A constant torque of 30m⋅N is applied to the rim of a 6.0-kg uniform disk of radius 0.20m .What is the angular speed of the disk about an axis through its center after it rotates 2.0 revolutions from rest?
v = a t
where a = alpha = angular acceleration
= torque/I
here I is moment of inertia of uniform disc = (1/2) m r^2
so I = (1/2)(6)(.04) = .12
and
a = 30/.12 = 250 radians/s^2
then v = 250 t
and theta = (1/2)a t^2 = 4 pi radians
so
125 t^2 = 4 pi
t = .317 second
then v = 250(.317) = 79.3 radians/second
To find the angular speed of the disk about an axis through its center, we can use the concept of torque and angular momentum.
Step 1: Determine the moment of inertia of the disk.
The moment of inertia (I) of a uniform 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.
In this case, the disk has a mass of 6.0 kg and a radius of 0.20 m. Plugging these values into the formula, we can calculate the moment of inertia:
I = (1/2) * 6.0 kg * (0.20 m)^2
I = 0.12 kg * m^2
Step 2: Calculate the torque applied to the disk.
The torque (τ) applied to the disk is given by the formula:
τ = r * F
where r is the distance from the axis of rotation to the point where the force is applied, and F is the force applied.
In this case, the torque is given as 30 m⋅N, and the radius is 0.20 m. Plugging these values into the formula, we can calculate the torque:
τ = (0.20 m) * (30 m⋅N)
τ = 6.0 N⋅m
Step 3: Calculate the initial angular momentum of the disk.
The initial angular momentum (L) of the disk is given by the formula:
L = I * ω
where ω is the angular speed of the disk.
Since the disk starts from rest, the initial angular speed is zero.
Step 4: Calculate the final angular momentum of the disk.
The final angular momentum (L') of the disk is given by the formula:
L' = I * ω'
where ω' is the final angular speed of the disk.
We are asked to find ω' after the disk rotates 2.0 revolutions from rest. Since one revolution is equivalent to 2π radians, the disk rotates 2.0 * 2π = 4π radians.
Given the torque (τ) and the time (t) for which it is applied (2.0 revolutions), we can calculate the change in angular momentum (ΔL) using torque-angular momentum relationship:
τ * t = ΔL
(6.0 N⋅m) * (4π rad) = ΔL
Step 5: Calculate the final angular speed of the disk.
The final angular speed (ω') can be obtained by rearranging the formula for angular momentum:
ΔL = L' - L
(6.0 N⋅m) * (4π rad) = (0.12 kg * m^2) * ω' - (0 kg * m^2) * 0
Simplifying the equation, we can find ω':
(6.0 N⋅m) * (4π rad) = (0.12 kg * m^2) * ω'
ω' = (6.0 N⋅m) * (4π rad) / (0.12 kg * m^2)
Calculating this expression will give us the final angular speed of the disk about an axis through its center.