Suppose that Superman wants to stop Earth so it does not rotate. He exerts a force on Earth F⃗ SonE at Earth's equator tangent to its surface for a time interval of 1 year. The mass of Earth is 5.98×1024kg, the radius is 6.37×106m. Assume that Earth is a solid sphere with its mass distributed uniformly.What magnitude force must he exert to stop Earth's rotation?
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To calculate the magnitude of the force that Superman must exert to stop Earth's rotation, we need to use the concept of angular momentum.
Angular momentum is given by the equation:
L = Iω
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia for a solid sphere can be calculated using the formula:
I = (2/5) * m * r^2
where m is the mass of the sphere and r is its radius.
Now, let's break down the problem into steps:
Step 1: Calculate the moment of inertia of the Earth.
Given:
Mass of Earth (m) = 5.98 × 10^24 kg
Radius of Earth (r) = 6.37 × 10^6 m
Using the formula for moment of inertia:
I = (2/5) * m * r^2
I = (2/5) * (5.98 × 10^24 kg) * (6.37 × 10^6 m)^2
Step 2: Calculate the initial angular velocity of the Earth.
Since Earth is currently rotating, it has an initial angular velocity. The formula to calculate angular velocity is:
ω = 2π / T
where T is the time taken for one complete rotation. In this case, T is equal to 24 hours (or 86400 seconds).
ω = 2π / 86400 s
Step 3: Calculate the initial angular momentum of the Earth.
Using the formula for angular momentum:
L = I * ω
Step 4: Calculate the change in angular momentum.
The change in angular momentum is equal to the final angular momentum (which is zero) minus the initial angular momentum.
ΔL = 0 - L
Step 5: Calculate the force required to cause this change in angular momentum.
The force required is equal to the change in angular momentum divided by the time interval (1 year or 3.154 × 10^7 seconds).
F = ΔL / Δt
Now you have all the necessary calculations to find the magnitude of the force Superman must exert to stop Earth's rotation.