A 68.6 kg solid aluminum cylindrical wheel of radius 0.45 m is rotating about its axle in frictionless bearings with angular velocity ω = 45.5 rad/s.
If its temperature is then raised from 21.0 ∘C to 88.0 ∘C, what is the fractional change in ω?
To determine the fractional change in angular velocity (ω) of the rotating aluminum wheel when its temperature is raised, we need to consider the conservation of angular momentum.
Step 1: Calculate the moment of inertia (I) of the aluminum wheel. The moment of inertia of a solid cylinder rotating about its axis is given by the formula:
I = (1/2) * m * r^2
where m is the mass of the cylinder and r is its radius.
Given:
Mass of the wheel (m) = 68.6 kg
Radius of the wheel (r) = 0.45 m
Plugging the values into the formula:
I = (1/2) * 68.6 kg * (0.45 m)^2
I = 6.6585 kg*m^2 (rounded to four decimal places)
Step 2: Use the principle of conservation of angular momentum.
According to the principle of conservation of angular momentum, the initial angular momentum (L1) is equal to the final angular momentum (L2). Mathematically, this can be represented as:
L1 = L2
The angular momentum (L) of a rotating object is given by the formula:
L = I * ω
where ω is the angular velocity of the object.
Step 3: Calculate the initial angular momentum (L1) of the wheel.
Given:
Temperature at the initial state (T1) = 21.0 °C
To calculate L1, we need to convert the given temperature to kelvin (K) using the formula:
T(K) = T(°C) + 273.15
Plugging in the values:
T1(K) = 21.0 °C + 273.15 = 294.15 K
With temperature information, we can use the formula for the coefficient of linear expansion (α) of aluminum:
α = 23 × 10^(-6) per degree Celsius
The change in length (ΔL) of the aluminum wheel due to the temperature change is given by:
ΔL = α * L * ΔT
where ΔT is the change in temperature.
Since the wheel is rotating about its axis, the change in radius (Δr) can be approximated as:
Δr = ΔL / 2
Now we can calculate the change in angular velocity (Δω) using the formula:
Δω = (Δr / r) * ω
where r is the radius of the wheel and ω is the initial angular velocity.
Given:
Temperature at the final state (T2) = 88.0 °C
Angular velocity (ω) = 45.5 rad/s
Step 4: Calculate the final angular momentum (L2) of the wheel.
To determine L2, we follow the same steps as in Step 3, but using the final temperature (T2) instead.
Step 5: Calculate the fractional change in angular velocity (Δω/ω).
Finally, we can find the fractional change in ω using the formula:
Fractional change in ω = (Δω / ω)
By plugging in the values obtained from Steps 3 and 4, we can obtain the solution.