- •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
168 |
6 Motion in Two and Three Dimensions |
You should in general think of the “Solver” as a call to a numerical function that returns the position and velocity vectors given the functional form of the acceleration and the initial conditions. When you grow up to become a professional physicist, you will have built your own set of methods and tools, numerical and analytical, to “Solve” problems. In particular, we advice you to use a fourth order Runge-Kutta method as your preferred method of numerical integration, although in this book we will focus on clarity and simplicity instead, and typically use Euler-Cromer’s method, unless this produces significant errors.
6.3.1 Example: Feather in the Wind
In this example you learn to find the position and velocity from the acceleration, when the acceleration is given by a simplified mathematical model, and when it is given by a differential equation.
You are planning to reproduce the introductory film in Forrest Gump—by capturing the motion of a feather caught in the wind. You plan to start with a feather dropped from a lift, at a height h above the ground. A slight wind is blowing. Our task is to find the motion of the feather, given its acceleration.
Sketch: As always, a good sketch solves half the problem. For simplicity we assume that the motion is two-dimensional. Figure 6.12 shows a sketch of the path of the feather, including the velocity of the wind.
Simplified model—Free fall: The simplest, and least realistic, model for the falling feather is to assume that it falls without air resistance. We release the feather with a horizontal velocity equal to that of the wind, v0 = w, and then assume that it has constant acceleration, a = −gj. We can then find the motion by integration.
Fig. 6.12 Sketch of a feather falling to the ground. The velocity field, w, of the air is illustrated by the arrows
a = −g j |
(6.94) |
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y |
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FD |
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w |
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r0 |
v |
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v-w |
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v |
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r(t) |
G |
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x |
6.3 Calculation of Motion |
169 |
and we release the feather at time t0 = 0 s wt r(0) = h j with v(0) = v0 = w. We find the velocity by direct integration:
v(t ) = v(t0) + t |
a dt = w + t |
−g j dt = w − gt j . |
(6.95) |
t0 |
0 |
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|
We can simplify this a bit further if the wind is horizontal, w = w i:
v(t ) = w i − gt j . |
(6.96) |
We find the position by integrating once more over time:
t |
t ) dt |
= |
r |
t |
|
w i |
− |
gt j |
|
dt |
r(t ) = r(t0) + t0 |
v(1 2 |
|
0 + 0 |
( |
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) |
(6.97) |
= h j + wt i − 2 gt j .
This gives us a complete solution for this simplified model, which is useful as a comparison when we address the full model.
Realistic model: A more realistic model includes two additional effects: Air resistance means that the acceleration depends on the velocity of the feather and the velocity of the wind, and the wind typically varies near the ground. A better model for the acceleration of the feather is:
a = −g j − C |v − w| (v − w) , |
(6.98) |
where w is the velocity of the wind. A realistic model for the wind is
|
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w(r) = w0 1 − e−y/b |
, |
(6.99) |
In this case, b = 5 m and w0 = 3 m/s i is the velocity of the wind. We drop the feather from rest from a position r0 = h j, with h = 10 m.
Unfortunately, we cannot solve this equation exactly, but it is not difficult to solve numerically. We apply Euler-Cromer’s method. We find the velocity and position at time ti + t from the position and velocity at ti using:
v(ti + |
t ) v(ti ) + |
t a(ti , vi ) |
(6.100) |
r(ti + |
t ) r(ti ) + |
t v(ti + t ) . |
(6.101) |
In addition, you are told that C = 30.0 m−1 for the feather.
Notice that the acceleration depends not only on the velocity v(t ) of the feather, but also on its position, because the velocity of the air, w, depends on the position of the feather. We therefore first calculate w(r) and then calculate a(t , r, v). This is implemented in the following program. Notice the use of vector notation to find the acceleration.
170
Fig. 6.13 The trajectory of the falling feather
y [m]
6 Motion in Two and Three Dimensions
10
5
0
0 |
5 |
10 |
15 |
20 |
25 |
30 |
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x [m] |
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# Physical |
constants |
|
h = 10.0 |
# |
m |
C = 30.0 |
# |
mˆ-1 |
w0 = 3.0 |
# |
m/s |
b = 5.0 |
# |
m g = 9.8 # m/s |
#Numerical constants time = 20.0
dt = 0.001
n = int(ceil(time/dt)) t = zeros((n,1),float) r = zeros((n,2),float) v = zeros((n,2),float) a = zeros((n,2),float)
#Initial conditions t[0] = 0.0
r[0,:] = array([0.0,h]) v[1,:] = array([0.0,0.0])
#Find the motion for i in range(n-1):
w = w0*(1.0-exp(-r[i,1]/b))*array([1,0])
a[i,:] = -g*array([0,1])-C*norm(v[i,:]-w)*(v[i,:]-w) v[i+1,:] = v[i,:] + a[i,:]*dt
r[i+1,:] = r[i,:] + v[i+1,:]*dt t[i+1] = t[i] + dt
#Plot motion
i = find(r[:,1]>0.0) plot(r[i,0],r[i,1],’-k’);
axis(’equal’), xlabel(’x [m]’); ylabel(’y [m]’);
The expression we have used for w(r) is only value when y > 0, and the feather hits the ground when y = 0 m. We therefore only plot the trajectory when y > 0. This is ensured using the find command.
The resulting path of the feather is illustrated in Fig. 6.13. Notice that the feather almost immediately follows the wind—the drag forces rapidly reduces differences between the velocity of the feather and the velocity of the surrounding air.
We can compare these results with the simplified, analytical solution we found above. The simplified solution is illustrated by a dotted line, which clearly was not a good solution, because the vertical acceleration is greatly reduced when the feather moves fast in the horizontal direction.