1. Monday(10/19): Introduction to Chap. 8, Rotational Motion, and

Angular Quantities. HW: Read pages 197-200 and solve prob. 72, 73, 75,

76 and 78 on page 223.

2. Tuesday(10/20): Rotational Dynamics, Torque, and Moment of Inertia.

HW: Read pages 201-207 and solve prob. 81, 82, 84, 87, and 88 on

page 224.

3. Wednesday(10/21): Lab on the Rotational Motion. HW: Process lab

data and write lab report (Due Friday).

4. Thursday(10/22): Newton's 2nd Law for Rotational Motion,

Equilibrium, and Center of Mass. HW: Read pages 208-217 and solve

prob. 89, 91, 94, 97, and 99 on pages 224-5.

5. Friday(10/23): Rotating frames of Reference, Centrifugal "Force",

and The Coriolis "Force". HW: Finish all Ch. 8 Homework Problems.

6. Monday(10/26):  Review Rotational Kinematics. Check homework.

HW: Complete Chapter Review Handouts.

7. Tuesday(10/27): Review Rotational Dynamics. Check homework.

HW: Complete Chapter Review Handouts.

8. Wednesday(10/28): TEST on Ch.8 - Rotational Mechanics.

HW: Go to web-site for notes on Ch.9 - Momentum and Its Conservation.

1. When an object spins about an axis, it is said to undergo rotary motion.

The axis of rotation is the line about which the rotation occurs. For

example, the earth rotates on its axis. (1 rev. = 360^o = 2π rad.)

2. Circular motion occurs when the entire object revolves around a single

point. We say that the earth revolves around the Sun. To find the velocity

we can use v = 2πr/T , with T being the rotational period, or time for one

complete round trip.

3. Additionally, any point on a rotating object is moving in a circular path,

demonstrating circular motion with a velocity which therefore must be

tangent to the circle.

4. It is not practical to use linear motion quantities to analyze rotary or

circular motion, so we use the rotational quantities angular displacement,

angular velocity, and angular acceleration.

5. Angular displacement is given by the Greek letter, theta (θ), and

measured in radians (rad.). A radian is the measure of a central angle

which intercepts an arc equal in length to the radius of the circle.

6. For angular velocity we use omega (ω) which then would be measured in

rad/sec., and angular acceleration, alpha (α), is in rad/s² . Remember, to

convert degrees to radians, use the conversion factor that 180^o = π rad .

7. The same five linear motion equations are then transformed into the

rotational motion equations using these new quantities.

8. These now become:

Δθ = ω_avg·Δt , ω_avg= (ω_i+ω_f)/2 , ω_f = ω_i+α·Δt , ω_f² = ω_i²+2α·Δθ , and

the last one is Δθ = ω_i·Δt + ½α·(Δt)² .

9. Uniform circular motion occurs when an acceleration of constant

magnitude is perpendicular to the tangential velocity and the object

maintains a constant speed but is accelerated toward the center

of the circle.

10. This introduces the concept of centripetal (center seeking)

acceleration, a_c = v²/r .

11. If a particle moves along a curved path in such a way that the

magnitude and direction of v change with time, the particle has an

acceleration vector that can be described with two component vectors.

12. The radial component vector arises from the change in direction of v ,

which is the centripetal acceleration, a_c = v²/r and the tangential

component vector a_t = Δv/Δt, is based on the change in magnitude of v .

13. The total acceleration can be found with the vector sum of these

two accelerations which occur at right angles, so we use the

Pythagorean theorem a_total = √(a_c² + a_t²) , and θ = tan^-1(a_c / a_t).

Part II: Rotational Dynamics.

1. Recall that we have equations which relate angular and linear quantities.

For example, for arc length, s = r·θ ; for linear velocity, we have, v = ω·r,

and acceleration, a = α·r .

motion produces torque, given by the Greek letter Tau, τ, lowercase.

Torque can be computed with the equation, τ = F·r·sin(θ).

3. In the torque equation, r is the perpendicular distance from the line-of-

action of the force and the point of rotation. The angle θ is measured

between the force and the distance from the axis.

4. Torque can either start, stop, or change the direction of rotation and is

measured in Newton·meters, Nm. This tells us as we study Rotational

Dynamics that the cause of rotation is torque.

5. The Moment of Inertia of an object is a measure of its resistance to

changes in rotational motion. Just think of the ice skater, arms-out versus

arms-in.

6. Our textbook has a table of values for the Moment of Inertia of various

objects on page 206. We use the variable I to indicate this quantity which

is sometimes called Rotational Inertia or even Moment of Inertia.

In general, this is given by I = Σmr² .

7. For an object to be in complete equilibrium, it must be in both rotational

and translational equilibrium. For the rotational, this implies that the

Clockwise Torque (CWT) must equal the Counter Clockwise Torque (CCWT).

8. Translational equilibrium implies that the sum of the forces in the x

direction is zero, and the sum of the forces in the y direction is also zero.

9. All rotating objects must: (i) obey Newton's second law, (ii) possess

angular momentum which still operates under the conservation law, and

(iii) have rotational kinetic energy which can do work and also obey the

conservation law in the absence of any external forces.

10. Newton's 2nd Law for rotational motion states that the angular

acceleration of an object is directly proportional to the applied torque, but

varies inversely with the rotational inertia. The equation is τ = I·α .

11. And still, we need these steps to solve any problem in Physics:

(i) read the problem and identify the given variables

(ii) determine what you are asked to solve for

(iii) find the correct motion formula to use

(iv) use algebra to isolate the unknown

(v) substitute-in the given information and simplify.

Here are the answers to the homework problems:

Page 223: #72. .600 rad, #73. 51 rad/s,  #75. (a) 197 rad/s  (b) 492 rad

                #76. -7.54 rad/s²,   #78. (a) 2.73   (b) 1.65

Page 224: #81. 23 N,  #82. 3.8 N·m,  #84. .050 kg·m², 

                #87. F_right = 63 N, F_left = 37 N,   #88. 1.16 m