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 find the net torque on the system, we need to calculate the torque produced by each force and then sum them up.
Torque is calculated using the formula:
Torque = Force * Lever Arm
The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force.
Let's calculate the torque produced by each force:
For force F1:
- The magnitude of the force is 80 N.
- The lever arm is the radius of the larger wheel, R = 6 meters.
- So the torque produced by F1 is 80 N * 6 meters = 480 Nm (counterclockwise).
For force F2:
- The magnitude of the force is 80 N.
- The lever arm is the radius of the larger wheel, R = 6 meters.
- So the torque produced by F2 is 80 N * 6 meters = 480 Nm (counterclockwise).
For force F3:
- The magnitude of the force is 40 N.
- The lever arm is the radius of the smaller wheel, r = 2 meters.
- So the torque produced by F3 is 40 N * 2 meters = 80 Nm (clockwise).
For force F4:
- The magnitude of the force is 30 N.
- The lever arm is the radius of the smaller wheel, r = 2 meters.
- So the torque produced by F4 is 30 N * 2 meters = 60 Nm (clockwise).
Now let's add up the torques:
Total torque = Torque F1 + Torque F2 + Torque F3 + Torque F4
= 480 Nm (counterclockwise) + 480 Nm (counterclockwise) + 80 Nm (clockwise) + 60 Nm (clockwise)
= 960 Nm (counterclockwise) + 140 Nm (clockwise)
To determine the net torque, we subtract the clockwise torque from the counterclockwise torque:
Net torque = Counterclockwise torque - Clockwise torque
= 960 Nm - 140 Nm
= 820 Nm (counterclockwise)
Therefore, the net torque on this system is 820 Nm counterclockwise.
To find 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 formula:
T = r * F * sin(θ)
Where:
- T is the torque produced by the force in Newton-meters (Nm)
- r is the radius of the wheel the force is applied to in meters (m)
- F is the magnitude of the force in Newtons (N)
- θ is the angle between the force and a line drawn radially from the center of the wheel to the point where the force is applied (measured in radians)
Now let's calculate the torque produced by each force:
For F1:
- r = 2 meters
- F = 80 N
- Since it is applied tangentially to the wheel, θ = 0 degrees = 0 radians
- T1 = r * F1 * sin(θ1) = 2 * 80 * sin(0) = 0 Nm
For F2:
- r = 2 meters
- F = 80 N
- Since it is applied tangentially to the wheel, θ = 0 degrees = 0 radians
- T2 = r * F2 * sin(θ2) = 2 * 80 * sin(0) = 0 Nm
For F3:
- r = 2 meters
- F = 40 N
- Since it is applied tangentially to the wheel, θ = 0 degrees = 0 radians
- T3 = r * F3 * sin(θ3) = 2 * 40 * sin(0) = 0 Nm
For F4:
- r = 6 meters
- F = 30 N
- Since it is applied tangentially to the wheel, θ = 0 degrees = 0 radians
- T4 = r * F4 * sin(θ4) = 6 * 30 * sin(0) = 0 Nm
Now, we can calculate the net torque:
Net torque = T1 + T2 + T3 + T4 = 0 Nm + 0 Nm + 0 Nm + 0 Nm = 0 Nm
Therefore, the net torque on this system is 0 Nm.