Moment of Inertia In general, when a force is applied to a body, the motion of the system can be broken down into translational motion of the center of mass and rotational motion about the center of mass. In this experiment, a weight is attached to a cylinder that will rotate about its axis as the weight descends (Fig. 1). Since the axis is fixed, there is no translational motion. Newton s second law will also apply to rotating body. However, the law is simplified if the variables are rotational variables. Newton s second law becomes τ = Iα (1) where τ is the torque expressed in dyne-cm or N-m, I is the moment of inertia in g cm 2 or Kg m 2, and α is the angular acceleration in radians/sec 2. The moment of inertia, like mass in linear motion, is the inertial property of the system which opposes a change in angular motion. It is possible to calculate the moment of inertia of a body if the distribution of the mass is known. The moment of inertia of objects used in this experiment is given as follows: R R Solid Cylinder I = ½ MR 2 (2) Thin Ring I = MR 2 (3) For the experimental set up as shown in Figure 1, the linear acceleration (a) of a descending weight (mg) is measured by using a smart pulley. If the string does not slip on the step pulley, then the linear acceleration of the weight will be the same as the tangential acceleration of a point on the rim of the step pulley. The tangential acceleration is related to the angular acceleration by Equation (5) α = a r (5) *where r is the radius of the step pulley the string is wrapped around. 48
As shown in Fig. 1, the tension (T) of Fig. the string 1 produces a torque (T r) on the cylinder. The equation of motion of the cylinder is given by Eqn. 6. Likewise Eqn 7 is the equation of motion for the descending mass. τ = Tr = Iα (6) mg T = ma (7) Solving Eqn 7 for T and substituting into Eqn. 6 will give an equation for the torque (τ). τ = mr(g a) (8) If torque (Eqn. 8) is plotted versus angular acceleration (Eqn. 5), then a straight line whose slope is the moment of inertia should result. The radius r in the equations is the radius of the pulley the string is wrapped around, not the radius of the larger platter. Procedure Record the mass of the hanger and the radius of the step pulley. Thread the string through the holes in the step pulley so that the pulley will wind up on the middle pulley. Stretch the string over the smart pulley and attach it to a hanging mass. Make sure the string is as parallel to the disc as possible. Open Logger Pro/ Open/ Experiments/Probe and Sensors/Photogates/ Pulley.xmlb. Place 40 g on the 5g weight hanger. On the computer screen, click the mouse on the "START" button and release the cart. Click "STOP" on the 49
computer screen when the weight hits the floor. Click on the "Velocity Graph", then click on the R= icon at the top of the graphs. The computer program automatically calculates the slope and draws the best fitting straight line through the data points. The slope of the graph is the acceleration of the mass. Record this value in the data table. Repeat the procedure in steps (2) and (3) with 80g, 120g, 160g and 200g on the 5g-weight hanger. Measure the mass of cylinder and its radius. Calculate its moment of inertia using Equation Show your work in the data section. Place the cylinder on top of the spindle and repeat the five measurements done in steps 2-5 above. 6. Measure the mass of ring and its outside radius and inside radius. The average radius is the sum of the outside radius and inside radius divided by two. Calculate its moment of inertia using Eqn. Show your work in the calculation section. Take the cylinder off and place the ring on top of the spindle. Repeat the five measurements done in steps 2-5 above. 7. Calculate the torque using Equation 8 and the angular acceleration using Equation Record these values in the tables. 8. Plot torque versus angular acceleration for disc with step pulley only, the disc with step pulley plus the cylinder, and the disc with step pulley plus the ring. In each graph, use a linear curve fit to find the slope of the curve and click set the intercept to zero box (The box is just above the display equation on graph box. The slope of the curve is the moment of inertia of the system. Therefore, the experimental value of the moment of inertia of the cylinder is just the difference between the slope for the disc with the step pulley plus the cylinder and the disc with step pulley only. Find the experimental values for the moment inertia of the cylinder disk and the ring. Find a percent error between the experimental value and the calculated value. 50
Data: Disc with the step pulley Radius of step pulley (Middle pulley) = Moment of inertia of Step Pulley (Slope of graph) = Disc with step pulley plus solid cylinder Mass of Cylinder = Radius of Cylinder (R)= Calculated Moment of Inertia = Slope = Exp. value of Moment of Inertia of the cylinder = % error = Show work for the calculated moment of inertia of a cylinder 51
Disc with step pulley plus ring Mass of Ring = Outside Radius= Inside Radius = Average Radius of Ring (R) = Calculated Moment of Inertia = Exp. value of Moment of Inertia of the ring = Slope = % error = Show work for the calculated moment of inertia of a ring 52