hours * 60 minutes/hour * 60 seconds/minute = 86,400 seconds

Since the Earth covers a circumference of 2*pi*r in one rotation, the velocity (v) can be calculated by dividing the circumference by the time:

v = (2 * pi * r) / t

where t is the time taken for one rotation. So, substituting in the values:

v = (2 * pi * 6,400,000 m) / 86,400 s

Now, you can calculate the value of v.