A disk with a mass of 18 kg, a diameter of 50 cm, and a thickness of 8 cm is mounted on a rough horizontal axle as shown on the left in the figure. (There is a friction force between the axle and the disk.) The disk is initially at rest. A constant force, F = 70 N,is applied to the edge of the disk at an angle of 37°, as shown on the right in the figure. After 2.0 s, the force is reduced to F = 27 N,and the disk spins with a constant angular velocity.
What is the angular velocity after 2 sec?
Figure moment of inertia I for the disk.
wi=0
Torque= I*angacceleration
70*.25*trigfunction= I acceleration
I don't understand the figure, so you will need to account for the angle. It is probably cos 37
find acceleration.
then
wf=wi+I acceleration*time
figure wf at time=2
Now, after two sec, figure the new angular acceleration. wi now is w at 2 seconds
wf(t)=w(2)i + I newacclearation *t
and you get a function of time.
I'm not getting the correct answer this way. Maybe I'm doing something wrong.
To determine the angular velocity of the disk after 2 seconds, we can use the principle of conservation of angular momentum.
The formula for angular momentum is:
L = Iω,
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
In this case, the moment of inertia of a disk rotating about its central axis can be calculated using the formula:
I = (1/2)MR²,
where M is the mass of the disk and R is its radius.
Given that the mass of the disk is 18 kg and the diameter is 50 cm, we can calculate the radius as half the diameter:
R = (50 cm) / 2 = 25 cm = 0.25 m.
The moment of inertia for the disk is therefore:
I = (1/2)(18 kg)(0.25 m)² = 0.5625 kg·m².
Now, let's consider the initial and final angular momenta. Initially, the disk is at rest, so its initial angular momentum is zero:
L₁ = 0.
After applying the force for 2 seconds, the disk spins with a constant angular velocity, which we need to find. Let's denote this angular velocity as ω₂.
The final angular momentum after 2 seconds is given by:
L₂ = Iω₂.
According to the principle of conservation of angular momentum, the initial and final angular momenta should be equal:
L₁ = L₂.
Since L₁ = 0, we can set L₂ = 0 to find the angular velocity:
Iω₂ = 0.
Simplifying the equation, we have:
ω₂ = 0 / I.
Therefore, the angular velocity after 2 seconds is zero.