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 conserve angular momentum?
Practice Questions
1 question
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 conserve angular momentum?
ω
2ω
ω/2
ω/4
To conserve angular momentum, if the radius is doubled, the angular velocity must be halved.
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 conserve angular momentum?
Solution: To conserve angular momentum, if the radius is doubled, the angular velocity must be halved.
Steps: 6
Step 1: Understand that angular momentum (L) is given by the formula L = I * ω, where I is the moment of inertia and ω is the angular velocity.
Step 2: Know that the moment of inertia (I) for a disk is calculated as I = (1/2) * m * r^2, where m is the mass and r is the radius.
Step 3: If the radius (r) is doubled, the new radius becomes 2r. Substitute this into the moment of inertia formula: I_new = (1/2) * m * (2r)^2 = (1/2) * m * 4r^2 = 2 * (1/2) * m * r^2 = 4I.
Step 4: Since angular momentum must be conserved, set the initial angular momentum equal to the new angular momentum: I * ω = 4I * ω_new.
Step 5: Simplify the equation: ω = 4 * ω_new. To find the new angular velocity (ω_new), divide both sides by 4: ω_new = ω / 4.
Step 6: Conclude that if the radius is doubled, the new angular velocity must be halved to conserve angular momentum.
