Simple Nature - Light and Matter

zero total torque.Conversely we can have zero total force and nonzero total torque.A merry-go-round’s engine needs to supply a nonzero torque on itto bring it up to speed, but there is zero total force on it. If therewas not zero total force on it, its center of mass would accelerate!Relationship between force and torqueHow do we calculate the amount of torque produced by a givenforce? Since it depends on leverage, we should expect it to dependon the distance between the axis and the point of application ofthe force. I’ll work out an equation relating torque to force for aparticular very simple situation, and give a more rigorous derivationon page 284, after developing some mathematical techniques thatdramatically shorten and simplify the proof.Consider a pointlike object which is initially at rest at a distancer from the axis we have chosen for defining angular momentum.We first observe that a force directly inward or outward, along theline connecting the axis to the object, does not impart any angularmomentum to the object.A force perpendicular to the line connecting the axis and theobject does, however, make the object pick up angular momentum.Newton’s second law givesa = F/m ,and using a = dv/ dt we find the velocity the object acquires aftera time dt,dv = F dt/m .We’re trying to relate force to a change in angular momentum, sowe multiply both sides of the equation by mr to givem dv r = F dt rDividing by dt gives the torque:dL = F dt r .dLdt = F rτ = F r .If a force acts at an angle other than 0 or 90 ◦ with respect to theline joining the object and the axis, it would be only the componentof the force perpendicular to the line that would produce a torque,τ = F ⊥ r .Although this result was proved under a simplified set of circumstances,it is more generally valid: 22 A proof is given in example 28 on page 284o / The geometric relationshipsreferred to in the relationshipbetween force and torque.Section 4.1 Angular Momentum In Two Dimensions 255