PHYS182A_195L-WorkEnergyTheorem_

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PHYSICS 182A/195L LAB REPORT - LAB 9: Work Energy Theorem Lab 9: Work Energy Theorem San Diego State University Department of Physics Physics 182A/195L TA: Lab partner 1: Ariana Gomes Lab partner 2: Karylle Querimit, Sarah Valenzuela Date: 10/31/23 Score: Data has been entered in blue. Theory Work is the change in energy of a system (one or more objects with mass) as the result of a force applied to that system. How exactly is force related to energy? Let’s think of an example. Recall that if a mass is held at a height h above the earth, then it has a potential energy . In order to move an object from the ground ( ) to a height , then it must have at least force acting on it while it moves from to (otherwise gravity would pull it back down). If we simply replace with in the expression for energy, we get: That is, the potential energy of the object is equal to the force we had to apply to it, times the distance , that it moved while the force was applied. More generally, any constant force applied to an object over some distance will change the energy of that object: This change in energy could be potential energy or kinetic energy, depending on the circumstances. A change in energy of a system due to a force is called Work. 1 Department of Physics
What if the force is not constant? A situation with non-constant force requires calculus. For a force that changes with , the work done by that force is This is one way to write the work-energy theorem , which tells us exactly how to compute the work done on a system by a changing force . Constant force Let’s use the work-energy theorem to calculate the work done on a cart with mass by a constant force. To generate a constant force, we will use a hanging mass attached to the cart with a pulley system. This way, a known hanging mass can be used to calculate the force applied to the system. Note that the force acting on the cart is not , it is the tension in the string. Using the free-body diagram above, and the fact that both masses must have the same acceleration, we can determine that the tension acting on the cart is: (For a detailed derivation, see the appendix.) Therefore, the work done on the cart by the pulley system is : Here, is the distance that the cart has moved. (It is also the distance that the hanging mass has fallen. Why?). Linear force Next, we’ll use the work-energy theorem to calculate the work done on a cart with mass by a linear force . To generate a force which changes linearly with x, we use a simple spring, which obeys Hooke's law:
PHYSICS 182A/195L LAB REPORT - LAB 9: Work Energy Theorem Here, is the spring constant and is the distance the spring is stretched from equilibrium. Suppose that the cart starts from some initial displacement , where the spring is taut. The cart is then released, so that the spring pulls it to the right. The force applied by the spring is changing linearly, since as the cart changes it’s x-position, the length of the spring changes and so does the force it applies. The total work done on the cart by the spring from the carts initial position to its final position is given by the work-energy theorem: This integral can be computed with the power-rule from calculus: An integral of a function is nothing but the area under the curve. Therefore, the work done by the force can also be seen as the area under the curve. In this case, the curve is a linear function with slope . The area is just the area of a triangle with base length and height , so 3 Department of Physics
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