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.
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?
1 answer
1 answer