A ceiling fan consists of a small cylindrical disk with 5 thin rods coming from the center. The disk has mass md = 2.9 kg and radius R = 0.22 m. The rods each have mass mr = 1.4 kg and length L = 0.83 m.
a. What is the moment of inertia of each rod about the axis of rotation?
b. What is the moment of inertia of the disk about the axis of rotation?
c. What is the moment of inertia of the whole ceiling fan?
d. When the fan is turned on, it takes t = 3.3 s and a total of 12 revolutions to accelerate up to its full speed.
What is the magnitude of the angular acceleration?
e. What is the final angular speed of the fan?
f. What is the final rotational energy of the fan?
g. Now the fan is turned to a lower setting where it ends with half of its rotational energy as before. The time it takes to slow to this new speed is also t = 3.3 s.
What is the final angular speed of the fan?
h. What is the magnitude of the angular acceleration while the fan slows down?
a. To find the moment of inertia of one rod about the axis of rotation, we can use the parallel axis theorem, which states:
I = I_cm + mr*L^2
where I_cm is the moment of inertia of the rod with respect to its center of mass, mr is the mass of the rod, and L is the distance from the center of mass to the axis of rotation. For a thin rod, the moment of inertia about the center of mass is given by:
I_cm = (1/12) * mr * L^2
Substituting the mass and length values, we get:
I_cm = (1/12) * 1.4 * 0.83^2 = 0.512033333 kg*m^2
Now we can find the moment of inertia of the rod about the axis of rotation:
I = 0.512033333 + 1.4 * (0.83/2)^2 = 0.512033333 + 0.48005 = 0.992083333 kg*m^2
b. The moment of inertia of the disk about the axis of rotation is given by:
I_disk = (1/2) * md * R^2
Substituting the mass and radius values, we get:
I_disk = (1/2) * 2.9 * 0.22^2 = 0.13954 kg*m^2
c. To find the moment of inertia of the whole ceiling fan, we need to sum the moments of inertia of the disk and all 5 rods:
I_total = I_disk + 5 * I_rod
I_total = 0.13954 + 5 * 0.992083333 = 0.13954 + 4.960416667 = 5.0999567 kg*m^2
d. The fan takes 12 revolutions to accelerate, so its angular displacement is:
Δθ = 12 * 2 * π = 24 * π rad
The time it takes to accelerate is t = 3.3 s. We can now use one of the equations of rotational motion to find the angular acceleration:
Δθ = ω_i * t + 0.5 * α * t^2
where ω_i is the initial angular speed, which is 0 because the fan starts from rest. Rearranging the equation, we get:
α = (2 * Δθ) / (t^2)
Substituting the values, we get:
α = (2 * 24 * π) / (3.3^2) = 16.74740764 rad/s^2
e. To find the final angular speed of the fan, we can use another equation of motion:
ω_f^2 = ω_i^2 + 2 * α * Δθ
Since the initial angular speed is 0, we get:
ω_f^2 = 2 * 16.74740764 * 24 * π
ω_f = sqrt(2001.718488) = 44.74242941 rad/s
f. The final rotational energy of the fan is given by:
E_f = 0.5 * I_total * ω_f^2
Substituting the values, we get:
E_f = 0.5 * 5.0999567 * 44.74242941^2 = 5049.999416 J
g. At the lower setting, the fan has half the rotational energy as before:
E_low = 0.5 * E_f = 0.5 * 5049.999416 = 2524.999708 J
We can now find the final angular speed at the lower setting:
ω_low^2 = (2 * E_low) / I_total
ω_low = sqrt((2 * 2524.999708) / 5.0999567) = 31.6227766 rad/s
h. To find the magnitude of the angular acceleration while the fan slows down, we can use the angular displacement equation again:
Δθ = ω_i * t + 0.5 * α_slow * t^2
Since the fan makes the same number of revolutions while slowing down, the angular displacement is the same as before:
Δθ = 24 * π rad
We now have:
24 * π = 44.74242941 * 3.3 + 0.5 * α_slow * 3.3^2
α_slow = -8.373703819 rad/s^2
The magnitude of the angular acceleration while the fan slows down is 8.374 rad/s^2.
To solve this problem, we need to use the formulas related to moment of inertia and angular acceleration. Let's go step by step through the questions.
a. The moment of inertia of each rod about its axis of rotation can be calculated using the formula for the moment of inertia of a slender rod rotating about one end:
I_rod = (1/3) * mr * L^2
Substituting the given values, we get:
I_rod = (1/3) * 1.4 kg * (0.83 m)^2
b. The moment of inertia of the disk about its axis of rotation can be calculated using the formula for the moment of inertia of a disk:
I_disk = (1/2) * md * R^2
Substituting the given values, we get:
I_disk = (1/2) * 2.9 kg * (0.22 m)^2
c. To find the moment of inertia of the whole ceiling fan, we need to consider the moments of inertia of the disk and the rods. Since they are rotating about the same axis, we can simply add their moments of inertia:
I_total = I_disk + 5 * I_rod
d. The angular acceleration can be calculated using the following formula:
α = (Δω) / Δt
First, we need to find the change in angular velocity (Δω). There are 12 revolutions, which means the fan rotates 12 * 2π radians. The time taken (Δt) is given as 3.3 seconds.
Δω = (12 * 2π) / 3.3 s
Now substitute the values and solve for α.
e. The final angular speed (ω_final) of the fan can be calculated using the formula:
ω_final = ω_initial + α * t
Since the fan starts from rest (ω_initial = 0), the equation simplifies to:
ω_final = α * t
Substitute the value of α obtained from the previous step and the given time (t = 3.3 s) to find ω_final.
f. The final rotational energy (E_rot_final) of the fan can be calculated using the formula:
E_rot_final = (1/2) * I_total * ω_final^2
Substitute the values of I_total and ω_final obtained from previous steps to find E_rot_final.
g. Given that the fan ends with half of its rotational energy as before, we can calculate the final angular speed (ω_final_new) using the formula for rotational energy:
E_rot_final_new = (1/2) * I_total * ω_final_new^2
Substitute the values of E_rot_final_new (which is half of E_rot_final) and I_total, and solve for ω_final_new.
h. To find the magnitude of the angular acceleration while the fan slows down, we need to calculate the change in angular velocity (Δω_new). The formula for angular acceleration is the same as before:
α_new = Δω_new / Δt
Since the fan starts with the final angular speed obtained in the previous step (ω_final_new), the change in angular velocity (Δω_new) is equal to -ω_final_new. Substitute the values and solve for α_new.
I hope this helps! Let me know if you have any further questions.