A disk rotates about its axis with an angular velocity of ω. If its radius is do
Practice Questions
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
Questions & Step-by-Step Solutions
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?
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.
Angular Momentum Conservation – Angular momentum (L) is conserved in a closed system, and is given by the product of moment of inertia (I) and angular velocity (ω). When the radius of a rotating object changes, the moment of inertia changes, which affects the angular velocity to keep angular momentum constant.
Moment of Inertia – The moment of inertia for a disk is proportional to the square of its radius (I = 1/2 m r^2). Doubling the radius increases the moment of inertia by a factor of four.
