For the tension to be the same on both the dynamics cart and on the hanging mass, the acceleration of each must also be the same. This tension is responsible for accelerating the cart. Since both the hanging mass and the dynamics cart are connected by the same piece of string, which is assumed not to stretch, the same tension acts on both. From the forces illustrated in Figure 2, the following equation can be written down using Newton s second law, ΣF H = m H g T = m H a H (3) In this equation, all of the variables have the same meaning with the addition that F H is the total force on the hanging weight, m H is the mass of the hanging weight, and a H is the acceleration of the hanging weight. To obtain the tension in the string, consider the forces acting on the hanging mass. To solve for the acceleration of the cart, from Equation 2, we need to know the tension in the string, T. This means that the normal force, n, must be balanced by the weight of the cart, M c g, so that n = M c g. Since we do not expect the cart to lift off of the air track or collapse into the track, we can say that there is no net force acting in the vertical, or y, direction. For the cart, this equation is, ΣF cartx = T = M C a C (2) In this equation, F cartx is the total force on the cart in the horizontal, or x-direction, T is the tension in the string, which always pulls away from the mass in the direction of the string, M C is the mass of the dynamics cart, and a C is the acceleration of the cart in the horizontal direction. With this convention in place, equations for the total force acting on each object can be written down. This direction is indicated in both Figures 1 and 2. A useful rule is to say that the direction of motion is the positive direction. Figure 1: Forces on Dynamics Cart with Hanging Mass Figure 2: Free-Body Diagram of Forces on Cart and Hanging Massģ Since force is a vector quantity, it is necessary to decide what are the positive and negative directions so that each force can be labeled as being either positive or negative. Free-body diagrams of the forces on the cart and hanging mass are provided in Figure 2. The forces acting on the cart and hanging mass are labelled in Figure 1. The experimental setup is shown in Figure 1. For this kind of problem, it is useful to draw a diagram of the forces acting on each of the masses involved in the problem. Flat Track with Hanging Mass In the first case of this lab exercise, a cart is attached by a piece of string to another mass which is hung over the table supporting the cart track by a pulley so that as the hanging mass falls, it pulls the cart along the track. To use this equation to calculate the acceleration, all of the forces acting on an object must be added, in a vector sense, to get the total force acting on the object. The standard representation for vectors is the symbol such as F for force with a small directional arrow over it. (1) In this equation, ΣF is the sum of the forces acting on an object, m is the mass ofĢ the object, and a is the acceleration of the object. This relationship was first postulated by Isaac Newton in his Second Law of Motion, ΣF = m a. Mass, or inertial mass, is a measure of the resistance of an object to motion. This resistance to motion, which is a measure of the amount of stuff something is made of, is known as mass. The difference between the Bug and the Tank is, primarily, that the Tank is made of more stuff and is harder to get started. The missing factor that relates the force to the acceleration can be deduced by considering that any one of us would prefer to push a VW Bug rather than an army Tank. Since the car goes from a state of being at rest to a state of motion, the car must be accelerated, since acceleration is a change in velocity (v i = 0 v f = v for an acceleration value a). Theoretical Background In order to get a stalled car moving, one must push or exert a force on the car. Equipment List PASCO Dynamics Cart with Track, Set of Hanging Masses, Two Photogates, PASCO Smart Timer, PASCO Photogate Sail, Table Clamp with Rod, Dynamics Track, Attach- ment to connect Rod to Track, Two Photogate Track Braces, Angle Indicator for Track, Ruler, Nylon String, Bubble Level, mass scale. Students will calculate a theoretical acceleration value using their derived equation, and then compare their results to an experimental value. A formulated analysis of forces acting on a dynamics cart will be developed by the student. 1 Newton s Second Law Objective The Newton s Second Law experiment provides the student a hands on demonstration of forces in motion.
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