After 2.0 seconds of the net torque acting on the disk, its final angular velocity is 50.0 rad/s.
Given:
Mass of the disk (m) = 1.0 kg
Force F1 = 40.0 N
Angle formed by F1 (θ1) = 30 degrees
Force F2 = 5.0 N
Radius of the disk (r) = 2.0 m
Time (t) = 2.0 seconds
Step 1: Calculate Torques (τ1 and τ2)
τ1 = r * F1 * sin(θ1)
τ1 = 2.0 m * 40.0 N * sin(30 degrees)
τ1 is approximately 40.0 Nm
τ2 = r * F2
τ2 = 2.0 m * 5.0 N
τ2 = 10.0 Nm
Step 2: Find Net Torque (τ)
τ = τ1 + τ2
τ = 40.0 Nm + 10.0 Nm
τ = 50.0 Nm
Step 3: Calculate Angular Acceleration (α)
Use the rotational analog of Newton's second law:
τ = I * α
The moment of inertia (I) for a disk is 1/2 * m * r^2:
τ = 1/2 * m * r^2 * α
Substitute known values:
50.0 Nm = 1/2 * 1.0 kg * (2.0 m)^2 * α
Solve for α:
α = 50.0 Nm / (1/2 * 1.0 kg * (2.0 m)^2)
α = 25.0 rad/s^2
Step 4: Calculate Final Angular Velocity (ω)
Use the kinematic equation for rotational motion:
ω = ω0 + α * t
Given that the disk starts from rest (ω0 = 0) and t = 2.0 seconds:
ω = 0 + 25.0 rad/s^2 * 2.0 s
ω = 50.0 rad/s