Engineering Expert Witness Blog

      Last time we introduced the mathematical formula for torque, which is most simply defined as a measure of how much a force acting upon an object causes that object to rotate around a pivot point.   When manipulated, torque can produce a mechanical advantage in gear trains and tools, which we’ll see later.   The formula is:

Torque = Distance × Force × sin(ϴ)

      We learned that the factors Distance and Force are vectors, and sin(ϴ) is a trigonometric function of the angle ϴ which is formed between their two vectors.   Let’s return to our wrench example and see how the torque formula works.

      Vectors have both a magnitude, that is, a size or extent, and a direction, and they are typically represented in physics and engineering problems by straight arrows.   In our illustration the vector for distance is represented by an orange arrow, while the vector for force is represented by a red arrow.   The orange distance vector has a magnitude of 6 inches, while the red force vector has a magnitude of 10 pounds, which is being supplied by the user’s arm muscle manipulating the nut.   That muscle force follows a path from the arm to the pivot point located at the center of the nut, a distance of 6 inches.

      Vector arrows point in a specific direction, a direction which is indicative of the way in which the vectors’ magnitudes — in our case inches of distance vs. pounds of force — are oriented with respect to one another.   In our illustration the orange distance vector points away from the pivot point.   This is according to engineering and physics convention, which dictates that, when a force vector is acting upon an object to produce a torque, the distance vector always points from the object’s pivot point to the line of force associated with the force vector.   The angle, ϴ, that is formed between the two vectors in our example is 90 degrees, as measured by any common, ordinary protractor.

      Next we must determine the trigonometric value for sin(ϴ).   This is easily accomplished by simply entering “90” into our calculator, then pressing the sin button.   An interesting fact is that when the angle ϴ ranges anywhere between 0 and 90 degrees, the values for sin(ϴ) always range between 0 and 1.   To see this in action enter any number between 0 and 90 into a scientific calculator, then press the sin button.

      For our angle of 90 degrees we find that,

sin(90) = 1

      Thus the formula for torque in our example, because the sin(ϴ) is equal to 1, simply becomes the product of the magnitudes of the Distance and Force vectors:

Torque = Distance × Force × sin(90)

Torque = Distance × Force × 1

Torque = Distance × Force

      Next time we’ll insert numerical values into the equation and see how easily torque can be manipulated.

_______________________________________