A flywheel is a solid disk that rotates about an axis that is perpendicular to the disk at its center. Rotating flywheels provide a means for storing energy in the form of rotational kinetic energy and are being considered as a possible alternative to batteries in electric cars. The gasoline burned in a 330-mile trip in a typical midsize car produces about 2.00 109 J of energy. How fast would a 10-kg flywheel with a radius of 0.25 m have to rotate to store this much energy? Give your answer in rev/min.
2e9 = 1/2 I ω^2
Have to look up I for a solid disc, solve for ω
To find the speed at which the flywheel must rotate to store the given amount of energy, we can use the formula for rotational kinetic energy:
Rotational kinetic energy (KE) = (1/2) * I * ω^2
Where:
KE is the rotational kinetic energy,
I is the moment of inertia of the flywheel,
ω (omega) is the angular velocity of the flywheel.
To find the moment of inertia of a solid disk like the flywheel, we can use the formula:
I = (1/2) * m * r^2
Where:
I is the moment of inertia,
m is the mass of the flywheel, and
r is the radius of the flywheel.
First, let's calculate the moment of inertia:
I = (1/2) * m * r^2
= (1/2) * 10 kg * (0.25 m)^2
= 0.3125 kg·m^2
Next, we can rearrange the formula for rotational kinetic energy to solve for angular velocity:
Rotational kinetic energy (KE) = (1/2) * I * ω^2
2 * KE = I * ω^2
ω^2 = (2 * KE) / I
ω = √((2 * KE) / I)
Substituting the given value for the rotational kinetic energy (2.00 x 10^9 J) and the calculated value for the moment of inertia (0.3125 kg·m^2):
ω = √((2 * 2.00 x 10^9 J) / 0.3125 kg·m^2)
= √((4.00 x 10^9 J) / 0.3125 kg·m^2)
≈ √(1.28 x 10^10) rad/s
Finally, to find the value in rev/min, we can convert from radians per second to revolutions per minute:
ω (rev/min) = (ω (rad/s) * 60) / (2π)
= [(√(1.28 x 10^10) rad/s) * 60] / (2π)
≈ 51277 rev/min
Therefore, a 10 kg flywheel with a radius of 0.25 m would need to rotate at approximately 51277 rev/min to store 2.00 x 10^9 J of energy.