A flywheel has a moment of inertia of 2.6 kg·m2. What is the average power needed to spin it up from rest to an angular velocity of 100 rad/s in 30 seconds?
power= work/time= 1/2 I w^2/30 in watts.
So would it be 1/2*2.6^2/30
To find the average power needed to spin up the flywheel, we need to calculate the work done to increase its angular velocity and divide it by the time taken. Here's how to do it step by step:
Step 1: Calculate the change in angular velocity.
The initial angular velocity (ωi) is 0 rad/s (from rest), and the final angular velocity (ωf) is 100 rad/s. So, the change in angular velocity (Δω) is:
Δω = ωf - ωi
= 100 rad/s - 0 rad/s
= 100 rad/s
Step 2: Calculate the rotational kinetic energy.
Rotational kinetic energy (KE_rot) is given by the formula:
KE_rot = (1/2) * I * ω^2
Where:
I is the moment of inertia
ω is the angular velocity
Plugging in the values:
KE_rot = (1/2) * 2.6 kg·m^2 * (100 rad/s)^2
= (1/2) * 2.6 * 10^2 kg·m^2/s^2
= 130 kg·m^2/s^2
Step 3: Calculate the work done.
The work done (W) to increase the angular velocity is equal to the change in rotational kinetic energy:
W = ΔKE_rot = KE_rot - KE_rot_initial
Since the flywheel starts from rest, the initial rotational kinetic energy (KE_rot_initial) is 0. Therefore:
W = KE_rot - 0
= KE_rot = 130 kg·m^2/s^2
Step 4: Calculate the average power.
Average power (P_avg) is given by the formula:
P_avg = W / t
Where:
W is the work done
t is the time taken
Plugging in the values:
P_avg = 130 kg·m^2/s^2 / 30 s
≈ 4.33 kg·m^2/s^3
Therefore, the average power needed to spin up the flywheel is approximately 4.33 kg·m²/s³.