Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Hunting oscillation wikipedia , lookup
Hooke's law wikipedia , lookup
Eigenstate thermalization hypothesis wikipedia , lookup
Kinetic energy wikipedia , lookup
Internal energy wikipedia , lookup
Mass versus weight wikipedia , lookup
Rubber elasticity wikipedia , lookup
PHYSICS EXPERIMENTS — 131 13-1 Experiment 13 Elastic Potential Energy of a Stretched Object The purpose of this experiment is to determine the work done in stretching an elastic object. This work done may, alternatively, be thought of as an increase in elastic potential energy stored in the object. This determination is then used to make predictions on the vertical motion of the stretched object, utilizing the concept of transformation of energy from elastic potential to gravitational potential. Preliminaries. Stretching any object, whether a rubber band or a steel spring, requires a force felas. Application of this force over a distance results in work W done to store energy as elastic potential energy, Uelas, in the stretched object. If the object is stretched from a length X0 to a length X, then W = U elas = at the bottom. As the hanging mass is not moving, felas is simply equal to the hanging weight Mg by Newton’s First Law. The stretch y = X−X 0 is determined as M is varied and the data represented in a graph of felas versus y. See Figure 2, in which a smooth curve has been fitted to the individual data points. Typically, this graph will have a linear section near the origin, (once the slack is taken out of the rubber). This is known as the elastic region, where Hooke's Law holds for small elongations. f elas X ∫ f elasdx X0 In any real-world system, there is no simple formula available to describe the stretch force and an analytic solution to the integral is not possible. You will measure and graph the stretch force felas on a thick rubber band as a function of the stretch distance. The elastic potential energy can then be determined from the area under the curve of this graph. The technique for measuring the stretch force is shown in Figure 1. The rubber band is suspended vertically and a mass M is hung (a) (b) Ball Xo X M Figure 1. Determination of elastic force y Figure 2. Elastic force vs stretch of an elastic cord The area under the curve between y=0 and y=L, where L is the maximum amount that the object is stretched can be determined in many ways. One method is to approximate the area by a series of rectangles. The range from y = 0 to y = L is divided into rectangles of equal width Δy. Example: if L = 55 cm, then 11 rectangles could be used since this gives each rectangle a simple width of Δy = 55cm/11 = 5 cm. The height of each rectangle is picked to give a reasonable fit to the real area. The height has units of force and is read off the vertical axis: f1, f2, f3, ... Some rectangles are shown in Figure 3. 13-2 PHYSICS EXPERIMENTS —131 f elas Procedure. f4 f3 f2 f1 Δy Δy L y . Figure 3. Approximating the area under the curve The height of each rectangle is fi and the area of each rectangle is fiΔy. The total area is Σ(fiΔy) = ΔyΣfi. We have “numerically integrated” the curve to yield Uelas = ΔyΣfi. Remember, the fi are read from the graph, and are not the measured data points of felas. The rubber band may be used as a slingshot. If it is held vertically, stretched an amount y=L and suddenly released, then the rubber band is capable of projecting a mass up into the air. In this projection, some of the stored Uelas is converted into kinetic energy of the projected object. This kinetic energy gets converted into gravitational potential energy as the object rises into the air and slows. The gravitational potential energy is given by Ugrav = mgh, where m is the object mass, g is the acceleration due to gravity, and h is the maximum height reached by the object above the launch point. Exactly how high the projected object flies depends on more than just the original elastic potential energy. The maximum height attained also depends on the retarding effects of air friction, the exact manner in which the rubber band is released, energy lost to heat in stretching the rubber band, and a host of other effects. These effects are difficult to account for, but they all act to reduce the amount of energy transferred to gravitational potential energy by an amount Wloss. Conservation of energy gives mgh = Uelas - Wloss. You will determine the rise height h to determine Wloss . • Put the band over the corner of the ruler, pull back, and release so that the ball at the end of the band is launched vertically. Then by trial determine how much you should stretch the band so that the ball just makes it to the ceiling. Record the distance L band had to be stretched to reach the ceiling and measure the height h from the initial position of the ball to the ceiling. • Mount the rubber band vertically. Determine and record its relaxed, unstretched length X0. Now suspend some mass M from the band and determine the stretched length X. Repeat for increasing values of M until the band is stretched the same amount L that you had when it was projected and just made it to the ceiling. Record the data in a table. • Compute the stretch force felas and the stretch y. Add these values to the table. • Make a plot of felas vs y. Find the work W=Uelas done in stretching the band a distance L by determining the area under the curve. • Predict how high H the object should have flown if there were no energy losses, Wloss = 0. • Ignoring Wloss is unrealistic. Use conservation of energy to determine Wloss and the fraction f of the original energy that does not go into gravitational potential energy. f = Wloss/Uelas = (Uelas-mgh)/Uelas {Show that f = 1- (h/H).}