- •Preface
- •Contents
- •1 Introduction
- •1.1 Physics
- •1.2 Mechanics
- •1.3 Integrating Numerical Methods
- •1.4 Problems and Exercises
- •1.5 How to Learn Physics
- •1.5.1 Advice for How to Succeed
- •1.6 How to Use This Book
- •2 Getting Started with Programming
- •2.1 A Python Calculator
- •2.2 Scripts and Functions
- •2.3 Plotting Data-Sets
- •2.4 Plotting a Function
- •2.5 Random Numbers
- •2.6 Conditions
- •2.7 Reading Real Data
- •2.7.1 Example: Plot of Function and Derivative
- •3 Units and Measurement
- •3.1 Standardized Units
- •3.2 Changing Units
- •3.4 Numerical Representation
- •4 Motion in One Dimension
- •4.1 Description of Motion
- •4.1.1 Example: Motion of a Falling Tennis Ball
- •4.2 Calculation of Motion
- •4.2.1 Example: Modeling the Motion of a Falling Tennis Ball
- •5 Forces in One Dimension
- •5.1 What Is a Force?
- •5.2 Identifying Forces
- •5.3.1 Example: Acceleration and Forces on a Lunar Lander
- •5.4 Force Models
- •5.5 Force Model: Gravitational Force
- •5.6 Force Model: Viscous Force
- •5.6.1 Example: Falling Raindrops
- •5.7 Force Model: Spring Force
- •5.7.1 Example: Motion of a Hanging Block
- •5.9.1 Example: Weight in an Elevator
- •6 Motion in Two and Three Dimensions
- •6.1 Vectors
- •6.2 Description of Motion
- •6.2.1 Example: Mars Express
- •6.3 Calculation of Motion
- •6.3.1 Example: Feather in the Wind
- •6.4 Frames of Reference
- •6.4.1 Example: Motion of a Boat on a Flowing River
- •7 Forces in Two and Three Dimensions
- •7.1 Identifying Forces
- •7.3.1 Example: Motion of a Ball with Gravity
- •7.4.1 Example: Path Through a Tornado
- •7.5.1 Example: Motion of a Bouncing Ball with Air Resistance
- •7.6.1 Example: Comet Trajectory
- •8 Constrained Motion
- •8.1 Linear Motion
- •8.2 Curved Motion
- •8.2.1 Example: Acceleration of a Matchbox Car
- •8.2.2 Example: Acceleration of a Rotating Rod
- •8.2.3 Example: Normal Acceleration in Circular Motion
- •9 Forces and Constrained Motion
- •9.1 Linear Constraints
- •9.1.1 Example: A Bead in the Wind
- •9.2.1 Example: Static Friction Forces
- •9.2.2 Example: Dynamic Friction of a Block Sliding up a Hill
- •9.2.3 Example: Oscillations During an Earthquake
- •9.3 Circular Motion
- •9.3.1 Example: A Car Driving Through a Curve
- •9.3.2 Example: Pendulum with Air Resistance
- •10 Work
- •10.1 Integration Methods
- •10.2 Work-Energy Theorem
- •10.3 Work Done by One-Dimensional Force Models
- •10.3.1 Example: Jumping from the Roof
- •10.3.2 Example: Stopping in a Cushion
- •10.4.1 Example: Work of Gravity
- •10.4.2 Example: Roller-Coaster Motion
- •10.4.3 Example: Work on a Block Sliding Down a Plane
- •10.5 Power
- •10.5.1 Example: Power Exerted When Climbing the Stairs
- •10.5.2 Example: Power of Small Bacterium
- •11 Energy
- •11.1 Motivating Examples
- •11.2 Potential Energy in One Dimension
- •11.2.1 Example: Falling Faster
- •11.2.2 Example: Roller-Coaster Motion
- •11.2.3 Example: Pendulum
- •11.2.4 Example: Spring Cannon
- •11.3 Energy Diagrams
- •11.3.1 Example: Energy Diagram for the Vertical Bow-Shot
- •11.3.2 Example: Atomic Motion Along a Surface
- •11.4 The Energy Principle
- •11.4.1 Example: Lift and Release
- •11.4.2 Example: Sliding Block
- •11.5 Potential Energy in Three Dimensions
- •11.5.1 Example: Constant Gravity in Three Dimensions
- •11.5.2 Example: Gravity in Three Dimensions
- •11.5.3 Example: Non-conservative Force Field
- •11.6 Energy Conservation as a Test of Numerical Solutions
- •12 Momentum, Impulse, and Collisions
- •12.2 Translational Momentum
- •12.3 Impulse and Change in Momentum
- •12.3.1 Example: Ball Colliding with Wall
- •12.3.2 Example: Hitting a Tennis Ball
- •12.4 Isolated Systems and Conservation of Momentum
- •12.5 Collisions
- •12.5.1 Example: Ballistic Pendulum
- •12.5.2 Example: Super-Ball
- •12.6 Modeling and Visualization of Collisions
- •12.7 Rocket Equation
- •12.7.1 Example: Adding Mass to a Railway Car
- •12.7.2 Example: Rocket with Diminishing Mass
- •13 Multiparticle Systems
- •13.1 Motion of a Multiparticle System
- •13.2 The Center of Mass
- •13.2.1 Example: Points on a Line
- •13.2.2 Example: Center of Mass of Object with Hole
- •13.2.3 Example: Center of Mass by Integration
- •13.2.4 Example: Center of Mass from Image Analysis
- •13.3.1 Example: Ballistic Motion with an Explosion
- •13.4 Motion in the Center of Mass System
- •13.5 Energy Partitioning
- •13.5.1 Example: Bouncing Dumbbell
- •13.6 Energy Principle for Multi-particle Systems
- •14 Rotational Motion
- •14.2 Angular Velocity
- •14.3 Angular Acceleration
- •14.3.1 Example: Oscillating Antenna
- •14.4 Comparing Linear and Rotational Motion
- •14.5 Solving for the Rotational Motion
- •14.5.1 Example: Revolutions of an Accelerating Disc
- •14.5.2 Example: Angular Velocities of Two Objects in Contact
- •14.6 Rotational Motion in Three Dimensions
- •14.6.1 Example: Velocity and Acceleration of a Conical Pendulum
- •15 Rotation of Rigid Bodies
- •15.1 Rigid Bodies
- •15.2 Kinetic Energy of a Rotating Rigid Body
- •15.3 Calculating the Moment of Inertia
- •15.3.1 Example: Moment of Inertia of Two-Particle System
- •15.3.2 Example: Moment of Inertia of a Plate
- •15.4 Conservation of Energy for Rigid Bodies
- •15.4.1 Example: Rotating Rod
- •15.5 Relating Rotational and Translational Motion
- •15.5.1 Example: Weight and Spinning Wheel
- •15.5.2 Example: Rolling Down a Hill
- •16 Dynamics of Rigid Bodies
- •16.2.1 Example: Torque and Vector Decomposition
- •16.2.2 Example: Pulling at a Wheel
- •16.2.3 Example: Blowing at a Pendulum
- •16.3 Rotational Motion Around a Moving Center of Mass
- •16.3.1 Example: Kicking a Ball
- •16.3.2 Example: Rolling down an Inclined Plane
- •16.3.3 Example: Bouncing Rod
- •16.4 Collisions and Conservation Laws
- •16.4.1 Example: Block on a Frictionless Table
- •16.4.2 Example: Changing Your Angular Velocity
- •16.4.3 Example: Conservation of Rotational Momentum
- •16.4.4 Example: Ballistic Pendulum
- •16.4.5 Example: Rotating Rod
- •16.5 General Rotational Motion
- •Index
14.1 Rotational State—Angle of Rotation |
441 |
Fig. 14.4 Plot of the angle, θ (t ), for the rotational motion in Fig. 14.1. Dashed curve shows how the rod would have continued to rotate if it had not hit the ground—such as if it fell off a cliff
|
8 |
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
θ |
4 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
|
7 |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
|
|
|
|
|
t (S) |
|
|
|
Periodicity of the State θ (t )
The angle, θ , describes a unique configuration of the rod for values from 0 to 2π (measured in radians). What happens when θ (t ) increases beyond 2π ? When θ reaches 2π the rod has rotated a full revolution, and the rod is in the same position as it was when θ was equal to 0. We cannot discern these positions: The position of the rod when θ = 2π is exactly the same as when θ = 0. However, it is customary to only use angles between 0 and 2π to describe the rotational position. This means that if the angle is larger than 2π we subtract 2π from the angle. This is seen in Fig. 14.4: When the angle θ (t ) reaches 2π , it continues at θ = 0. Similarly, when the angle decreases below 0, we add 2π to the angle, so that it continues at 2π . You are, of course, free to choose to describe the motion using an angle θ that increases also beyond 2π , but then you have to remember that the motion is periodic so that higher values does not represent new positions.
14.2 Angular Velocity
During rotation, the angle θ (t ) changes with time. How can we characterize how fast the rod rotates? By the angular velocity, which is defined as the rate of the change of the angle in analogy with the (translational) velocity, which is the rate of change of
the position. During the time interval from t to t + |
t , the angle changes from θ (t ) |
||||
to θ (t + t ). We define the average angular velocity over the time |
t as: |
||||
ω¯ = |
θ (t + t ) − θ (t ) |
= |
Δθ |
, |
(14.5) |
|
|
||||
|
t |
t |
|
When the time interval becomes small, we find the instantaneous angular velocity for the rotational motion, which we in the following call the angular velocity:
442 14 Rotational Motion
Angular velocity: |
|
|
|
|
|
Δθ |
|
d θ |
˙ |
|
|
|
|
|
|
|
|
ω = lim |
|
= = θ . |
(14.6) |
||
t →0 t |
|
d t |
|
|
|
|
|
|
|
|
|
Figure 14.4 shows the angle, θ (t ), and the angular velocity, ω(t ) = d θ /d t for the rotational motion in Fig. 14.1. Since the angular velocity is the time derivative of the angle, we interpret the angular velocity as the slope of the θ (t ) curve (just as we did for the translational velocity). We see that the motions in Fig. 14.1 has a constant, positive angular velocity.
Test your understanding: Can you sketch θ (t ) and ω (t ) for a rod that is rotating equally fast in the opposite direction?
Velocity of a Point on a Rotating Body
As the rod rotates, every part of the rod moves in a circle around the rotation axis. What is the velocity of a small part of the rotating rod, and how can we relate it to the angular velocity? Let us address the motion of a small part, P , of the rotating body directly. We have illustrated its motion during a small time interval t , in Fig. 14.5. The distance from P to the rotation axis is R. The small part P moves along a circular path around the rotation axis with R as the radius. During the small
(A)
|
S |
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
. |
|
|
|
|
|
|
|
|
(B) |
|
|||
0 |
|
|
|
S |
|
|
|
|
|||||
|
|
|
. |
4 |
|
|
|
|
|
|
|
||
0 |
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
S |
|
|
|
|
|
|
|
|
|
|
|
|
|
.3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
S |
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
. |
|
|
|
|
|
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
|
|
|
S |
|
|
|
|
|
|
|
|
|
|
|
|
|
=R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
S |
|
|
|
|
|
|
Δθ |
|
.1 |
|
|
|
|
|||||
|
|
|
0 |
|
|
|
|
|
|||||
|
|
|
(0 |
|
|
|
Δθ(0.4S) |
|
|||||
|
|
|
|
|
4 |
|
|
|
|
||||
|
|
|
|
|
. |
|
|
|
Δθ(0.2S) |
U T |
|||
|
|
|
|
|
) |
|
|
|
|||||
|
|
|
|
|
|
S |
|
|
|
|
|
|
|
Δθ(0.4S) |
|
|
|
|
|
|
0.0S |
|
Δθ(0.1S) |
|
|||
|
|
|
|
|
|
|
|
|
|
|
|||
|
R |
|
|
|
|
|
|
|
|
|
R |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Fig. 14.5 a Illustration of the motion of a small part, P , of a rod rotating around an axis through the origin. b Illustration of the velocity vector for P as the time interval t decreases
14.2 Angular Velocity |
443 |
time interval t , the rod has rotated an angle Δθ from the orientation θ (t ) to the new orientation θ (t + t ) = θ (t ) + Δθ . How far has P moved? It has moved the arc length s = RΔθ along its circular path. The speed of the small part P is therefore:
v = |
s |
= R |
Δθ |
. |
(14.7) |
|
|
||||
|
t |
t |
|
If we let the time interval t become infinitesimally small, we find the speed of the point P to be:
v = |
d s |
= |
d |
( R θ ) = R |
d θ |
= R ω . |
(14.8) |
|
|
|
|||||
|
d t d t |
|
d t |
|
The speed of a point on the rod is therefore proportional to the angular velocity of the rotation, but also proportional to the distance R to the rotational axis: Points further away from the rotation axis rotate with higher speeds.
What is the direction of the velocity vector for P ? Fig. 14.5 shows that when the time interval t becomes smaller, the change in angle Δθ also becomes smaller, and the direction of the displacement vector from P at time t to P at time t + t approaches that of a tangent to the circle. Excatly the same result we found earlier when we studied circular motion. The velocity vector is therefore parallel to the tangent to a circle of radius R, and points in the direction of the tangential unit vector uˆ T . The velocity of the point P is therefore:
v = R ω uˆ T . |
(14.9) |
Motion with Constant Angular Velocity
If an object rotates with a constant angular velocity, we can find the speed of the point P from the distance traveled during one complete revolution, s = 2π R, divided by the time of one revolution, call the period T :
v = |
s |
= |
2π R |
, |
(14.10) |
|
|
||||
|
T |
|
T |
|
where R is the distance from P to the rotation axis. We also know that the velocity is v = Rω, therefore we find that:
v = |
2π |
R = ω R ω = |
2π |
. |
(14.11) |
|
|
||||
|
T |
|
T |
|
The angular velocity is often also called the angular frequency.