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**Unformatted text preview: **Connexions module: m14095 1 Work - kinetic energy theorem * Sunil Kumar Singh This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † Abstract The kinetic energy of a particle changes by the amount of work done on it. Work is itself energy, but plays a speci c role with respect to other forms for energy. Its relationship with di erent energy forms will automatically come to the fore as we investigate them. In this module, we shall investigate the relationship between work and kinetic energy. To appreciate the connection between work and kinetic energy, let us consider a block, which is moving with a speed "v" in a straight line on a rough horizontal plane. The kinetic friction opposes the motion and eventually brings the block to rest after a displacement say "r". A block is brought to rest by friction Figure 1: Friction applies in opposite direction to displacement . Here, kinetic friction is equal to the product of coe cient of kinetic friction and normal force applied by the horizontal surface on the block, F k = μ k N = μ k mg Kinetic friction opposes the motion of the block with deceleration, a, : a = F k m = μ k mg m = μ k g * Version 1.16: Sep 12, 2009 11:22 am GMT-5 † http://creativecommons.org/licenses/by/2.0/ http://cnx.org/content/m14095/1.16/ Connexions module: m14095 2 Considering motion in x-direction and using equation of motion for deacceleration, v 2 2 = v 1 2- 2 ar , we have : 0 = v 2- 2 ar v 2 = 2 ar = 2 μ k gr Thus, kinetic energy of the block in the beginning of motion is : K = 1 2 mv 2 = 1 2 x m 2 μ k gr = μ k mgr A close inspection of the expression of initial kinetic energy as calculated above reveals that the expression is equal to the magnitude of work done by the kinetic friction to bring the block to rest from its initial sate of motion. The magnitude of work done by the kinetic friction is : W F = F k r = μ k mgr = K This brings up to a new de nition of kinetic energy : De nition 1: Kinetic energy Kinetic energy of a particle in motion is equal to the amount of work done by an external force to bring the particle to rest. 1 Work - kinetic energy theorem Work - kinetic energy theorem is a generalized description of motion - not speci c to any force type like gravity or friction. We shall, here, formally write work - kinetic energy theorem considering an external force. The application of a constant external force results in the change in kinetic energy of the particle. For the time being, we consider a "constant" external force. At the end of this module, we shall extend the concept to variable force as well. Let v i be the initial speed of the particle, when we start observing motion. Now, the acceleration of the particle is : A force moves the block on a horizontal surface Figure 2: Force does work on the block....

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