If the radius of a rotating object is halved while keeping the angular velocity constant, what happens to the linear velocity at the edge?

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Q: If the radius of a rotating object is halved while keeping the angular velocity constant, what happens to the linear velocity at the edge?

Solution: Linear velocity v = rω. If r is halved and ω remains constant, v also halves.

Steps: 7

Step 1: Understand the formula for linear velocity, which is v = rω, where v is linear velocity, r is radius, and ω is angular velocity.
Step 2: Identify that in this scenario, the radius (r) is being halved. This means if the original radius is r, the new radius will be r/2.
Step 3: Note that the angular velocity (ω) remains constant, meaning it does not change.
Step 4: Substitute the new radius into the formula: v = (r/2)ω.
Step 5: Compare the new linear velocity with the original: The original linear velocity was v = rω, and the new linear velocity is v = (r/2)ω.
Step 6: Realize that (r/2)ω is half of rω, so the new linear velocity is half of the original linear velocity.
Step 7: Conclude that if the radius is halved while keeping angular velocity constant, the linear velocity at the edge also halves.