ENERGY BALANCE

Energy Balance

Energy Balance

Introduction

Energy gives us one more tool to use to analyze physical situations. When forces and accelerations are used, you usually freeze the action at a particular instant in time, draw a free-body diagram, set up force equations, figure out accelerations, etc. With energy the approach is usually a little different. Often you can look at the starting conditions (initial speed and height, for instance) and the final conditions (final speed and height), and not have to worry about what happens in between. The initial and final information can often tell you all you need to know.

The concept of work in physics is much more narrowly defined than the common use of the word. Work is done on an object when an applied force moves it through a distance. In our everyday language, work is related to expenditure of muscular effort, but this is not the case in the language of physics. A person that holds a heavy object does no physical work because the force is not moving the object through a distance. Work, according to the physics definition, is being accomplished while the heavy object is being lifted but not while the object is stationary. Another example of the absence of work is a mass on the end of a string rotating in a horizontal circle on a frictionless surface. The centripetal force is directed toward the center of the circle and, therefore, is not moving the object through a distance; that is, the force is not in the direction of motion of the object. (However, work was done to set the mass in motion.) Mathematically, work is W = F · x, where F is the applied force and x is the distance moved, that is, displacement. Work is a scalar. The SI unit for work is the joule (J), which is newton-meter or kg m/s2.

Discussion and Analysis

If work is done by a varying force, the above equation cannot be used. Figure 1 shows the force-versus-displacement graph for an object that has three different successive forces acting on it. The force is increasing in segment I, is constant in segment II, and is decreasing in segment III. The work performed on the object by each force is the area between the curve and the x axis. The total work done is the total area between the curve and the x axis.

Kinetic energy is the energy of an object in motion. The expression for kinetic energy can be derived from the definition for work and from kinematic relationships. Consider a force applied parallel to the surface that moves an object with constant acceleration.

From the definition of work, from Newton's second law of motion, and from kinematics, W = Fx = max and v f 2 = v o 2 + 2 ax, or a = ( v f 2 - v o 2)/2 x. Substitute the last expression for acceleration into the expression for work to obtain W = m ( ...