OpenStax College Physics for AP® Courses, Chapter 9, Problem 2 (Test Prep for AP® Courses)

This is College Physics Answers with Shaun Dychko. Five forces of equal size labeled A through E are applied to this object and the object is pinned at this position P here— this is a pivot point— and the question is what force will produce the greatest torque? Well, torque depends on the distance from the pivot point to the point where the force is applied— that's the lever arm r— and then multiplied by the size of the force, which is the same for each of them so that's not a distinguishing feature between the torque's produced by each... they all have the same force and then we also multiply by sin of the angle between the force and this lever arm that connects the point of application of the force to the pivot point. So the angle here for force A is here and it's some fairly large angle and we can see that this distance here is not very large compared to say... this distance here... this is the largest length lever-arm and it's nearly 90 degrees; 90 degrees produces the maximum in sin Θ so the closer Θ is to 90, the greater it will be and its maximum is... sin of 90 is 1 and that's the maximum for sin Θ. And so the answer's gonna be (d) because it has the longest length lever-arm and it has the angle closest to 90 although I guess to be more precise, the lever-arm between force E and P is of greater length but this angle is almost 180 degrees like if it went to here that would be 180 so it's missing... it's off from 180 by only this amount so that's close enough to being 180 and the sin of 180 is 0 so this large angle, which is very much more than 90... almost 180 is going to more than compensate for the longer lever-arm. So despite the longer lever-arm for force E, this sin of such a large angle... such an angle close to 180 will make this overall torque small compared to the torque produced by force D. Force C and force B produce no torque at all because these forces are 180 degrees exactly to the lever arm.