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Home Conservation Laws Conservation of Angular Momentum Relationship between force and torque  
See also: Torque distinguished from force, The torque due to gravity, Discussion  Angular Momentum  
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Relationship between force and torqueHow do we calculate the amount of torque produced by a given force? Since it depends on leverage, we should expect it to depend on the distance between the axis and the point of application of the force. We'll derive an equation relating torque to force for a particular very simple situation, and state without proof that the equation actually applies to all situations.
Consider a pointlike object which is initially at rest at a distance r from the axis we have chosen for defining angular momentum. We first observe that a force directly inward or outward, along the line connecting the axis to the object, does not impart any angular momentum to the object. A force perpendicular to the line connecting the axis and the object does, however, make the object pick up angular momentum. Newton's second law gives
and assuming for simplicity that the force is constant, the constant acceleration equation a = Δv/Δt allows us to find the velocity the object acquires after a time Δt, Δv = FΔt
We are trying to relate force to a change in angular momentum, so we multiply both sides of the equation by mr to give
Dividing by Δt gives the torque:
If a force acts at an angle other than 0 or 90 °with respect to the line joining the object and the axis, it would be only the component of the force perpendicular to the line that would produce a torque,
Although this result was proved under a simplified set of circumstances, it is more generally valid:
relationship between force and torqueThe rate at which a force transfers angular momentum to an object, i.e., the torque produced by the force, is given by τ  = rF , where r is the distance from the axis to the point of application of the force, and F? is the component of the force that is perpendicular to the line joining the axis to the point of application.
The equation is stated with absolute value signs because the positive and negative signs of force and torque indicate different things, so there is no useful relationship between them. The sign of the torque must be found by physical inspection of the case at hand. From the equation, we see that the units of torque can be written as newtons multiplied by meters. Metric torque wrenches are calibrated in N·m, but American ones use footpounds, which is also a unit of distance multiplied by a unit of force. We know from our study of mechanical work that newtons multiplied by meters equal joules, but torque is a completely different quantity from work, and nobody writes torques with units of joules, even though it would be technically correct.
Sometimes torque can be more neatly visualized in terms of the quantity r? shown in figure q, which gives us a third way of expressing the relationship between torque and force: τ = rF . Of course you would not want to go and memorize all three equations for torque. Starting from any one of them you could easily derive the other two using trigonometry. Familiarizing yourself with them can however clue you in to easier avenues of attack on certain problems.


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