Lecture 10 Chapter 10 Rotational Energy, Moment of Inertia, and Torque Demo SHOW DIFFERENT OBJECTS ROTATING DOWN AN INCLINED PLANE More Demos

Chapter 10 Rotation Angular variables analogous to linear motion for constant Angle Angular velocity Angular acceleration Kinetic energy including rotation Moment of Inertia Parallel axis theorem Torque Rolling without Slipping

EXAMPLE -ROTATION WITH CONSTANT ANGULAR ACCELERATION Restrict discussion to a fixed axis of rotation but also applies if the axis is in translation as long as the axis of rotation goes throuigh the center of mass

Rotation with constant angular acceleration: Consider some string wound around a cylinder with a weight attached to the string. There is no slippage between string and cylinder. θ v s r mg T ma

Red dot indicates a spot on the cylinder that is rotating as I apply a force to the massless string Front view Isometric view ω θ v s r

Define Angle (radians) θ v s r Conversion from degrees to radians is 0.0174 radians per degree or 57.3 degrees per radian For s=r theta to be valid theta must be in radians. Pull a string off a spool and show that the length of string pulled out is the arc length thru which a point on the rim moved Differentiate s=r theta All points in a rigid body have the same omega and alpha. at is the tangential componet and is zero unless the point on the rim speeds up or slows down ar is the radial component or centripetal acceleration component and is zero unless the object is turning. s and θ are not vectors

Define Angular Velocity ω Take derivative holding r constant Tangential velocity Angular velocity: Pseudo vector For s=r theta to be valid theta must be in radians. Pull a string off a spool and show that the length of string pulled out is the arc length thru which a point on the rim moved Differentiate s=r theta All points in a rigid body have the same omega and alpha. at is the tangential componet and is zero unless the point on the rim speeds up or slows down ar is the radial component or centripetal acceleration component and is zero unless the object is turning.

Use Right hand rule to get direction of ω v ω r v Counterclockwise is + for angular displacement θ and angular velocity ω.

Also called the tangential acceleration Define Angular Acceleration Also called the tangential acceleration is called the angular acceleration α is in the same or opposite direction as ω Recall there is also the radial acc.

Two Kinds of Acceleration θ s Tangential acceleration at ar Radial acceleration ω at ar a Radial and tangential accelerations are perpendicular to each other

Angular Quantities Conceptual Example : Is the lion faster than the horse? On a rotating carousel or merry-go-round, one child sits on a horse near the outer edge and another child sits on a lion halfway out from the center. (a) Which child has the greater linear velocity? (b) Which child has the greater angular velocity? Answer: The horse has a greater linear velocity; the angular velocities are the same.

Angular Quantities Objects farther from the axis of rotation will move faster. Figure 10-6. Caption: A wheel rotating uniformly counterclockwise. Two points on the wheel, at distances RA and RB from the center, have the same angular velocity ω because they travel through the same angle θ in the same time interval. But the two points have different linear velocities because they travel different distances in the same time interval. Since RB > RA, then vB > vA (because v = Rω).

Angular Quantities If the angular velocity of a rotating object changes, it has a tangential acceleration: Even if the angular velocity is constant, each point on the object has a centripetal acceleration: Figure 10-7. Caption: On a rotating wheel whose angular speed is increasing, a point P has both tangential and radial (centripetal) components of linear acceleration. (See also Chapter 5.)

For constant acceleration We have an analogous set of formulas for angular variables

Example Example 10-5: Given ω as function of time. A disk of radius R = 3.0 m rotates at an angular velocity ω = (1.6 + 1.2t) rad/s, where t is in seconds. At the instant t = 2.0 s, determine: (a) the angular acceleration. Is it constant? (b) the speed v and the components of the acceleration a of a point on the edge of the disk © the initial angular velocity (d) the angular displacement after 2.00 s. Assume initial displacement=0 Solution: a. The angular acceleration is the derivative of the angular velocity: 1.2 rad/s2. b. V = 12.0 m/s; atan = 3.6 m/s2; aR = 48 m/s2.

We call mr2 the moment of inertia I. How do we define kinetic energy of a rotating body? Consider what is the kinetic energy of a small rigid object moving in a circle? v ar θ We call mr2 the moment of inertia I. r Kinetic energy of rotation and rotational inertia Kinetic energy of rotation Suppose we have 2 bodies? How do we define it?

Calculation of Moment of Inertia Figure 10.19 Calculation of Moment of Inertia Example 10-8 Two Cases

Suppose our rotating body is a rigid rod v s θ I is called the moment of inertia about an axis through the end of the rod.

Evaluation of the rotational inertia for a rod rotating about an axis through the end perpendicular to the length L . θ

Now consider rod rotating about an axis through the center of mass of the rod θ L v .

Parallel Axis Theorem . Notice that the difference General relation v θ L v . r com General relation

Determining Moments of Inertia Example : Parallel axis. Determine the moment of inertia of a solid cylinder of radius R0 and mass M about an axis tangent to its edge and parallel to its symmetry axis. Figure 10-25. Solution: ICM = ½ MR02, so I = 3/2 MR02.

Perpendicular-axis Theorem The perpendicular-axis theorem is valid only for flat objects. Figure 10-27. Caption: The perpendicular-axis theorem.

Torque: What is it? A longer lever arm is very helpful in rotating objects. Figure 10-12. Caption: (a) A tire iron too can have a long lever arm. (b) A plumber can exert greater torque using a wrench with a long lever arm.

Torque. It is similar to force but it also depends on the axis of rotation. Why do we have to define torque? Use towel to open stuck cap on jar Door knob far from hinge Screw driver with large fat handle Lug wrench to unscrew nuts on rim for tires

Consider Torque τ for Opening a Door the lever arm for FA is the distance from the knob to the hinge; the lever arm for FD is zero; the lever arm for FC is as shown. Lever Arm Figure 10-13. Caption: (a) Forces acting at different angles at the doorknob. (b) The lever arm is defined as the perpendicular distance from the axis of rotation (the hinge) to the line of action of the force.

Torque Figure 10-14. Caption: Torque = R┴ F = RF┴.

Newton’s 2nd law for Rotation τ=Iα Suppose I have small mass at the end of a massless rod

Newton’s 2nd law for rotation Suppose the rod does have mass and we have the same axis of rotation, then you simply add the moment of inertia

Work Energy Theorem for Rotations

Figure 10.21 Do Example 10-9

Figure 10.22 Do Example 10-10

What is the acceleration of the mass m What is the acceleration of the mass m? Now we can take into account the rotation of the pulley? v +x -y M r θ ma T mg

Frictionless Sideways Atwood machine with a pulley with mass +x -y Now take into account the rotation of the pulley. new equation T1 Mp T2 a

Now include friction between block M and surface +x -y new equation Ia = (T2-T1) R T1 Mp T2 a

Rotational Kinetic Energy When using conservation of energy, both rotational and translational kinetic energy must be taken into account. All these objects have the same potential energy at the top, but the time it takes them to get down the incline depends on how much rotational inertia or kinetic energy they have. Figure 10-34.

Translational and Rotational Kinetic Energy h V V = 0 Pure translation, K =1/2 MV2 at the bottom of the plane Translation and rolling down inclined plane. at the bottom of the plane

First we have to ask what is rolling without slipping? Linear speed of the center of mass of wheel is ds/dt without slipping The angular speed ω about the com is dθ/dt. From s=θR we get ds/dt = dθ/dt R or vcom = ω R for smooth rolling motion

Rolling can be considered rotating about an axis through the com while the center of mass moves. At the bottom P is instantaneously at rest. The wheel also moves slower at the bottom because pure rotation motion and pure translation partially cancel out See photo in Fig 12-4. Also tire tracks are clear in the snow and are not smudged.

What is the acceleration of: a) An object in free fall? b) A block sliding down a frictionless inclined plane? c) A block with with friction? d) A sphere rolling down an inclined plane? e) The tip of a tall pole when it hits the ground

Figure 10.36

What is the acceleration of a sphere smoothly rolling down an inclined plane? Positive x

What is the acceleration of a sphere smoothly rolling down an inclined plane? Find torque about the com τnet= Iα Positive x Solve for fs

This will predict which objects will roll down the inclined faster. Solve for This will predict which objects will roll down the inclined faster.

Let θ= 30 deg Sin 30 = 0.5 shape Icom 1+ Icom/MR2 acom sphere 2/5 MR2 1.4 0.71x g/2 disk 1/2MR2 1.5 0.67 x g/2 pipe MR2 2.0 0.50 x g/2

Yo-yo rolls down the string as if it were inclined plane at 90 degrees Instead of friction, tension in the string holds it back The moment of inertia Icom is that of the yo-yo itself.

Figure 11.50

Rules τ=Iα is useful to solve problems. Valid when: When the axis of rotation is fixed in an inertial reference frame When the axis of rotation passes through the center of mass and is fixed in direction even if it is accelerating

Other topics When does a bowling ball stop sliding and roll without slipping Why does a rolling sphere slow down?