To answer questions 6, 7, and 8, we need to calculate the net torque and use it to find the angular acceleration and rotational energy.
6) To find the z-component of the net torque about the z-axis, we need to sum the torques due to each force. The torque due to a force is given by the cross product of the force and its position vector from the origin.
The position vectors for each force are:
1) F1: (R, 0, 0)
2) F2: (0, -R, 0)
3) F3: (-Rcos(θ), Rsin(θ), 0)
The torques are given by:
τ1 = r1 x F1 = (R, 0, 0) x (315, 0, 0) = (0, 0, 315R)
τ2 = r2 x F2 = (0, -R, 0) x (0, -315, 0) = (0, 0, 0)
τ3 = r3 x F3 = (-Rcos(θ), Rsin(θ), 0) x (315cos(θ), 315sin(θ), 0)
= (0, 0, -315Rcos(θ)sin(θ))
The z-component of the net torque is the sum of these torques:
τz_net = τ1z + τ2z + τ3z = 315R - 315Rcos(θ)sin(θ)
7) To find the magnitude of the angular acceleration about the z-axis, we can use Newton's second law for rotational motion, which states that the net torque is equal to the moment of inertia multiplied by the angular acceleration:
τz_net = I * α
Since the disk is a uniform disk, the moment of inertia can be calculated as:
I = (1/2) * m * R^2
Therefore, the angular acceleration is:
α = τz_net / I = (315R - 315Rcos(θ)sin(θ)) / ((1/2) * m * R^2)
8) To find the rotational energy of the disk after the forces have been applied for t = 1.7 s, we can use the formula for rotational kinetic energy:
Rotational Energy = (1/2) * I * ω^2
Since the disk starts from rest, the initial angular velocity ω is 0. We can use the angular acceleration calculated in question 7 to find the final angular velocity ω_f:
ω_f = α * t
Then, we can calculate the rotational energy:
Rotational Energy = (1/2) * I * ω_f^2