A disk rotates about its axis with an angular velocity of ω. If its radius is doubled, what will be the new angular velocity to maintain the same linear velocity at the edge?
Practice Questions
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Q1
A disk rotates about its axis with an angular velocity of ω. If its radius is doubled, what will be the new angular velocity to maintain the same linear velocity at the edge?
ω/2
ω
2ω
4ω
The linear velocity v = rω. If the radius is doubled, to maintain the same v, the angular velocity must remain ω.
Questions & Step-by-step Solutions
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Q
Q: A disk rotates about its axis with an angular velocity of ω. If its radius is doubled, what will be the new angular velocity to maintain the same linear velocity at the edge?
Solution: The linear velocity v = rω. If the radius is doubled, to maintain the same v, the angular velocity must remain ω.
Steps: 7
Step 1: Understand that the linear velocity (v) at the edge of the disk is given by the formula v = rω, where r is the radius and ω is the angular velocity.
Step 2: Identify that if the radius (r) is doubled, the new radius becomes 2r.
Step 3: Write the formula for the new linear velocity with the new radius: v' = (2r)ω'.
Step 4: To maintain the same linear velocity (v = v'), set the two equations equal: rω = (2r)ω'.
Step 5: Simplify the equation by dividing both sides by r (assuming r is not zero): ω = 2ω'.
Step 6: Solve for the new angular velocity (ω'): ω' = ω/2.
Step 7: Conclude that to maintain the same linear velocity at the edge when the radius is doubled, the new angular velocity must be half of the original angular velocity.
