10/05/09 - Work and Energy

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10/05/09 - Work and Energy

			Work

			In physics, work has a special definition: a force must be exerted to move a mass some distance (displacement). If the object moves in the direction of the force, then the work done is simply the product of the force exerted and the distance moved (displacement): W = F D. The unit of work is a Newton-meter, which is called a Joule. A force of 5 N exerted over a displacement of 3 m does (5N)(3m) = 15 J of work. Note that work is not a vector, but both force and displacement are vectors.

			If the object moves in a different direction than the applied force, only the vector component of the force in the direction of the motion does work: W = F D cos θ.

			Kinetic Energy

			Kinetic energy literally means energy of motion. Any mass which is moving has kinetic energy. The amount of kinetic energy depends on both the mass and the speed of the moving object. In fact, kinetic energy (KE) for a mass m moving at speed v equals (1/2)mv2. Note that kinetic energy is not a vector.

			Work and Kinetic Energy

			If we apply a constant unbalanced force F to a mass and it has a displacement of D in the direction of the force, then we've done an amount of work W = F D. The constant unbalanced force F causes the mass to accelerate at a constant rate a = F/m. The constant acceleration causes the velocity of the mass to increase linearly, as shown in the graph below:

			This says that the displacement is the area under the curve and D = (1/2)(at)(t) = (1/2)(at)2/a = (1/2)v2/a = (1/2)v2/(F/m) = (1/2)mv2/F. This means that the work W = F D = (1/2)mv2 = KE. In other words, the work that was done on the object increased its kinetic energy. In fact, all of its kinetic energy came from the work that was done on the mass.

			The kinetic energy in the moving mass can then do work on another object by exerting a force on it and moving it some distance. Hence, the original work that was done is not lost, it was stored in the moving object as kinetic energy and can be recovered at a later time.

			This leads to the concept of Conservation of Energy: Energy is neither created nor destroyed, it simply changes form.

			Work and Potential Energy

			Potential energy is stored energy and comes in many forms.

			Gravitational potential energy

			If we lift a mass m by a height h, we need to exert a force on it at least equal to its weight mg. Hence the work we do to lift it is W = F D = mg(h) = mgh.

			If we then let the object fall from height h back to the ground, that potential energy is converted into kinetic energy as the object accelerates downward.

			From the graph we see that -h = (1/2)(-gt)(t) = -(1/2)v2/g. Hence the potential energy (PE) = mgh = (1/2)mv2. As the mass falls under the influence of gravity, its potential energy is transformed to kinetic energy, and in the absence of friction or air resistance, no energy is lost. The work we did initially to raise the mass, was stored as potential energy, which was then released as kinetic energy, the same amount of kinetic energy we would have gotten if we had applied the work to the mass to accelerate it directly.

			Homework:

			Read the first part of Chapter 6, pp. 165-182.

			Answer the questions in HW 10 on WebAssign: http://webassign.net/login.html

			Link to Exam 1-5 Solutions

			http://www.civerson.com/P210/Chapters_1_5_Exam_ANS.pdf