PHYS200 Lab 6

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Athabasca University, Athabasca *

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Feb 20, 2024

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Mellino1 Athabasca University PHYS 200: Introductory Physics Lab 6: Rolling Motion

Mellino2 Introduction When a flat surface makes an angle with the horizontal, it is called an incline. The incline is a plane where a body of mass can be placed, and forces can act on that object. The forces that act on that object are the weight (directed downwards), the normal force (perpendicular to the surface), and the force of friction (parallel with the incline). When the object, such as a box, is on the plane, the force of friction must not be stronger than the force of gravity that is parallel to the incline (mgsintheta). On the other hand, a circular object will roll down by rotational motions caused by the shape of the object. The following equations presented in the lesson demonstrate the geometry and angle as components to an object’s linear acceleration: Equation 1 A = ½(g)sin theta (hoop) Equation 2 A = 2/3(g) sin theta (solid cylinder) Equation 3 A = 3/5(g)sin theta (spherical shell) Equation 4 A = 5/7(g)sin theta (solid sphere) In rolling motion, gravitational potential energy is transformed into kinetic energy as the object begins to roll. Therefore, this lab will analyze an object’s rotational motion and simultaneous translational motion. At the

Mellino3 end, the equation’s validity will be investigated with the results of the experiment. Procedure For the purposes of this lab, the materials used are a medium sized piece of cardboard about 30cm in length as an inclined surface. Each of the different size objects will also be used in this lab. The items required are:  A hoop  A solid cylinder  A spherical shell  A solid sphere  Measuring tape  A camera  Cardboard as the inclined surface To observe the motion of each of these objects, books will be used to prop up the carboard and form the surface required. The set up will be of an angle less than 10 degrees to ensure that the proper rolling motion is achieved.

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Mellino4 Photo 1. Shows the materials required. Each of the shapes chosen are a roll of tape as a hoop, a food can as the solid cylinder, a golf ball as the solid sphere, and a plastic ball as the spherical shell. Photo 2. The set up prior to releasing the objects to roll down the incline.

Mellino5 Photo 3. The angle from the horizontal of the incline is 10 degrees according to the tracker. Data Collection and Analysis

Mellino6 Photo 4. The graph represents the motion of the spherical shell rolling down the incline. The total time that the can took to roll down is 1.168 seconds. The line of best fit provides a quadratic equation in the form x = A*t^2 + Bt + C Where the parameters are x = (-9.1 x 10^-1)t^2 + (8 x 10^-2)t + (-1.2 x 10^-1) Photo 5. The solid sphere (golf ball) rolling down the incline is demonstrated by the photo.

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Mellino7 Photo 6. The graph presents the data of the golf ball rolling down the incline in 1.018s. The line of best fit given by the data is given by the equation x = A*t^2 + B*t + C x = (-8.8 x 10^-1)t^2 + (-1.1 x 10^-1) + (-4.5 x 10^-2) Photo 5. For the hoop, after several attempts with the roll of tape it was not a suitable choice for a hoop due to the uneven surface of the tape causing the object to roll at a slight angle. Therefore, switching for a perfect circle allowed for more even results.

Mellino8 Photo 6. The graph displays the data collected from the hoop rolling down the incline over a time period of 1.252s. The fit equation in the form x = A*t^2 + B*t + C is X = (-6.5 x 10^-1)t^2 + (6 x 10^-2)t + (-6.2 x 10^-2) Photo 7. The graph presents the line of best fit for the data of the solid cylinder rolling down the incline at approximately 10 degrees from the horizontal in 0.968s.

Mellino9 Where the fit equation is x = (-8.5 x 10^-1)t^2 + (-3.3 x 10^-1)t + (1.0 x 10^-1) Given the equations and the time for each object to make it’s way down the incline, the following values are found: X Shell sphere = -1.268m X Solid sphere = -1.067m X Shell cylinder = -1.006m X Solid cylinder =-1.016m Analysis Using the data given and the line of best fit equation ( y = At^2 + Bt + C) with its corresponding parameters, the next step is to find the experimental linear acceleration for each of the objects used in this lab. The kinematic equation can be used to find the acceleration: X = xo + vot + 1/2 at^2 In this equation, C represents the original position of the object, xo. In all cases, it should be 0 for time and distance, but due to limits in the accuracy in setting up the exact distance at 0, the inaccuracy can be seen by how C is slightly off from 0 in each of the cases. Spherical Shell: The parameters A, B and C are given from the graph from the previous step. A = (-9.1 x 10^-1)t^2 B = (8 x 10^-2)t C = (-1.2 x 10^-1) X = xo + vot + 1/2 at^2 (-1.268) = (-1.2 x 10^-1) + (8 x 10^-2)(1.168) + ½(a) (- 9.1 x 10^-1)(1.168)^2 a = 1.978m/s^2

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Mellino10 Solid sphere: A= (-8.8 x 10^-1) B = (-1.1 x 10^-1) C = (-4.5 x 10^-2) X = xo + vot + 1/2 at^2 -1.067 = (-4.5 x 10^-2) + (-1.1 x 10^-1)(1.018) + 1/2a(- 8.8 x 10^-1)(1.018)^2 A = 1.99m/s^2 Shell cylinder: A = (-6.5 x 10^-1) B = (6 x 10^-2) C = (-6.2 x 10^-2) X = xo + vot + 1/2 at^2 -1.006 = (-6.2 x 10^-2) + (6 x 10^-2)(1.252) + 1/2a(-6.5 x 10^-1)(1.252)^2 A = 2.00m/s^2 Solid cylinder: A = (-8.5 x 10^-1) B = (-3.3 x 10^-1) C = (1.0 x 10^-1) X = xo + vot + 1/2 at^2 -1.016 = (1.0 x 10^-1) + (-3.3 x 10^-1)(0.968) + 1/2a(- 8.5 x 10^-1)(0.968)^2 A = 2.00m/s^2 Using the angle from the set up 10.00 above the horizontal. Using this value, the results can be compared with the values given by the initial formulas for acceleration given in the introduction of the lab.

Mellino11 Spherical shell: A = 3/5(g)sin theta A = 3/5(9.8)sin(10) = 1.02m/s^2 The difference between predicted acceleration and the experimental value is 0.957m/s^2. Solid sphere: a = 5/7(g)sin theta A = 5/7(9.8)sin(10) = 1.22m/s^2 The difference between predicted acceleration and the experimental value is 0.774m/s^2. Shell cylinder: a = 1/2gsin theta A = ½(9.8)sin(10) = 0.851m/s^2 The difference between predicted acceleration and the experimental value is 1.14m/s^2. Solid Cylinder: 2/3(g)sin theta A = 2/3(9.8)sin(10) = 1.13m/s^2 The difference between predicted acceleration and the experimental value is 0.86m/s^2. Conclusion When comparing the experimental results to the expected values, there was a discrepancy between nearly every one of the values obtained for the acceleration. The first and most likely reason for this, is selecting points that were not fully accurate in the tracker. This could have caused the parameters to be off in all of the calculations. Also, the heavy objects caused inconsistencies, as well as the actual texture of the objects. For example, as mentioned previously, one of the initial materials I planned to use (the tape) did not end up giving accurate results after all and had to be changed for a different object. In the future, it would be interesting to

Mellino12 redo this lab with objects of similar mass / texture to ensure that it doesn’t cause any discrepancies. Also, when using tracker, it can be difficult to line up the axes perfectly, so getting a better angle if I had someone to help me record may have helped to get more accurate results as well. Overall, the lab demonstrates the objects acceleration due to the moments of inertia. Questions 1. For the lab analysis: Use the law of conservation of mechanical energy to prove that the linear acceleration of a hoop rolling down an inclined plane (of angle ) is given by Equation L6.1 L6.1. Refer to Figure 10.12 in the textbook for the relevant moment of inertia . 2. Show similar proofs of Equations L6.2 L6.2, L6.3 L6.3, and L6.4 L6.4.

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