A wheel-and-axle system shown consists of two coaxial wheels of radius r = 2 meters and R = 6 meters. The forces applied to the system as shown in the diagram are:

F1 = 80 N
F2 = 80 N
F3 = 40 N
F4 = 30 N

If counterclockwise is positive, what is the net torque on this system?

To determine the net torque on the system, we need to calculate the torque produced by each force and then sum them up.

The torque produced by a force (T) can be calculated using the equation: T = r * F * sin(theta), where r is the radius at which the force is applied, F is the magnitude of the force, and theta is the angle between the position vector and the force vector.

Let's calculate the torque produced by each force:

1. Force F1:
- The radius at which F1 is applied is equal to the radius of the smaller wheel, r = 2 meters.
- The magnitude of F1 is 80 N.
- The angle between the position vector and the force vector is 90 degrees, as F1 is applied tangentially to the wheel.
- Therefore, the torque produced by F1 is: T1 = r * F1 * sin(theta1) = 2 * 80 * sin(90°) = 160 Nm.

2. Force F2:
- The radius at which F2 is applied is equal to the radius of the larger wheel, R = 6 meters.
- The magnitude of F2 is 80 N.
- The angle between the position vector and the force vector is also 90 degrees.
- So, the torque produced by F2 is: T2 = R * F2 * sin(theta2) = 6 * 80 * sin(90°) = 480 Nm.

3. Force F3:
- F3 is applied at the center of the larger wheel, so the radius is 0.
- The magnitude of F3 is 40 N.
- No torque is produced by F3 since the radius is 0.

4. Force F4:
- F4 is also applied at the center of the larger wheel, so the radius is 0.
- The magnitude of F4 is 30 N.
- No torque is produced by F4.

Now, we can calculate the net torque by summing up the individual torques:
Net Torque = T1 + T2 + T3 + T4 = 160 Nm + 480 Nm + 0 Nm + 0 Nm = 640 Nm.

Therefore, the net torque on the system is 640 Nm in the counterclockwise direction.