Answer :
The total kinetic energy of one wheel is approximately 2734 joules, calculated using the moment of inertia and linear velocity.
Here's how to find the total kinetic energy of one wheel:
Step 1: Find the moment of inertia of the wheel.
We can treat the wheel as a solid disk for this calculation. The moment of inertia (I) of a solid disk is:
I = (1/2) * MR²
where:
M is the mass of the wheel (20.9 kg)
R is the radius of the wheel (diameter / 2)
First, calculate the radius:
R = diameter / 2 = 1.1 m / 2 = 0.55 m
Now, plug the values into the moment of inertia formula:
I = (1/2) * 20.9 kg * (0.55 m)² = 3.44 kgm²
Step 2: Find the linear velocity of a point on the rim of the wheel.
The car's speed (37.9 m/s) represents the linear velocity (v) of the center of the wheel. However, for the kinetic energy of a rotating object, we need the velocity of a point on its rim.
The rim of the wheel traces a larger circle compared to the center. The circumference of this circle is the distance a point on the rim travels in one rotation. We can relate this distance to the linear velocity of the car.
Circumference of the rim's path (c) = 2 * pi * R = 2 * pi * 0.55 m ≈ 3.46 m (pi ≈ 3.14)
Since the car travels at a constant speed (37.9 m/s), a point on the rim also travels at this speed along its larger circular path.
Step 3: Relate linear velocity of the rim to the velocity at the center.
The linear velocity of a point on the rim ([tex]\[v_{\text{rim}}[/tex]) is related to the linear velocity at the center (v) by the ratio of the circumferences:
[tex]\[v_{\text{rim}}[/tex] = v * (c / (2 * pi * R))
[tex]\[v_{\text{rim}}[/tex] = 37.9 m/s * (3.46 m / (2 * pi * 0.55 m)) ≈ 21.7 m/s
Step 4: Find the kinetic energy.
The kinetic energy (KE) of a rotating solid disk is:
KE = (1/2) * I * ω²
where:
ω (omega) is the angular velocity of the wheel (rad/s)
However, we don't have the angular velocity directly. We can relate it to the linear velocity of the rim using the relationship between linear and angular velocity for rotational motion:
[tex]\[v_{\text{rim}}[/tex] = ω * R
Since we already found [tex]\[v_{\text{rim}}[/tex] and R, we can solve for ω:
ω = [tex]\[v_{\text{rim}}[/tex] / R = 21.7 m/s / 0.55 m ≈ 39.6 rad/s
Now, plug the moment of inertia (I) and angular velocity (ω) into the kinetic energy formula:
KE = (1/2) * 3.44 kgm² * (39.6 rad/s)² ≈ 2734 J
Therefore, the total kinetic energy of one wheel is approximately 2734 joules.
