Chapter 9 Lecture Chapter 9: Rotational Motion 2016 Pearson Education, Inc. Goals for Chapter 9 To study angular velocity and angular acceleration. To examine rotation with constant angular acceleration. To understand the relationship between linear and angular quantities. To determine the kinetic energy of rotation and

the moment of inertia. To study rotation about a moving axis. 2016 Pearson Education, Inc. Rotational Motion Rigid body instead of a particle Rotational motion about a fixed axis Rolling motion (without slipping) Rotational Motion A New Way to Measure Angular Quantities Figure 9.2 The radian is dimensionless as a

ratio of lengths 2016 Pearson Education, Inc. 3 word dictionary Concepts of rotational motion Distance d angle Velocity angular velocity Acceleration angular acceleration Angle How many degrees are in one radian ? (rad is the unit of choice for rotational motion) ratio of two lengths (dimensionless)

Factors of unity or 1 radian is the angle subtended at the center of a circle by an arc with length equal to the radius. Angular velocity Other units are; We Have a Sign Convention Figure 9.7 2016 Pearson Education, Inc. Angular acceleration Relationship between linear and angular quantities

Tangential component of acceleration; For; Radial component Magnitude of Angular Quantities Kinematical variables to describe the rotational motion: Angular position, velocity and acceleration l ( rad ) R d lim

t 0 t dt ( rad/s ) ave d lim t 0 t dt

2 ( rad/s ) ave Rotational Motion Angular Quantities: Vector Kinematical variables to describe the rotational motion: Angular position, velocity and acceleration z Vector natures l

( rad ) R d k lim t 0 k k ( rad/s ) t dt d k lim t 0 k k ( rad/s 2 ) t dt x

Rotational Motion R.-H. Rule y R from the Axis (O) Solid Disk Solid Cylinder Rotational Motion Kinematical Equations for constant angular acceleration

Conversion : x , v , a 1 2 (1) 0 0t t 2 (2) 0 t 2 2 0 (3) 2 ( - 0 ) Note : constant Rotational Motion Rotational Motion

Rotation of a Bicycle Wheel Figure 9.8 at t=3.00 s? 3 complete revolutions with an additional 0.34 rev 123 at t=3.00 s? OR using See Example 9.2 in your text

2016 Pearson Education, Inc. Relationship between Linear and Angular Quantities l (1) t (like l vt ) R l ( R )t v R dv d (2) a tan R dt dt

a tan R (3) a rad atan arad v2 (R)2 2 R R R Rotational Motion Q9.4

Clicker question A DVD is rotating with an everincreasing speed. How do the centripetal acceleration arad and tangential acceleration atan compare at points P and Q? A. P and Q have the same arad and atan. B. Q has a greater arad and a greater atan than P. C. Q has a smaller arad and a greater atan than P. D. Q has a greater arad and a smaller atan than P. E. P and Q have the same arad, but Q has a greater atan than P. 2016 Pearson Education, Inc.

Q9.2 Clicker question A DVD is initially at rest so that the line PQ on the discs surface is along the +x-axis. The disc begins to turn with a constant = 5.0 rad/s2. At t = 0.40 s, what is the angle between the line PQ and the +x-axis? A. 0.40 rad B. 0.80 rad C. 1.0 rad D. 1.6 rad E. 2.0 rad

2016 Pearson Education, Inc. Clicker question On a merry-go-round, you decide to put your toddler on an animal that will have a small angular velocity. Which animal do you pick? a) Any animal; they all have the same angular velocity. b) One close to the hub. c) One close to the rim. 2016 Pearson Education, Inc. vtan= same = r11= r22

Tooth spacing is the same 2 = 1 2r1 2r2 = N1 N2 N1 N2 Rotational Motion Q9.5

Clicker question Compared to a gear tooth on the rear sprocket (on the left, of small radius) of a bicycle, a gear tooth on the front sprocket (on the right, of large radius) has A. a faster linear speed and a faster angular speed. B. the same linear speed and a faster angular speed. C. a slower linear speed and the same angular speed. D. the same linear speed and a slower angular speed. E. none of the above. 2016 Pearson Education, Inc. Energy in rotational motion and

moment of inertia: With moment of Inertia; For large it is better faster and not so much heavier Example: fly wheel. [kg m2] For general shapes; (c - factors) Hula hoop = all point on the circumference Example 9-3 Given; ; ; Now;

Energy in rotational motion and moment of inertia; Finding the moment of inertia for common shapes Q9.6 Clicker question You want to double the radius of a rotating solid sphere while keeping its kinetic energy constant. (The mass does not change.) To do this, the final angular velocity of the sphere must be A. four times its initial value. B. twice its initial value. C. the same as its initial value. D. half of its initial value.

E. one-quarter of its initial value. 2016 Pearson Education, Inc. Q9.7 Clicker question The three objects shown here all A. Object A is rotating fastest. have the same mass and the B. Object B is rotating fastest. same outer radius. Each object C. Object C is rotating fastest. is rotating about its axis of symmetry (shown in blue). All

D. Two of these are tied for three objects have the same fastest. rotational kinetic energy. Which E. All three rotate at the same object is rotating fastest? speed. A. 2016 Pearson Education, Inc. B. C.

Q-RT9.1 Clicker question Objects A, B, and C all have the same mass, all have the same outer dimension, and are all uniform. A. B. C. Rank these objects in order of their moment of inertia about an axis through its center (shown in blue), from largest to smallest.

a) BCA 2016 Pearson Education, Inc. b) ACB c) CBA d)ABC Q-RT9.2 Clicker question Objects A, B, and C all have the same mass, all have the same

outer dimension, and are all uniform. Each object is rotating about an axis through its center (shown in blue). All three objects have the same rotational kinetic energy. A. B. C. Rank these objects in order of their angular speed of rotation, from fastest to slowest. a) BCA 2016 Pearson Education, Inc.

b) ACB c) CBA d)ABC Note: In rotational motion the moment of inertia depends on the axis of rotation. It is not like a mass a constant parameter of an object Example 9.6: An abstract sculpture a) For axis BC, disks B and C are on axis; b) For axis through A perpendicular to the plane;

c) If the object rotates with around the axis through A perpendicular to the plane, what is ? Moments of Inertia & Rotational Energy Example 9.7 2016 Pearson Education, Inc. Q9.8 Clicker question A thin, very light wire is wrapped around a drum that is free to rotate. The free end of the wire is attached

to a ball of mass m. The drum has the same mass m. Its radius is R and its moment of inertia is I = (1/2)mR2. As the ball falls, the drum spins. At an instant that the ball has translational kinetic energy K, what is the rotational kinetic energy of the drum? A. K B. 2K E. none of these 2016 Pearson Education, Inc.

C. K/2 D. K/4 Problem 9.31: a) Each mass is at a distance from the axis. (axis is through center) b) Each mass is m from the axis. along the line AB) c) Two masses are on the axis and two are from the axis. (axis is alongCD) The value of I depends on the location of the axis. =1.4kg

=0.28 kg Problem 9.34: Rotational Motion Which object will win? All moments of inertia in the previous table can be expressed as; (c - number) Compare; a) For a thin walled hollow cylinder b) For a solid cylinder etc. Conservation of energy;

Small bodies wins over large bodies. On a horizontal surface , what fraction of the total kinetic energy is rotational? a) a uniform solid cylinder . b) a uniform sphere c)a thin walled hollow sphere d)a hollow cylinder with outer radius R an inner radius R/2On 9.49. Set Up: Apply Eq. (9.19). For an object that is rolling without slipping we have vcm R . Solve: The fraction of the total kinetic energy that is rotational is (1/2) I cm 2 1 1

2 2 (1/2) Mvcm (1/2) I cm 2 1 ( M /I cm )vcm / 2 1 ( MR 2 /I cm ) (a) I cm (1/2) MR 2 , so the above ratio is 1/3 (b) I cm (2/5) MR 2 so the above ratio is 2/7. K= (c) I cm (2/3) MR 2 so the ratio is 2/5. (d) I cm (5/8) MR 2 so the ratio is 5/13 Reflect: The moment of inertia of each object takes the form I MR 2 . The ratio of rotational

kinetic energy to total kinetic energy can be written as increases. Small bodies win over large bodies Rotational Motion 1 . 1 1/ 1 The ratio increases as

9.40 A light string is wrapped around the outer rim of a solid uniform cylinder of diameter 75 cm that can rotate about an axis through its center. A stone is tied to the free end of the string .When the string is released from rest the stone reaches a speed of 3.5m/sec after having fallen 2.5 m What is the mass of the cylinder?? Use coordinates where +y is upward. Take the origin at the final position of the stone, so for the stone yf 0 and yi 250 m The cylinder has no change in gravitational potential energy. The cylinder has rotational kinetic energy and the stone has translational kinetic energy. Let m be the mass of the stone and let M be the mass of the cylinder. For the cylinder 1 I MR 2 The speed of the stone and the angular speed of the cylinder are related by R 2 Solve: Conservation of energy says U i Ki U f K f Ki 0 and U f 0, so U i K f The conservation of energy

1 1 expression becomes mgyi m 2 I 2 2 2 1 1 1 2 1 1 1 I MR 2 ( /R ) 2 M 2 , so mgyi m 2 M 2 and 2 2 2 4 2

4 M 2m 2 gyi 2 2 2 2 300 kg 2 980 m/s 250 m 350 m/s 180 kg 2 350 m/s 2

Rotational Motion Clicker question You are preparing your unpowered soapbox vehicle for a soapbox derby down a local hill. You're choosing between solid-rim wheels and wheels that have transparent hollow rims. Which kind will help you win the race? a) Either kind; it doesn't matter. b) The solid-rim wheels. c) The hollow-rim wheels. 2016 Pearson Education, Inc.

solid hollow Conservation of energy in a well Find the speed v of the bucket and the angular velocity w of the cylinder just as the bucket hits the water Energy conservation; Note: free fall velocity for M<< Clicker question Two identical uniform solid spheres are attached by a solid uniform thin rod. The rod lies on a line connecting the centers of mass of the two spheres. Axes A, B, C, and D are in the same plane as the centers of mass of the spheres and of the rod. For the combined object of two spheres plus rod, rank the objects moments of inertia about the four axes, from largest to smallest.

A B C D 2016 Pearson Education, Inc. An Airplane Propeller Example 9.4 Refer to Figure 9.13 and the worked problem on page 264. 2016 Pearson Education, Inc.