A rotating disc has an angular velocity of ω. If the radius of the disc is doubl
Practice Questions
Q1
A rotating disc has an angular velocity of ω. If the radius of the disc is doubled while keeping the mass constant, what happens to the angular momentum?
It doubles
It remains the same
It quadruples
It halves
Questions & Step-by-Step Solutions
A rotating disc has an angular velocity of ω. If the radius of the disc is doubled while keeping the mass constant, what happens to the angular momentum?
Step 1: Understand that angular momentum (L) is calculated using the formula L = Iω, where I is the moment of inertia and ω is the angular velocity.
Step 2: Know that the moment of inertia (I) for a disc is given by the formula I = (1/2) m r^2, where m is the mass and r is the radius.
Step 3: If the radius (r) of the disc is doubled, the new radius becomes 2r.
Step 4: Calculate the new moment of inertia (I') using the new radius: I' = (1/2) m (2r)^2 = (1/2) m (4r^2) = 4 * (1/2) m r^2 = 4I.
Step 5: Since the radius is doubled, the angular velocity (ω) will decrease. Specifically, if the radius doubles, the angular velocity will decrease by a factor of 2 (ω' = ω/2).
Step 6: Now substitute the new values into the angular momentum formula: L' = I'ω' = (4I)(ω/2) = 2Iω.
Step 7: Compare the new angular momentum (L') with the original angular momentum (L): L = Iω. Since L' = 2Iω, it shows that L' is actually twice the original L.
Step 8: Conclude that when the radius is doubled, the angular momentum increases.
Angular Momentum – Angular momentum (L) is the product of the moment of inertia (I) and angular velocity (ω), represented by the formula L = Iω.
Moment of Inertia – The moment of inertia (I) of a disc is dependent on its mass and the square of its radius, which means if the radius is doubled, I increases by a factor of 4.
Conservation of Angular Momentum – In a closed system, if no external torques act, the total angular momentum remains constant.
