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Physics 115: Spring 2005 Rolling Something Introduction The purpose of this laboratory is to investigate the dynamics of a wheel and axle rolling without slipping down an incline from two standpoints: the dynamical equation, τ = Iα; and conservation of energy. Equipment Brass disk on axle, inclined track, meter stick, stopwatch, scale, and calipers. Theory For a system consisting of a wheel mounted on an axis which is allowed to roll without slipping down an inclined plane, it is possible to calculate the acceleration of the wheel/axle system down the plane using either Newton's laws for translation and rotation or energy conservation. First, let us consider the analysis using Newton's laws. Refer to Figure 1. Since, for a rolling body, the axis of rotation is not fixed, we can consider the motion a combination of a translation of the center of mass plus a rotation about the center of mass. It is necessary, therefore, to consider Newton's second law for both the linear and the rotational motion involved in this system. Let us first consider the linear motion described by the equation ∑F = ma. If we consider the linear motion of the axis (which remains parallel to the incline), we see that the sum of the forces in this direction is just mgsinθ - f , where f is the (static) friction force between the inclined plane and the axle. Thus, the ∑ F = ma = mg sin θ − f above equation becomes: Likewise, in considering the rotational motion of the wheel with moment of inertia I, we have Στ = Iα. Since the rotational motion of the wheel about its axis is due to the friction force, this equation can be written: ∑τ = I α = f r where r is the radius of the axle. 1 We also know that the linear and rotational acceleration are coupled by the equation a = αr. From here, it is possible to combine Equations 1 and 2 to obtain the following result: a= mg sin θ I (m + 2 ) r By determining the moment of inertia of the object, you can calculate a theoretical value for the acceleration using equation 3. Since the acceleration is constant, you can measure the time it takes the wheel/axel system to travel a known distance and then use the equations for constant acceleration to find the acceleration experimentally. Alternatively, you can equate the total change of potential energy of the wheel/axle system mgh to the total change in kinetic energy. In this case, there is both kinetic energy ½mv2 associated with the linear motion and kinetic energy associated with the rotational motion ½Iω2. Procedure 1) Angle of the incline: From equation (3), you can see that knowing the angle of the incline that the disk will roll down is essential in determining the amount of force from gravity the disk will experience. To determine this angle accurately, carefully measure two heights along the incline, and the length between those heights. Use your knowledge of trigonometry to calculate the angle of the incline which the disk will roll down. Show your work below: Angle of Incline____________ 2) Calculate the maximum amount of error in your calculation of the Angle of Incline. (Hint: Using Propagation of error is quite difficult here. Try to sub in the max error and find the maximum effect in the results.) Show your work on back of this page. Maximum error in angle of incline____________ 2 3) Moment of Inertia: To find the total moment of inertia, I, of the system, you need to examine both the disk, and the axle on which it is attached. Each of these elements has a mass, and, as such, a moment of inertia which you must calculate separately. The formula for the moment of inertia of a solid disk of radius R is (4) I=mR2/2 Take apart the disk and the axle, and use the balance to determine the weight and radius of each: Weight of the axle____________ Radius of the axle____________(To measure this value as accurately as possible, use the vernier calipers provided instead of a typical ruler) (show work here) Moment of inertia, I, of the axle_______________ Weight of the disk____________ Radius of the disk____________ (show work here) Moment of inertia, I, of the disk_______________ Total moment of Inertia (Idisk + Iaxle)____________ 3 4) Determine a theoretical value for the acceleration of the disk/axle: Now use equation (3) to find a theoretical value for the acceleration of the disk/axle. Remember, the “r” in equation (3) refers to the radius of the axle, since the frictional force f acts on the axle, not the disk. Show your calculations below: atheoretical__________________ 5) Time to experiment!!!: Place the disk/axle combination at the top of the track and use a stick or ruler to hold it in place. Use a clean, quick release and allow the disk and axle to roll down the track, making sure that the axle is not slipping on the track. Make several measurements of the time it takes to reach the bottom of the track and discard any wildly inconsistent results. Take the average of your remaining measurements and use it in the calculations for final velocity and acceleration: Table: Part IV Time to reach bottom (seconds) Trial 1 Trial 2 Trial 3 Extra trials: Average time 4 6) Find the actual acceleration of the disk/axle: Using the equation x=vinitialt+.5at2 Calculate the experimental acceleration of the disk/axle: (5) aexperimental________________ 7) Find the final velocity of the disk/axle: Using the equation vfinal=aexperimental t Find the experimental final velocity of the disk/axle: (6) vfinal____________ 8) Remember, there is a difference between the rotational velocity, ω, and the translational velocity, v, which you just calculated. Come up with an expression for v in terms of ω and whatever other variables you need. Justify that expression. 5 9) Determine the rotational velocity, ω final, at the bottom of the incline: ω final________________________ 10) Is energy conserved? From the initial potential energy, mgh, and the final translational and rotational velocities, determine whether or not the total energy of the system was conserved as the disk/axle rolled down the incline (show work below): Is the total energy conserved_____________ If not, how much energy is lost____________ Is most of the kinetic energy in the form of rotation of the disk/axle about its center of mass, or in the form of translation___________ Calculate the relative percentages of each: Percentage of Kinetic Energy made up by Rotational Kinetic Energy_______ by Translational Kinetic Energy___________ 6 11) One last question: In the theory section, we included the forces due to friction in the analysis using Newton’s laws, but we did not include any friction in the analysis using conservation of energy. Why doesn’t friction have to be included in the energy analysis? 7