11.2
Net Torque Calculations
When a mechanic tries to remove a hex nut with a wrench, it is easier if the force is applied at the farthest end of the wrench handle. The lever arm…
Consider a meter scale pivoted at its center. Three forces of 3.0 N, 5.0 N, and 2.0 N are acting on it at distances of 0.25 meters, 0 meters, and 0.5 meters, respectively.
The angles made by the forces with the scale are 60°, 90°, and 30°, respectively.
Recall that the magnitude of the torque is expressed as the product of the lever arm and the sine component of the force.
As the values for the applied force, lever arm, and the angles are known, the torque due to each force can be calculated.
Therefore, the magnitude of torque τ1 is 0.650 Nm. Since the rotation of the scale is clockwise, due to F1, the torque τ1 is negative.
The force F2 is acting at the pivot point. Therefore, the torque τ2 is zero. The magnitude of the torque τ3 is equal to 0.5 Nm. Since the scale rotates counterclockwise, the torque is positive.
Therefore, the net torque on the scale is −0.15 Nm, directed into the plane of the paper.
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Q1: How do you calculate the magnitude of torque for a single force?
Torque magnitude equals the lever arm distance multiplied by the sine component of the force. The lever arm is the perpendicular distance from the pivot point to where the force is applied. Only the force component perpendicular to the lever arm contributes to rotation. For example, a 5.0 N force at 0.25 meters with a 60-degree angle produces a torque of 0.650 Nm.
Q2: Why does applying force perpendicular to the lever arm create more torque?
Only the perpendicular component of force generates torque about a pivot point. When force is applied perpendicular to the lever arm, the entire force magnitude contributes to rotation. Pushing at an angle reduces the effective perpendicular component, decreasing torque. This principle connects to the equation of rotational dynamics, which governs how torque produces angular acceleration.
Q3: What determines whether torque is positive or negative?
Torque sign depends on the direction of rotation about the pivot point. Clockwise rotation produces negative torque, while counterclockwise rotation produces positive torque. The sign is assigned after calculating magnitude using the cross product of the lever arm and force vector relative to your chosen coordinate system.
Q4: How do you find net torque when multiple forces act on an object?
Calculate individual torque for each force, then sum all torques algebraically. Include both magnitude and sign for each torque. If a force acts at the pivot point, its torque is zero. For example, three forces produce torques of -0.650 Nm, 0 Nm, and 0.5 Nm, yielding a net torque of -0.15 Nm directed into the plane.
Q5: Why does a force applied at the pivot point produce zero torque?
Torque requires a lever arm—the distance from the pivot point to the force application point. When force is applied directly at the pivot, the lever arm distance is zero. Since torque equals lever arm multiplied by the force component, zero lever arm results in zero torque regardless of force magnitude.
Q6: What is the first step in solving a net torque problem?
Establish a coordinate system with the pivot point or axis of rotation as the origin. This reference frame determines how you measure lever arm distances and calculate the cross product of the lever arm and force vector. A clear coordinate system ensures consistent sign assignment and accurate net torque calculation.
Q7: How does lever arm length affect the torque produced by a force?
Torque increases proportionally with lever arm length. A longer distance from the pivot point to the force application point produces greater torque for the same force magnitude. This is why applying force at the farthest end of a wrench handle is more effective than pushing near the pivot point, and understanding this principle is essential for work and power for rotational motion.