An early major objection to the idea that the earth rotates on its axis was that it would have
to turn so fast that people would be thrown off into space. Show the error in this logic by
calculating:
a. the speed of a 100 kg person standing at the equator. (The radius of the earth is
about 6400 km).
b. the net force necessary to keep the person in this motion.
c. how the normal force acting on the person compares to their weight.
To address the objection around people being thrown off into space due to Earth's rotation, we can use basic principles of physics. Let's go step by step to calculate the required values.
a. To determine the speed of a 100 kg person standing at the equator, we need the circumference of the Earth at the equator (since the equator experiences the maximum rotational velocity).
The formula to calculate the circumference of a circle is:
Circumference = 2 × π × radius
Given that the radius of the Earth is about 6400 km, we can substitute this value into the formula:
Circumference = 2 × π × 6400 km
To get the speed, we need to divide the circumference by the time it takes for one rotation, which is 24 hours or 86,400 seconds:
Speed = Circumference / Time
Speed = (2 × π × 6400 km) / 86,400 s
Convert km to meters and seconds:
Speed = (2 × π × 6,400,000 m) / 86,400 s
Now we can calculate the speed:
Speed ≈ 463 m/s
Therefore, the speed of a 100 kg person standing at the equator is approximately 463 m/s.
b. To find the net force required to keep the person in motion, we can use the following formula:
Force = mass × acceleration
Since the person is moving in a circular path, the acceleration is centripetal acceleration given by:
Acceleration = (velocity)^2 / radius
Plugging in the values:
Acceleration = (463 m/s)^2 / 6,400,000 m
Now we can calculate the force:
Force = 100 kg × [(463 m/s)^2 / 6,400,000 m]
Force ≈ 338 Newtons
Therefore, the net force necessary to keep the person in motion is approximately 338 Newtons.
c. To compare the normal force acting on the person to their weight, we need to recognize that the normal force counteracts the force of gravity. At the equator, the normal force is equal to the weight of the person.
Weight = mass × gravitational acceleration
Weight = 100 kg × 9.8 m/s^2
Weight = 980 Newtons
Comparing the normal force to weight:
Normal Force = 980 Newtons
Since the normal force is equal to the weight, the person experiences a normal force that is equivalent to their weight.