A string is wound around the edge of a solid 1.60 kg disk with a 0.130 m radius. The disk is initially at rest when the string is pulled, applying a force of 8.75 N in the plane of the disk and tangent to its edge. If the force is applied for 1.90 seconds, what is the magnitude of its final angular velocity?
Torque of force = 8.75 Newtons * 0.13 meters
alpha = angular acceleration = Torque / moment of inertia
= 8.75 * 0.13 / [ 1.60 * (1/2)(.13)^2]
omega = angular velocity = alpha * t = alpha * 1.90 seconds
To find the magnitude of the final angular velocity, we can use the principle of conservation of angular momentum. The angular momentum of the disk is given by the equation:
L = I * ω
where L is the angular momentum, I is the moment of inertia of the disk, and ω is the angular velocity.
The moment of inertia of a solid disk can be calculated using the formula:
I = (1/2) * m * r^2
where m is the mass of the disk and r is its radius.
In this case, the mass of the disk is given as 1.60 kg and the radius is given as 0.130 m.
Therefore, the moment of inertia I is:
I = (1/2) * 1.60 kg * (0.130 m)^2
Next, we can find the initial angular momentum by multiplying the moment of inertia by the initial angular velocity. Since the disk is initially at rest, the initial angular velocity is 0.
So, the initial angular momentum is:
L_initial = I * ω_initial = I * 0 = 0
According to the principle of conservation of angular momentum, the final angular momentum is equal to the initial angular momentum.
L_final = L_initial = 0
We can now find the final angular velocity by rearranging the equation for angular momentum:
L_final = I * ω_final
ω_final = L_final / I
Since L_final = 0 and I is given, we can substitute these values into the equation to find the final angular velocity.
ω_final = 0 / I = 0
Therefore, the magnitude of the final angular velocity is zero. This means that the disk does not rotate and remains at rest.