The Calculation of the Coefficient of Friction Using Projectile Motion - Report Example

Cover Page The calculation of the coefficient of Friction using projectile motion Aim of the experiment The aim of this experiment is to test and verify the relation that the frictional force acting on a body is proportional to the normal force acting between the surfaces. Hypothesis: The friction acting between two surfaces is proportional to the normal force acting between the two surfaces. If F is the force acting normally to the surface and the frictional force is f, then this implies that: Note that the relationship is hypothesized to be linear. Also, the two forces are perpendicular to each other. F acts normally downwards while f acts along the surface. Literature review The friction between two surfaces is proportional to the normal force acting on the surface. The constant of proportionality is called the coefficient of friction (Halliday, Resnick, & Krance, 2001). This is a constant that is dependent on the particular surface in question. The aim of this experiment is to verify the liner nature of this relationship and to determine the constant of proportionality hence. Description of the Experiment The arrangement of the experimental setup is as shown in the figure. 1) A 20 cm long steel tube of diameter 1 inch was adjusted on a table with pivot support so that it could be tilted. 2) The surface beneath the tube was lined with white paper on top of which a carbon paper was laid. 3) The distance between the tube and the table, the table and the ground were measured using a standard measuring tape. 4) A number of identical steel balls of measured mass (weighing machine) were taken. 5) A laser pointer was taken and adjusted so that the beam of the laser would pass through the axis of the tube. Analysis of the experimental setup The ball falls through the tube due to gravitational acceleration; however it also experiences friction while moving through the tube. We note that: While falling through the tube, the potential energy of the balls gets converted to kinetic energy. Also accounting for the loss due to friction, the conservation of energy gives that: Where W is the energy expended due to friction. (Kleppner & kolenklow, 2010) Where µ is the coefficient of friction between the ball and the surface of the tube. Thus, Solving for v, Solving for v, This gives the velocity of the ball while exiting the tube. After this, the ball is a projectile. The maximum distance travelled in the y direction must be equal to the height of the tube H. Thus, Or the time of projectile transit t is given by: And therefore the displacement of the ball in the x direction is given by: Also, solving for v in terms of x, it can be shown that: This value of velocity has been obtained by the direct application of Newton’s laws of motion. This must be the same as the value obtained by the direct energy calculation. Thus, Using the relation, But this must be proportional to the work done by the normal force acting between the ball and the surface, that is; Independent, dependent and control variables The independent variable is the variable that we change during the course of the experiment. The dependent variables change their values when we change the independent variable. Control variables are kept constant through the experiment so as to nullify any effects that they may have on the result. We note that the only variable that we change during the course of the experiment is the angle of tilt of the tube. In this experiment, this is adjusted by changing the value of L, which is related to the Tangent of the angle by the relation Ltan=H. This in turn changes both the distance of projectile fall x as well as the value of frictional force by the relation established above. Thus, the value of the energy lost due to friction W is a dependent variable and so is the distance of projectile fall (x). Both of these variables are related to the independent variable L by the relationship established above. The mass of the ball (m) and the height of the table (H) remain unaltered during the experiment and are the control variables. Similarly the height h, is adjusted by the tilt as well. Either of this can be the control variable, but in this experiment, the distance L is taken as the control variable due to the ease of using a laser pointer. This is a rather simple experiment to carry out as the number of apparatus required as small. Further, the calculations are not complex, and can be done in real time, greatly reducing the chances of error (as there is only one moving part). Units All the lengths are in cm, while then energy terms are in joules as shown in table Selection of the experimental setup We note that to measure the friction involved, a simple experimental setup is preferred. 1) The chief reason is that as the number of components and experimental parameters increase, the complexity of the system increases and the number of control variables and component forces required increases. In this setup, the only component is the tube and this remains the only surface that the ball comes into contact with. 2) Further, the manipulation and measurement of the independent and the dependent variable is easily accomplished. The tilt of the tube is adjusted by the laser pointer and the position of the ball is measured by the point at which the ball falls, using the measuring tape. 3) Also, the components are easily available and can set be set up quickly Apparatus required: Steel tube, steel balls, ruler, measuring tape, carbon paper, white paper, laser pointer. Procedure 1) A laser pointer is shown through the tube and the point where the pointer strikes the ground determines the length L. The angle is changed via adjusting L as . 2) The steel ball is kept at the inlet of the tube and is dropped ensuring that it begins from rest. 3) To make sure that there is sufficient friction between the tube and the ball, the inside of the tube is repeatedly scrubbed with sandpaper. 4) The ground near the point of approach of the ball is lined with white paper on top of which carbon paper is laid. Distances x and L are measured from the spot made by the carbon paper to the base of the tube. 5) The value of W thus calculated is then plotted versus mglcos, where ‘l’ is the length of the tube. The nature of the curve will give us the nature of the dependence of friction on the normal force of the ball. Tabulation of results 1) Mass of the ball m = 4.13 gm 2) H = 1.02 m = 102 cm 3) Length of the tube = 20 cm S No L (cm)  X (cm) W (joules) mglcos (joules) 1 176.78 30.00 31.20 10.16 7.15 2 163.34 32.00 32.40 9.95 7.01 3 140.49 36.00 36.30 9.49 6.68 4 121.65 40.00 40.30 8.99 6.33 5 102.08 45.00 46.20 8.30 5.84 . Discussion We have plotted the value of the energy expended due to friction W versus the work done due to the normal force. We note that the relationship between the two are linear with the slope given by  = 1.42. This is called the coefficient of friction between the two surfaces. This experimental design clearly makes use of Newton’s laws of mechanics to enumerate the test parameter. It can also be easily repeated by any investigator, as the number of steps required to carry out the experiment and to set up the design are very limited. The fact that the same experimental setup can be used by many different researches to yield the identical result is the demonstration of the validity of the result obtained. Result We do find that the frictional force acting between two surfaces is proportional to the normal force between them. Thus, our hypothesis is validated. Bibliography Halliday, D., Resnick, R., & Krance, K. (2001). Physics. Wiley. Kleppner, D., & kolenklow, R. (2010). Introduction to Mechanics. Cambridge University Press. University Physics with Modern Physics with Mastering Physics (12th Edition), Hugh D. Young, Roger A. Freedman, Lewis Ford, 2010 Read More