Introductory Material In earlier experiments, you studied linear motion. In this experiment you will study rotational motion. It's possible to draw analogies between linear and rotational motion as shown in Table 7-1. Linear motion Rotational Motion X position (m) 0 angle (rad) V velocity (m/s) w angular velocity (rad/s) a acceleration (m/s2) a angular acceleration (rad/s) Table 7-1. Comparison of similar quantities in linear and rotational motion. The kinematical equations when the angular acceleration a is constant are: 0 = 0+ wot + zat 2 @ = @ +at (7.1) a = do In Eq. (7.1), the initial values at time t = 0 are 0, wo, and o. Equation (7.1) is analogous to the equations for linear motion (for example the first equation above corresponds to x = xo + vet + 1/2 at for linear motion, where vo is the initial velocity and xo is the initial position of an object). Newton's Second Law, F = ma, also has an analogue in rotational dynamics: T = Id (7.2) where: T is the net torque acting on the object I is the object's moment of inertia (also known as rotational inertia). Torque has units of newton-meters (N.m) and is analogous to force; the moment of inertia has units of kilogram-meter (kg.m ) and is analogous to mass. Equation (7.2) states that the angular acceleration a of an object is proportional to the net torque (the sum of the torques) acting on the object. You will make use of Eq. (7.1) and Eq. (7.2) to determine the moment of inertia lo of a large wheel. You will spin the wheel by applying a known torque. You will then measure the angular position of the spinning wheel as a function of time and use Eq. (7.1) to determine the wheel's angular acceleration. The moment of inertia for the wheel can then be found using an equation derived from Eq. (7.2). M7-2 soos & MoodThe experimental setup is shown in Figure 7-1. The wheel is mounted horizontally in the same plane as the lab table. An accelerating torque r is applied by pulling a string around the wheel. The string is pulled with a known force because it hangs over a pulley with a known weight. The acceleration of gravity causes the mass to fall, pulling on the string, which in turn accelerates the wheel into a spin. Wheel, mounted in a horizontal plane String with tension T Rotary Motion Sensor Axis of rotation Holes for mounting additional masses in Part II Rotation of wheel, with angular velocity Mass m @ and angular hanging from acceleration a string Figure 7-1. Setup for experiment. Mass on the right hangs from a string, pulling the wheel so that the wheel spins about its axis of rotation. You will use the Rotary Motion Sensor to measure the angular velocity @ and angular acceleration a. The mass will fall, and as it falls it will pull the string and make the wheel spin. Once the mass hits the ground and stops, the applied torque is zero. If no other torques act on the wheel, then the wheel will spin forever with constant angular velocity. Of course you will observe that the wheel gradually decelerates to a stop. The wheel's rotational motion is affected by a frictional torque of acting in a direction opposite to the angular velocity w. The net torque acting on the wheel as the mass falls is the difference between rand , and Eq. (7.2) leads to the conclusion that the wheel's angular acceleration is: wok asviens of litt 290 T - If = 10 (7.3) M7-3T Inertia Here, r is found from the magnitude of the T string cross product of the position vector Ro and the tension T in the string t = ROT sing = ROT (7.4) R. As seen in Figure 7-2, the angle o between Ro and T is 90. Since the pulley is assumed to be massless, the tension is the same throughout the length of the string. Two forces act upon the mass: the Figure 7-2. This view from the top shows how gravitational force W and the string tension T (see torque is applied to the wheel. The string the free body diagram in Figure 7-3). According to applies a force T to the outside of the wheel. Newton's Second Law: Note that angle d between the vectors Ro and T is 90. This means that a torque t = Ro T is ma = W - T applied to the wheel. Substituting mg in for W and solving for T gives: T = mg - ma (7.6) Combining Eqs. (7.3), (7.4), and (7.6) yields the net torque acting on the wheel: T = MR (g - a) = Id+ if (7.7) The linear position x of the hanging mass is proportional to the angular position O of the wheel, x = CO, because their motion is linked through the string. For one revolution of the wheel, the angular position of the wheel changes by 2x radians. The string unwinds during this revolution, and the hanging mass drops a distance equal to W the wheel's circumference. The value of C is therefore: C = = = 1 rev (m) _2x Rom 0 1 rev (rad) 2x rad =Ro m/rad = x =R.0 (7.8) W BOY Figure 7-3. Side view: the hanging mass m In a similar way it can be shown that v = Row and a = Roddo. experiences two forces: W = mg downward, and Applying these relationships to Eq. (7.7) gives the equation tension T upwards. you will use during Part I of this experiment to find the wheel's moment of inertia, Io: mgRo - ma Ro = Ido + If (7.9) In the experiment, you will measure the angular acceleration o of the wheel for various hanging masses m. To analyze your experimental results, you will make a graph, with mgRo - In calculus-based you will recognize that the full expression uses a vector cross-product: t = Rox T M7-4mooRo on the vertical axis, and oo on the horizontal axis. Comparing Eq. (7.9) with y = mx + b (m is not mass in this equation), you can see that this equation has the form of a straight line, with a slope and an intercept. In the experiment, you will fit your data for mgR - ma.R VS. do to a straight line. From Eq. (7.9), you can see that the slope of this straight line is the wheel's moment of inertia, Io. That's how you will measure the moment of inertia, which is your main result. Your fit will wheel. also generate a value for the intercept in Eq. (7.9), which is the frictional torque of acting on the robin pleeno lengud Measuring the Angular Acceleration of the Wheel teirby er saur Wow nobision In previous experiments, you have measured an object's position as a function of time and then used the Notation for Part I kinematical equations for linear motion to find the object's linear acceleration. In this experiment, you will measure the quantity for the for the wheel pulley wheel's angular position O as a function of time and then use the kinematical equations for rotational motion to find the angular wheel's angular acceleration o. The "massless" pulley is position 9 Op part of a rotary motion sensor. As the string causes the pulley to spin about its axis, the rotary motion sensor reports angular the angular position Op of the pulley's axis to the computer as velocity W a function of time. Note that Op is reported in units of angular revolutions, not radians. acceleration ap One revolution of the wheel corresponds to several revolutions of the pulley because the two have different radii. Table 7-2. Notation used in Part I of this experiment. Their angular positions are related by a proportionality constant B, where 0 = BOp. Suppose the hanging mass falls a distance x, causing the pulley to make Np revolutions. Then using Eq. (7.8): (x/ Ro radians) x B =- -rad/rev (7.10) bary ow ploding Op (Np revolutions) NPR Mods You will determine the value of B in this experiment by measuring NP ous doidw over a distance x. B also relates the angular velocities and the angular accelerations of the wheel and the pulley: W = BWP do = BOP . axis of rotation If you attach a couple of objects together, their moments of Moment of a ring = Mr- inertia add. This is easiest when they have a common center of mass. Figure 7-4. Moment of inertia for a ring. M7-5Exp. M7: The Moment Of Inertia In Part II of this experiment, you will attach an additional ring to the wheel, and this will add to its moment of inertia. The ring itself has a radius r and a mass M, and its theoretical moment of inertia is: bail Idglanz I ring = Mr2 (7.11) singni To measure the experimental value, Iring , you'll measure the angular acceleration of the wheel twice, with and without the additional ring. In order for you to use these two results to yield Iring , we need to produce an equation to use that will yield the desired quantity Iring from measurements that you can make. The notation we'll use is this: is the angular acceleration without the additional ring. a is the angular acceleration with the additional ring COURT Let's rewrite Eq. (7.9) twice, with and without the additional ring: mgRo - make = (10 + Iring )a + If weroll (7.12) mg Ro - maoRo = Iodo+ If Subtracting the second equation from the first gives: atit atxs ah mRo(ao - a) = - Io(do - a) + alringon (7.13) linker Solving for Iring yields: the Iring = (10 + mR? ) do - a (7.14) Equation (7.14) is what you will use to calculate the experimental value for the moment of inertia of the extra ring, Iring . Tansibe Finally, in Table 7-3 we summarize the definitions of a few of the symbols we used above. You will use these symbols again in your worksheet. The last three quantities listed are torques, which are vectors that point in opposite directions. Their difference is the net torque acting on the wheel. In performing calculations using Eq. (7.14), be sure to double-check your result, because we've found that many students make arithmetic errors in this equation. M7-6Exp. M7: The Moment Of Inertia Notation: ap angular acceleration of rotary motion sensor. a = Bap : angular acceleration of wheel without the additional ring. a = Bap : angular acceleration of wheel with the additional ring. mgRo torque due to weight of hanging mass ma Roz torque due to tension in string without the additional ring ma Roz torque due to tension in string with the additional ring Table 7-3. List of quantities and notation used in this experiment. Pre-Laboratory Questions 1. ORDER OF MAGNITUDE ESTIMATION': The manufacturer of the Roto-Dyne wheel reports that the mass of the wheel's rotating components is 1.41 + 0.05 kg. Assuming the wheel is a uniform disk, calculate an estimate of the wheel's moment of inertia using the 2 formula I disc = - mrs. You may want to refer to the pictures of the Roto-Dyne wheel here and in the Instrument Glossary. State clearly all the assumptions that you make and show your calculations. 2. You are given a smaller wheel (Ro = 5.0 cm) that has the same moment of inertia as the Roto-Dyne. Suppose a 25-g mass hangs from a string wrapped around the wheel, and the mass is allowed to fall. Calculate the angular acceleration of the wheel using Eq. (7.9) under the assumption that there is no frictional torque. If the wheel starts at rest, how much will the wheel turn in 1.5 s? What will be the wheel's angular velocity then? 3. After reading the Experimental Procedure, identify the specific feature on a specific graph that you will use to measure the angular acceleration o of the wheel in the laboratory. send and Toames ofd'T Equipment List Short vistors awore Calipers* Meter Stick Computer* Rotary Motion Sensor* Cylindrical Masses (4) Roto-Dyne Wheel* Foam Pad a revolution Slotted Mass Set 5-g Hook Platform be negn String Logger Pro Software* * An asterisk indicates that an item is described in the Instrument Glossary. 3 Order of Magnitude Estimation is explained in the Introductory Material of Experiment M2. M7-7Roto-Dyne Wheel The Roto-Dyne wheel is ~0.4 m in diameter and made of glass-reinforced nylon. The wheel spins about its central axis on a low friction bearing. The outer rim of the wheel has a wide groove in which string or rope can be wrapped, and on the face of the wheel are tapped brackets for screwing in accessories, such as extra masses. G-10