A force of 100 N is applied tangentially to a disk of radius 10 cm and mass 0.5 kg. the disk is initially at rest. What is the angular acceleration of the disk? How many revolutions does the disk goes through after 10 sec. How much rotational work was done on the disk?
Angular acceleration = Torque/(moment of inertia)
The moment of inertia is (1/2) M R^2 for a disk.
After 10 seconds, the number of radians turned is
theta = (1/2)(angular acceleration)*(10 s)^2
Divide that by 2 pi for the number of revolutions.
Rotational work done is (torque)*(theta)
To find the angular acceleration of the disk, we need to use the formula:
τ = I α
where τ is the torque applied to the disk, I is the moment of inertia of the disk, and α is the angular acceleration.
First, let's find the moment of inertia of the disk. The moment of inertia of a disk is given by the formula:
I = (1/2) m r^2
where m is the mass of the disk and r is the radius of the disk.
Plugging in the values:
m = 0.5 kg
r = 0.1 m
I = (1/2) * 0.5 kg * (0.1 m)^2
I = 0.005 kg·m^2
Now, let's find the torque applied to the disk. The torque is given by the formula:
τ = rF
where r is the radius of the disk and F is the force applied tangentially to the disk.
Plugging in the values:
r = 0.1 m
F = 100 N
τ = 0.1 m * 100 N
τ = 10 N·m
Now, we can calculate the angular acceleration using the formula τ = I α:
α = τ / I
α = 10 N·m / 0.005 kg·m^2
α = 2000 rad/s^2
So, the angular acceleration of the disk is 2000 rad/s^2.
Next, let's find the number of revolutions the disk goes through after 10 seconds. The angular displacement (θ) can be calculated using the formula:
θ = ω0 t + (1/2) α t^2
where ω0 is the initial angular velocity, t is the time, and α is the angular acceleration. Since the disk is initially at rest, the initial angular velocity (ω0) is 0.
Plugging in the values:
ω0 = 0 rad/s
t = 10 s
α = 2000 rad/s^2
θ = 0 rad/s * 10 s + (1/2) * 2000 rad/s^2 * (10 s)^2
θ = 0 + 0.5 * 2000 rad/s^2 * 100 s^2
θ = 100,000 rad
Since there are 2π radians in one revolution, we can convert the angular displacement to revolutions:
Number of revolutions = θ / (2π radians/revolution)
Number of revolutions = 100,000 rad / (2π rad/rev)
Number of revolutions ≈ 15,915 revolutions
So, the disk goes through approximately 15,915 revolutions after 10 seconds.
Finally, let's calculate the rotational work done on the disk. The rotational work (W) is given by the formula:
W = τ θ
where τ is the torque applied to the disk and θ is the angular displacement.
Plugging in the values:
τ = 10 N·m
θ =100,000 rad
W = 10 N·m * 100,000 rad
W = 1,000,000 N·m
So, the rotational work done on the disk is 1,000,000 N·m.