Rotational Motion (Moment of Inertia) PDF

Rotational Motion (Moment of Inertia)  Objectives Distinguish between inertia and moment of inertia. Calculate the moment of inertia of various objects. Explain the meaning of the radius of gyration. Use the radius of gyration to solve for an object's moment of inerti...

Rotational Motion (Moment of Inertia)  Objectives Distinguish between inertia and moment of inertia. Calculate the moment of inertia of various objects. Explain the meaning of the radius of gyration. Use the radius of gyration to solve for an object's moment of inertia. Rotational Inertia and Rolling Which will roll down an incline with greater acceleration, a hollow cylinder or a solid cylinder of the same mass and radius? The answer is the cylinder with the smaller rotational inertia because the cylinder with the greater rotational inertia requires more time to get rolling. Rotational Inertia and Rolling A solid cylinder rolls down an incline faster than a hollow one, whether or not they have the same mass or diameter. Rotational Inertia and Rolling Inertia of any kind is a measure of “laziness.” The cylinder with its mass concentrated farthest from the axis of rotation—the hollow cylinder—has the greater rotational inertia. The solid cylinder will roll with greater acceleration.  Rotational Inertia and Rolling Any solid cylinder will roll down an incline with more acceleration than any hollow cylinder, regardless of mass or radius. A hollow cylinder has more “laziness per mass” than a solid cylinder. Rotational Inertia and Rolling think! A heavy iron cylinder and a light wooden cylinder, similar in shape, roll down an incline. Which will have more acceleration?  Rotational Inertia and Rolling think! A heavy iron cylinder and a light wooden cylinder, similar in shape, roll down an incline. Which will have more acceleration? Answer: The cylinders have different masses, but the same rotational inertia per mass, so both will accelerate equally down the incline. Their different masses make no difference, just as the acceleration of free fall is not affected by different masses. All objects of the same shape have the same “laziness per mass” ratio. Rotational Inertia and Rolling think! Would you expect the rotational inertia of a hollow sphere about its center to be greater or less than the rotational inertia of a solid sphere? Defend your answer.  Rotational Inertia and Rolling think! Would you expect the rotational inertia of a hollow sphere about its center to be greater or less than the rotational inertia of a solid sphere? Defend your answer. Answer: Greater. Just as the value for a hoop’s rotational inertia is greater than a solid cylinder’s, the rotational inertia of a hollow sphere would be greater than that of a same-mass solid sphere for the same reason: the mass of the hollow sphere is farther from the center. The shape of an object determines how easy or hard it is to spin Hinge For objects of the same mass, the longer one is tougher to spin → takes more torque  It matters where the hinge is The stick with the hinge at the end takes 4 times more torque to get it spinning than the stick with the hinge in the center.  Rotational Inertia (moment of inertia) Rotational inertia is a parameter that is used to quantify how much torque it takes to get a particular object rotating it depends not only on the mass of the object, but where the mass is relative to the hinge or axis of rotation the rotational inertia is bigger, if more mass is located farther from the axis.  How fast does it spin? For spinning or rotational motion, the rotational inertia of an object plays the same role as ordinary mass for simple motion For a given amount of torque applied to an object, its rotational inertia determines its rotational acceleration → the smaller the rotational inertia, the bigger the rotational acceleration Rotational Inertia Newton’s first law, the law of inertia, applies to rotating objects. An object rotating about an internal axis tends to keep rotating about that axis. Rotating objects tend to keep rotating, while non-rotating objects tend to remain non-rotating. The resistance of an object to changes in its rotational motion is called rotational inertia (sometimes moment of inertia). Rotational Inertia Just as it takes a force to change the linear state of motion of an object, a torque is required to change the rotational state of motion of an object. In the absence of a net torque, a rotating object keeps rotating, while a non-rotating object stays non-rotating.  Same torque, different rotational inertia Big rotational inertia Small rotational inertia spins spins slow fast Rotational Inertia Rotational Inertia and Mass Like inertia in the linear sense, rotational inertia depends on mass, but unlike inertia, rotational inertia depends on the distribution of the mass. The greater the distance between an object’s mass concentration and the axis of rotation, the greater the rotational inertia. Rotational Dynamics; Torque and Rotational Inertia Knowing that , we see that This is for a single point mass; what about an extended object? As the angular acceleration is the same for the whole object, we can write: Rotational Dynamics; Torque and Rotational Inertia The quantity is called the rotational inertia of an object. The distribution of mass matters here – these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation. Rotational Inertia Rotational inertia depends on the distance of mass from the axis of rotation.  Rotational Inertia By holding a long pole, the tightrope walker increases his rotational inertia.  Rotational Inertia A long baseball bat held near its thinner end has more rotational inertia than a short bat of the same mass. Once moving, it has a greater tendency to keep moving, but it is harder to bring it up to speed. Baseball players sometimes “choke up” on a bat to reduce its rotational inertia, which makes it easier to bring up to speed. A bat held at its end, or a long bat, doesn't swing as readily. Rotational Inertia The short pendulum will swing back and forth more frequently than the long pendulum.  Rotational Inertia For similar mass distributions, short legs have less rotational inertia than long legs.  Rotational Inertia The rotational inertia of an object is not necessarily a fixed quantity. It is greater when the mass within the object is extended from the axis of rotation. Rotational Inertia You bend your legs when you run to reduce their rotational inertia. Bent legs are easier to swing back and forth. Rotational Inertia think! When swinging your leg from your hip, why is the rotational inertia of the leg less when it is bent? Rotational Inertia think! When swinging your leg from your hip, why is the rotational inertia of the leg less when it is bent? Answer: The rotational inertia of any object is less when its mass is concentrated closer to the axis of rotation. Can you see that a bent leg satisfies this requirement? Written Work 1 Situation A long pole is rotated around three different rotation axes: central core axis, midpoint axis, and one end axis as shown in figure below. The pole is easiest to rotate about its central core axis, and it is hardest to rotate around its one end axis. Written Work Analysis 1. Which axis of rotation the pole obtains the greatest moment of inertia? 2. In which axis of rotation, the pole had the smallest moment of inertia? 3. How do the axes of rotation affect the rotation of the pole? (Hint: Relate it to the moment of inertia.) Written Work Answer 1. The pole obtains the greatest moment of inertia when the axis of rotation is on its one end. 2. The pole has the smallest moment of inertia when it is rotated about its central core 3. The closer the distribution of mass to the rotation axis, the lower its moment of inertia, hence the easier it is to rotate. As a result, it is much easier to rotate a pole about its central core than about its midpoint or one end. Rotational Inertia Single Point or Point-like objects moment of inertia can be generally express as: I= mr 2 Where: I = moment of inertia m = mass of the object r = perpendicular distance of the object from the axis of rotation Rotational Inertia Example An object having 0.1 kg is attached to a 0.5-m string and is rotated about a fixed axis. What is the moment of inertia of the object? Solution I = mr2 = (0.1 kg) (0.5 m)2 = 0.025 kg·m2 Rotating a 0.1 kg object moment of inertia is 0.025 kg·m2  Practice Problem 1 Calculate the moment of inertia of a 66.7cm-diameter bicycle wheel. The rim and tire have a combined mass of 1.25kg.  Practice Problem 1 Calculate the moment of inertia of a 66.7cm-diameter bicycle wheel. The rim and tire have a combined mass of 1.25kg. I = mr2 m = 1.25 kg r = (.667 m)/2 =.3335 m (They gave you the diameter) so I = (1.25 kg)(.3335 m)2 = 0.139 kgm2  Practice Problem 2 A small 1.05kg ball on the end of the light rod is rotated in a horizontal circle of a radius 0.900 m. Calculate the moment of inertia of the system about the axis of rotation  Practice Problem 2 A small 1.05kg ball on the end of the light rod is rotated in a horizontal circle of a radius 0.900 m. Calculate the moment of inertia of the system about the axis of rotation. The small ball can be treated as a particle for calculating the moment of inertia. I = mr2 I = (1.05 kg)(.900)2 = 0.8505 kgm2 Rotational Inertia Multiple-object System I= Σmr =(m 2 1 r2 1)+(m2 r2 2)+(m3 r2 3)… Rotational Inertia Example Three 0.1 kg objects are attached to a string and rotated about an axis. Objects 1, 2, and 3 are 0.5 m, 0.3 m, and 0.1 m, respectively, away from the axis of rotation. Calculate the moment of inertia of the system. Solution I = ∑𝑚𝑟2= (m1r21)+(m2r22)+(m3r23) = (0.1 kg) (0.5 m)2 + (0.1 kg) (0.3 m)2 + (0.1 kg) (0.1 m)2 = 0.035 kg·m2 The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation – compare (f) and (g), for example. The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation – compare (f) and (g), for example.  Practice Problem 3 A helicopter rotor blade can be considered a long thin rod, as shown in the figure below. If each of the three rotor helicopter blades is 3.75m long and has a mass of 160kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.  Practice Problem 3 A helicopter rotor blade can be considered a long thin rod, as shown in figure below. If each of the three rotor helicopter blades is 3.75m long and has a mass of 160kg, calculate the moment of inertia of the three rotor blades about the axis of rotation. I = 1/3ML2 so each rotor has a moment of inertia of: I = 1/3(160 kg)(3.75 m)2 = 1/32250 kgm2 = 750 kgm2 Three rotors would have three times this moment: I = (750 kgm2)3 = 2250 kgm2  Rotational Inertia and Gymnastics The human body can rotate freely about three principal axes of rotation. Each of these axes is at right angles to the others and passes through the center of gravity. The rotational inertia of the body differs about each axis. Rotational Inertia and Gymnastics The human body has three principal axes of rotation. Rotational Inertia and Gymnastics Longitudinal Axis Rotational inertia is least about the longitudinal axis, which is the vertical head-to-toe axis, because most of the mass is concentrated along this axis. A rotation of your body about your longitudinal axis is the easiest rotation to perform. Rotational inertia is increased by simply extending a leg or the arms. Rotational Inertia and Gymnastics An ice skater rotates around her longitudinal axis when going into a spin. a.The skater has the least amount of rotational inertia when her arms are tucked in. Rotational Inertia and Gymnastics An ice skater rotates around her longitudinal axis when going into a spin. a.The skater has the least amount of rotational inertia when her arms are tucked in. b.The rotational inertia when both arms are extended is about three times more than in the tucked position. Rotational Inertia and Gymnastics c and d. With your leg and arms extended, you can vary your spin rate by as much as six times. Rotational Inertia and Gymnastics Transverse Axis You rotate about your transverse axis when you perform a somersault or a flip. Rotational Inertia and Gymnastics A flip involves rotation about the transverse axis. a. Rotational inertia is least in the tuck position. Rotational Inertia and Gymnastics A flip involves rotation about the transverse axis. a. Rotational inertia is least in the tuck position. b. Rotational inertia is 1.5 times greater. Rotational Inertia and Gymnastics A flip involves rotation about the transverse axis. a. Rotational inertia is least in the tuck position. b. Rotational inertia is 1.5 times greater. c. Rotational inertia is 3 times greater. Rotational Inertia and Gymnastics A flip involves rotation about the transverse axis. a. Rotational inertia is least in the tuck position. b. Rotational inertia is 1.5 times greater. c. Rotational inertia is 3 times greater. d. Rotational inertia is 5 times greater than in the tuck position. Rotational Inertia and Gymnastics The rotational inertia of a body is with respect to the rotational axis. a.The gymnast has the greatest rotational inertia when she pivots about the bar. Rotational Inertia and Gymnastics The rotational inertia of a body is with respect to the rotational axis. a.The gymnast has the greatest rotational inertia when she pivots about the bar. b.The axis of rotation changes from the bar to a line through her center of gravity when she somersaults in the tuck position. Solve it! Find the moment of inertia of a disc of mass 3 kg and radius 50 cm about the following axes. a). axis passing through the center and perpendicular to the plane of the disc; b). axis touching the edge and perpendicular to the plane of the disc a.) b.) Solve it! Find the moment of inertia of a disc of mass 3 kg and radius 50 cm about the following axes. a). axis passing through the center and perpendicular to the plane of the disc; Solve it! Find the moment of inertia of a disc of mass 3 kg and radius 50 cm about the following axes. b). axis touching the edge and perpendicular to the plane of the disc Solve it! Find the moment of inertia about the geometric center of the given structure made up of one thin rod connecting two similar solid spheres as shown in Figure. Hint: The structure is made up of three objects. Solve it! Find the moment of inertia about the geometric center of the given structure made up of one thin rod connecting two similar solid spheres as shown in Figure. Hint: The structure is made up of three objects. Solve it! Find the moment of inertia about the geometric center of the given structure made up of one thin rod connecting two similar solid spheres as shown in Figure. Solve it! Find the moment of inertia about the geometric center of the given structure made up of one thin rod connecting two similar solid spheres as shown in Figure. The mass of the sphere, M = 5 kg and the radius of the sphere, R = 10 cm = 0.1 m The moment of inertia of the sphere about its center of mass is, Ic = 2/5 MR2 The moment of inertia of the sphere about geometric center of structure is’ Isph = Md2 Solve it! Find the moment of inertia about the geometric center of the given structure made up of one thin rod connecting two similar solid spheres as shown in Figure. Written Work 2 1. What is the rotational inertia (I) of the disk with a radius, R = 4 meters and a mass of 2 kg? 2. Three balls are attached to a cable and are being rotated. Ball A is 0.5 kg and is 1.0 m away from the axis of rotation. Ball B is 1.0 kg and placed 0.8 m away from the axis. Ball C, which is 0.5 m away from the axis, is 1.2 kg. Calculate the total moment of inertia of the balls. Written Work 2 3. How far is the object from its axis of rotation if it is 4 kg and has a moment of inertia 40 kg·m2? 4. The moment of inertia of the ball is 0.01 kg·m2 and is rotating around a 0.5-m string. What is the mass of the ball?