How To Calculate The Net Torque

Imagine this: You need to unscrew a bolt from a wooden plank. You find the correctly sized wrench and secure it to the bolt. In order to begin to loosen the wrench, you need to hold the handle and pull or push in a direction that is perpendicular to the handle of the wrench. Pushing along the direction of the wrench will not apply a torque to the bolt, and it will not loosen.

Torque is the influence that is calculated from forces that effect rotational movement, or cause rotation about an axis.

The formula for determining torque, ​**​τ​**​ is

\(\tau = r\times F\)

where ​**​r​​ is the lever arm and ​​F​​ is the force. Remember, ​​r​**​,​​ τ​​, and ​**​F​**​ are all vector quantities, thus the operation is not scalar multiplication, but a vector cross product. If the angle, ​θ​, between the lever arm and force is known, then magnitude of the torque can be calculated as

\(\tau = rF\sin{\theta}\)

The standard or SI torque unit is Newton meters, or Nm.

Net torque means calculating the resulting torque from ​n​ different contributing forces. Thus:

\(\Sigma^n_i \vec{\tau} = \Sigma^n_i r_i F_i sin(\theta)\)

Just like in kinematics, if the sum of the torques is 0, then the object is in rotational equilibrium, meaning it is neither accelerating nor decelerating.

The torque equation is jam-packed with important information about how torque is generated, and how to calculate a net torque. Understanding the terms in the equation will help you complete a general net torque calculation.

First, the axis of rotation is the point about which the rotation will occur. For the wrench torque example, the axis of rotation was through the center of the bolt, since the wrench will rotate around the bolt. For a see-saw, the axis of rotation is the middle of the bench, where the fulcrum is placed, and the children at the ends of the see-saw are applying the torque.

Next, the distance between the axis of rotation and the applied force is called the lever arm. Determining the lever arm can be tricky because it is a vector quantity, thus there are potentially many possible lever arms, but only one correct one.

Lastly, the line of action is an imaginary line that can be extended from the applied force in order to determine the lever arm.

The best way to start most physics problems is to draw a picture of the situation. Sometimes that picture is described as a free body diagram (FBD), where the object upon which the forces are acting is drawn, and the forces are drawn as arrows with their direction and magnitude labeled. Other important information to add to your FBD are a coordinate axes, and the axis of rotation.

For solving for net torque, an accurate free body diagram is critical.

Step 1: Draw the FBD and include a coordinate axes. Label the axis of rotation.

Step 2: Draw all of the forces that are acting on the body, using the information given to accurately place the forces relative to the axis of rotation.

Step 3: To determine the lever arm (which is likely given in the problem), extend the line of action from the force, such that the lever arm can be drawn through the axis of the rotation and perpendicular to the force.

Step 4: Information from the problem may give information about the angle between the lever arm and the force, such that the contribution to the torque can be calculated:

\(\tau_i=r_iF_i\sin{\theta_i}\)

Step 5: Add up each contribution from each of the N forces, to determine the net torque.