(a) Calculate the angular momentum of an ice skater spinning at 6.00 rev/s given his moment of inertia is 0.490 kg·m2.
(b) He reduces his rate of spin (his angular velocity) by extending his arms and increasing his moment of inertia. Find the value of his moment of inertia if his angular velocity drops to 1.35 rev/s.
(c) Suppose instead he keeps his arms in and allows friction with the ice to slow him to 3.00 rev/s. What average torque was exerted if this takes 14.0 seconds?
(a) To calculate the angular momentum of the ice skater, you can use the formula:
Angular momentum (L) = moment of inertia (I) x angular velocity (ω)
Given that the angular velocity (ω) is 6.00 rev/s and the moment of inertia (I) is 0.490 kg·m^2, you can substitute these values into the formula to calculate the angular momentum (L):
L = 0.490 kg·m^2 x 6.00 rev/s
To convert from revolutions per second to radians per second, you need to multiply by 2π (since there are 2π radians in one revolution):
L = 0.490 kg·m^2 x 6.00 rev/s x 2π rad/rev
Now, you can calculate the angular momentum:
L = 0.490 kg·m^2 x 6.00 rev/s x 2π rad/rev
= 36.720 kg·m^2·rad/s
Therefore, the angular momentum of the ice skater spinning at 6.00 rev/s is 36.720 kg·m^2·rad/s.
(b) To find the new moment of inertia (I) when the angular velocity (ω) is reduced to 1.35 rev/s, you can use the formula:
Angular momentum (L) = moment of inertia (I) x angular velocity (ω)
Given that the angular momentum is conserved (remains the same), you can equate the initial angular momentum (L1) with the final angular momentum (L2):
I1 x ω1 = I2 x ω2
Substituting the values and solving for the new moment of inertia (I2):
0.490 kg·m^2 x 6.00 rev/s = I2 x 1.35 rev/s
I2 = (0.490 kg·m^2 x 6.00 rev/s) / 1.35 rev/s
I2 = 2.1778 kg·m^2
Therefore, the value of the moment of inertia when the angular velocity drops to 1.35 rev/s is 2.1778 kg·m^2.
(c) To calculate the average torque exerted when the ice skater slows down from 6.00 rev/s to 3.00 rev/s in 14.0 seconds, you can use the formula:
Torque (τ) = Change in angular momentum (ΔL) / Change in time (Δt)
The change in angular momentum (ΔL) can be calculated by taking the difference between the final angular momentum (L2) and the initial angular momentum (L1):
ΔL = L2 - L1
Using the values from part (a):
ΔL = 0.490 kg·m^2 x 3.00 rev/s x 2π rad/rev - 0.490 kg·m^2 x 6.00 rev/s x 2π rad/rev
The change in time (Δt) is given as 14.0 seconds.
Now, you can substitute the values into the formula to calculate the average torque:
τ = ΔL / Δt
Finally, calculate the average torque:
τ = (0.490 kg·m^2 x 3.00 rev/s x 2π rad/rev - 0.490 kg·m^2 x 6.00 rev/s x 2π rad/rev) / 14.0 s
Therefore, the average torque exerted is the calculated value using the given values.