<h1>Text Preview</h1> <p> The block moving back and forth has the following characteristics: Equilibrium point ... Maximum acceleration to the left Maximum acceleration to the right Zero velocity Maximum velocity At this point the spring exerts the greatest push toward the right. At this point there is no force acting on the mass. Recall that work is defined as the force exerted on an object times the distance that the object moves. The area under the line is the width times the height. In this case the width is the distance moved and the height is the force exerted in moving it. You can see that the area is the work done on the block by the person pushing. The equation work equals force times distance is used when force is constant. What if the force varies with distance as it does in the next frame? Here's a graph of F versus x for the spring/mass system. Notice that the amplitude to the left is shown as x-sub-a. The force is at its maximum at negative x-sub-a and the force is zero at the rest position. Imagine the block moved all the way to the left and then released. The spring pushes with a maximum force initially. This force drops to zero at the origin. The area under the diagonal line is the work done by the spring on the block. Since the base of the triangle is x and its height is k-x, the work done by the spring is equal to one half times k times x squared. The areas of the individual columns, Fx times dx, are added to give the total area, which is equal to the total work done by the spring in moving the block from its initial position, x-initial, to its final position, x-final. This is done mathematically by integrating force times displacement from x-initial to x-final, or minus k times x from x equals minus x sub m to x = 0. We obtain the result that the work done by the spring is equal to one half times k times x sub m squared. From the left-most position, the force is positive (to the right) and the motion is positive. As the block continues past the rest point, the force becomes negative (directed to the left) while the motion continues in the positive direction. Since x-sub-m equals A, the formula for work done by the spring becomes Work equals one-half k times A-squared, and as the block moves from x-sub-I equals negative A-sub-max to x-sub-f equals positive A-sub-max, the work adds up to zero. Suppose that k = 52 Newtons per meter and that the block moves from x equals minus 5.0 centimeters to its unstretched or rest position. How much work is done by the spring? ... Use the formula shown here. ...</p> <p id="TextCopyright">Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education</p>