1. Chapter 10 - Rotational Kinematics
Physics Notes
Mr. Seaney

2. Rotary Motion: Motion around an Axis
Rotary motion is the motion of a body about an internal axis (wheel on an axle).
May be constant or variable
Rotary Motion Intro

3. 10.1 Angular Position
Angular Position: The amount of rotation from a reference position, described with a positive or negative angle.
Radians are typically used to describe angular position. The two lines used to measure the angle radiate from the point about which the wheel rotates.
The axis of rotation is a line also used to describe an object’s rotation. It passes through the wheel’s center, since the wheel rotates about that point, and it is perpendicular to the wheel.
The angle in radians equals the arc length s divided by the radius r.
2π radians equals one revolution around a circle, or 360°.
One radian equals about 57.3°.
To convert radians to degrees, multiply by the conversion factor 360°/2π. To convert degrees to radians, multiply by the reciprocal: 2π/360°.
The Greek letter θ (theta) is used to represent angular position.
Angular position

4. 10.2 Angular Displacement
Angular displacement: Change in angular position.
To calculate angular displacement, you subtract the initial angular position from the final position.
One revolution equals 2π radians of angular displacement.
The angular displacement is the total angle “swept out” during rotational motion from an initial to a final position.
If a wheel turns counterclockwise three complete revolutions, its angular displacement is 6π radians.
Angular displacement

5. 10.3 Angular Velocity
Angular velocity (ω) is the time rate of angular displacement measured in radians per second.
Average angular velocity equals the total angular displacement divided by the elapsed time
Instantaneous angular velocity refers to the angular velocity at a precise moment in time. It equals the limit of the average velocity as the increment of time approaches zero.
The sign of angular velocity follows that of angular displacement: positive for counterclockwise rotation and negative for clockwise rotation. The magnitude (absolute value) of angular velocity is angular speed.
Angular velocity

6. Angles in circular motion
Radians are units of angle
An angle in radians = arc length / radius
1 radian is just over 57º
There are 2π = 6.28 radians in a whole circle

7. Angular Velocity T = 2π/ω = 1/f f = ω/2π
Angular velocity ω is the angle turned through per second
ω = Δθ/Δt = 2π / T
2π = whole circle angle
T = time to complete one revolution
T = 2π/ω = 1/f
f = ω/2π

8. 10.4 Angular Acceleration Angular acceleration
Angular acceleration (α) is constant rate of change of angular velocity in radians per second squared (rad/s2)
Linear Rotary
Angular acceleration

9. 10. 5 Sample problem: a clock 10
10.5 Sample problem: a clock  Interactive checkpoint: a potter’s wheel

10. 10.7 Equations for rotational motion with constant acceleration
Sample problem: a carousel
 Interactive checkpoint: roulette
Interactive problem: launch the rocket

11. 10.11 Tangential Velocity
Tangential velocity: The instantaneous linear velocity of a point on a rotating object.
Tangential speed equals the product of the distance to the axis of rotation, r, and the angular velocity, ω.
The units for tangential velocity are meters per second.
The direction of the velocity is always tangent to the path of the object.
Tangential velocity

12. 10.12 Tangential acceleration
Tangential acceleration: A vector tangent to the circular path whose magnitude is the rate of change of tangential speed.
The magnitude of the tangential acceleration vector equals the rate of change of tangential speed.
The tangential acceleration vector is always parallel to the linear velocity vector. When the object is speeding up, it points in the same direction as the tangential velocity vector; when the object is slowing down, tangential acceleration points in the opposite direction.
The centripetal and tangential acceleration vectors are perpendicular to each other.
An object’s overall acceleration is the sum of the two vectors. To put it another way: The centripetal and tangential acceleration are perpendicular components of the object's overall acceleration.
Tangential acceleration can be calculated as the product of the radius and the angular acceleration.
Tangential acceleration

13. 10.13 Interactive checkpoint: a marching band

14. 10.14 Tangential and centripetal acceleration
The overall acceleration equals the vector sum of the centripetal and tangential accelerations.
The two vectors are perpendicular, so they form two legs of a right triangle.
The Pythagorean theorem can be used to calculate the magnitude of the overall acceleration.
The direction of the overall acceleration, measured from the centripetal acceleration vector (or the radius line), can be calculated using trigonometry.
Combining centripetal and tangential acceleration

15. 10.15 Vectors and angular motion
Right hand rule
10.16 Interactive summary problem: 11.6 seconds to liftoff

16. Precession
Spinning objects produce angular velocities perpendicular to the axis of rotation. The weight of the object in a gravitational field produces torque. This torque causes angular acceleration and the axis of rotation itself begins to rotate. This is precession.
Bicycle wheel suspended on a rope
Earth’s spin produces precession

17. Periodic Motion
Regular vibrations or oscillations repeat the same movement on either side of the equilibrium position f times per second (f is the frequency)
Displacement is the distance from the equilibrium position
Amplitude is the maximum displacement
Period (T) is the time for one cycle or or 1 complete oscillation
Intro to SHM

18. Producing time traces
2 ways of producing a voltage analogue of the motion of an oscillating system
SHM
Graph

19. The conical pendulum
The vertical component of the tension (Tcosθ) supports the weight (mg)
The horizontal component of tension (Tsinθ) provides the centripetal force

20. Time traces

21. Simple Harmonic Motion
Simple harmonic motion is linear motion in which the acceleration is proportional to the displacement from an equilibrium position and directed toward that position
Period is independent of amplitude
Same time for a large swing and a small swing
For a pendulum this only works for angles of deflection up to about 20º
Period & Frequency

22. Simple Harmonic Motion cont…
Gradient of displacement v. time graph gives a velocity v. time graph
Max veloc at x = 0
Zero veloc at x = max
Angular frequency

23. Simple Harmonic Motion cont…
Acceleration v. time graph is produced from the gradient of a velocity v. time graph
Max a at V = zero
Zero a at v = max
Amplitude

24. Simple Harmonic Motion cont…
Displacement and acceleration are out of phase
a is proportional to – x
interactive problem: match the curve
Phase and phase constant
Sample problem: graph equation
Velocity
Interactive checkpoint: particle speed
Acceleration
calculating period from acceleration
Hence the minus

25. Simple Harmonic Motion cont…
a = -ω²x equation defines SHM
T = 2π/ω
F = -kx eg. a trolley tethered between two springs
Summary of simple harmonic motion

26. Circular Motion and SHM
The peg following a circular path casts a shadow which follows SHM
This gives a mathematical connection between the period T and the angular velocity of the rotating peg
Simple harmonic motion and uniform circular motion
Period, spring constant, and mass
Interactive problem: match the curve again
Work and potential energy of a spring
Total energy
Interactive checkpoint: spring energy and period
Sample problem: falling block on a spring
T = 2π/ω

27. The Pendulum
Pendulum – an object suspended so that it can swing back and forth about an axis.
Rules of simple pendulums
Period is independent of mass
Period is independent of amplitude as long as the arc is small (<20º)
Period is directly proportional to the square root of the length.
Period is inversely proportional square root of the acceleration of gravity
torsional pendulum simple pendulum
Interactive problem: a pendulum
Period of a physical pendulum

28. Pendulums cont… Sample problem: meter-stick pendulum
Damped oscillations
Forced oscillations and resonance