Elementary Mechanics Using Matlab: A Modern Course Combining Analytical and Numerical Techniques(Undergraduate Lecture Notes in Physics)

This book was developed as a textbook for use in the course "Introduction to Mechanics" at the Department of Physics at the University of Oslo starting in 2007. I would like to thank Morten Hjorth-Jensen and Arnt Inge Vistnes for involving me in the of physics of the Computers in Science Education program.

Introduction

Physics
Mechanics
Integrating Numerical Methods
Problems and Exercises
How to Learn Physics
How to Use This Book

The text is identical, only the parts describing the specific programming languages are different in the two cases. Think about how you learn: What parts of the teaching activities do you find most useful for your learning.

Getting Started with Programming

A Matlab Calculator
Scripts and Functions
Plotting Data-Sets
Plotting a Function
Random Numbers
Conditions
Reading Real Data

Example: Plot of Function and Derivative

This will generate a file with a .m extension - we call such a file an m-file because it indicates that the file contains a matlab script/program. Alternatively, you can run the script by typing the name of the script (the name you gave it when you saved it) at the command line.

Table 2.1 Measurement of masses m i as a function of volumes V i

Summary

In order to use a numerical approximation for the derivative, we must perform an approximation for each of the values of x in the set of x. A script is a sequence of executable commands stored in a separate .m-file - All variables are subsequently available on the command line - You start the script by typing F5 in the editor.

Table 2.3 Line markers, colors and plotting symbols

Exercises

Seconds

Spherical mass

Angle

The function g(x;n) is given as:. a) Make a function value(x,n) which returns the value afg(x;n). c) Use the help function to find out how to place legends for each of the plots in the figure. a) Make a function logistic(x,r) which returns the value of x(i+1) given x(i)other's input. For a data-set(i),v(i), you can estimate the function corresponding to the integral ofv(t) with respect to the time interval of the iterative scheme:. c)Write a script to calculate the time integraly(ti) of the data set using this formula. d) Plot the position y(t) and the derivative v(t) as functions of time in two plots one above the other.

Units and Measurement

Standardized Units
Changing Units
Uncertainty and Significant Digits
Numerical Representation
Sphere mass and volume. A small steel sphere has a radius of 1.2 mm

Multiplication: The number with the least significant digits determines the number of significant digits of the result. For all practical purposes, you will be limited by the number of significant figures in your measurements.

Fig. 3.1 Measuring the width of a table with a pencil and a shoe

Motion in One Dimension

Description of Motion

Example: Motion of a Falling Tennis Ball

The first seconds of the race are illustrated by the four pictures in Fig.4.1. The displacement is read directly from the motion diagram as the length of the line ngax(1 s)tox(2 s).

Fig. 4.1 Top Illustrations from the 100 m final in the 2008 Olympic Games in Beijing, showing the position of the Usain Bolt during the first 3 s

Calculation of Motion

Example: Modeling the Motion of a Falling Tennis Ball

Given this expression for acceleration, we determine the velocity and position of the object. Solving this equation means finding the speed (t) and position (t) of the ball at any time. Solution of the simplified model: Since the acceleration is given and constant, we can find the velocity by direct integration of the acceleration:.

Fig. 4.7 Illustration of the motion of “The Rocket”. The accelerations are illustrated for the whole time interval (top figure) and the time-resolution is shown by the squares representing the measure-ment points (bottom figure)

Pedometer. Can you use the accelerometer in your phone as a pedometer?

Driving backwards. You drive in a train that is subject to constant acceleration

The speed of the ball at the beginning of the movement. f) Plot the average acceleration as a function of time for the ball. g)When are the maximum and minimum accelerations. Figure 4-13 shows the motion diagram for a car traveling on a straight road. a) Describe the movement of the car. b) Sketch the position as a function of time. c) Estimate the speed of the car during the movement. d) Estimate the acceleration of the car during the movement. Figure 4-14 shows the motion diagram for a motion. b) Propose a process that leads to this motion diagram.

Fig. 4.12 Random motion of a grain of dust

Archery. As an expert archer you are able to fire off an arrow with a maximum velocity of 50 m/s when you pull the string a length of 70 cm

We first consider the no-dispersion case. a) Write a program to find the motion of an electron using the Euler-Cromer method to find velocity and position from acceleration. The acceleration of the ball is given as:. a) Write the equation you need to solve to find the motion of the ball. This means that we assume that the coefficient of friction does not depend on the speed of the block.

Fig. 4.16 The velocity of a particle moving along the x-axis

What Is a Force?

In this chapter we will show you that the acceleration of an object is related to the forces acting on the object. There is a gravitational force from the Earth on the book, which pulls the book downwards. The sum of the forces of all these small springs is the force of the table on the book.

Fig. 5.1 The structured problem solving approach

Identifying Forces

The direction of the normal force from the floor on the ball depends on the direction of the floor as illustrated in Fig.5.3d. To illustrate the normal force acting on the ball, we draw a vector starting at the point of contact, acting in the direction of the normal force, and with a length related to its size, as illustrated in Fig.5.3 c. When you select the direction of the axis, you also select the positive direction for the axis.

Fig. 5.3 Illustration of a ball bouncing off the floor

Newton’s Second Law of Motion

Example: Acceleration and Forces on a Lunar Lander

Vector equation: Newton's second law is a vector equation: Acceleration is in the direction of the force and acceleration is proportional to the force. It turns out that Newton's second law is valid if we choose a certain point called the center of mass of the object (or any point on the object that does not move relative to the center of mass). Free body diagram: We illustrate the module deduction in a sketch, as shown in Fig.5.4.

Fig. 5.4 a Sketch of descent of the reentry module, b free-body diagram of the module during reentry, and c during weighing

Force Models

Additional material: We can find the time when the drag force becomes less than FDC =106N by first finding the minimum where FD(ti) is less than FDC and then finding the corresponding ti. You can get ideas for how when you learn about air resistance later in this chapter.

Force Model: Gravitational Force

For a body on the surface of the Earth, the gravitational acceleration is about g =9.81 m/s2, while for a body on the surface of the Moon, the gravitational acceleration is gm =0.17g. The table of gravitational acceleration on the surface of various bodies in the solar system can be found in table 5.1. 96 5 Forces in One Dimension There are local differences along the Earth's surface due to variations in the spherical shape of the Earth, to topographic variations, and to differences in density in the Earth's crust.

Table 5.1 The acceleration of gravity on the surface of various objects in the Solar system

Force Model: Viscous Force

Example: Falling Raindrops

At higher velocities, the fluid flow becomes turbulent near the surface, the flow becomes irregular and the force is approximately proportional to the square of the velocity. For high speeds, we see that the drag coefficient becomes approximately constant, and the force is therefore proportional to the square of the velocity. We can find the mass of the drop by assuming it is spherical and made of water.

Fig. 5.7 Illustration of a drag force on an object due to the motion of the object relative to the surrounding fluid

Force Model: Spring Force

Example: Motion of a Hanging Block

Middle) Showing the force from the wall on the sphere as a function of displacement. For the force from the spring, we use the spring force model:. where the elongation ΔL depends on the position,y, of the block. The figures above illustrate the position of the block and the forces acting on the block. The red arrows show spring force and the blue arrows show gravity.

Fig. 5.11 Illustration of an experiment to measure the force needed to extend a spring a distance Δ L, and a plot of the force, F (Δ L )

Newton’s First Law

Newton’s Third Law

Example: Weight in an Elevator

For clarity, we have placed the gravitational force from the box on the Earth, at a point near the Earth's surface. We see that the position of the person on the weight fluctuates around the stationary solution. The contact force of a fluid on a moving object depends on the velocity of the object relative to the fluid.

Fig. 5.19 Illustration of two crates lying on the ground. Various steps in the process of designing the free-body diagram are illustrated

Exercises Exercises

In the army. You are told by a friend in the army that the force you feel when you fire a gun is the same as the force felt by a sandbag hit by the bullet because the
Whiplash. Explain why a properly adjusted headrest will reduce the chance of whiplash injury if your car is hit from behind
Parachute. If you jump from a plane you quickly reach the terminal velocity
Parachuter. A person jumps from an airplane, falling freely for several seconds before he pulls the cord of his parachute and the parachute unfolds
Forces on a drop of water. A drop of water is hanging from a faucet
Forces on an anchor. Susan is standing on the floor in a boat, pulling a rope attached to an achor
Firing a bullet. A bullet of mass 0.1 kg is fired through a 1 m rifle barrel

A drop of water hangs from a tap. a) Identify the forces acting on the droplet and draw a free-body diagram of the droplet. One sphere is hollow and the other is solid. a) Draw a free body diagram for one of the spheres. The acceleration of gravity is g. a) Draw a free body diagram of the person when the bungee cord is taut.

Motion in Two and Three Dimensions

Vectors

When two vectors are parallel and point in the same direction, the dot product is equal to the product of the magnitudes. The value of the dot product is independent of the unit vectors used to decompose the vectors. A common application of the dot product is to find the component of a vector along the direction given by a vector b.

Description of Motion

Example: Mars Express

We use the word speed for a velocity vector, and the word velocity for the magnitude of a vector velocity. 3Here we have implicitly assumed that the time derivatives of the unit vectors are equal to zero. More insight can be gained from a motion diagram or by calculating the velocity and acceleration of the module along the trajectory.

Fig. 6.2 Illustration of a cheetah chasing a Thompson gazelle, and an illustration of the two- two-dimensional motion and the two-two-dimensional motion diagram

Calculation of Motion

Example: Feather in the Wind

How can you use these readings to determine the probe's speed and position during its flight. Using these models, you can use Newton's second law to find the acceleration of an object. Finally, you find the object's velocity and position vector as a function of time based on the expression you have for the acceleration and the initial position and velocity values.

Fig. 6.9 Plot of the components of the accelerations recorded by the accelerometer in the probe, and a table listing the first 10 vales

Frames of Reference

Example: Motion of a Boat on a Flowing River

This corresponds to the vector velocity of the ball measured from the car in Figure 6.14. Driving at different speeds: The speed of the boat relative to the river, vb(t), is recorded by the speedometer. Therefore, we can use (6.108) to find the speed of the boat relative to the ground.

Fig. 6.14 An illustration of a person throwing a ball straight up from a convertible driving at a constant velocity as seen from a person on the sidewalk

A plane in crosswinds. You are trying to steer an airplane towards the north

Determine the velocity vector as a function of time. b) Determine the position vector as a function of time. c) Given an interpretation of the movement in its two distinct phases. Another similar experiment was performed, where the motion data was given in motion2.d.13. h) Find the positionr2 of the motion and plot it on the same graph as the motion in motion1.d. i) Can you give a physical interpretation of the movement? j) Where is the magnitude of the acceleration the maximum. Third, we solve the equations of motion and determine the position and velocity of the objects as functions of time.

Identifying Forces

We know a force due to the deformation of the wheel and the ground. The only long-range force is gravity from the Earth, which acts on the machine and points toward the center of the Earth. Choose a coordinate system and draw the axes of the coordinate system on the same figure as the system.

Newton’s Second Law

In this case, the net force is zero even if none of the forces are pointing in the same direction, as shown by the graphical vector summation on the right. Net force is a vector sum: The net force is a vector sum of all external forces. Note that the net force can be zero even if none of the force vectors point in the same direction, as illustrated in Figure 7-3.

Fig. 7.3 A sphere is hanging from two ropes that are attached to the roof. In this case, the net force is zero even when none of the forces point in the same direction, as shown by the graphical vector summation to the right

Force Model—Constant Gravity

Example: Motion of a Ball with Gravity

Identify and Sketch: In this exercise we deal with the motion of the ball, described by the positioner(s) as a function of time. The origin is placed on the ground, directly below the initial position of the ball att =t0. We have now found the complete solution for the motion of a ball subject only to gravity.

Fig. 7.4 The motion of a ball thrown across the lecture room

Force Model—Viscous Force

Analyzing: We see that the movement in the x and y direction is independent of each other. If the ground is flat, it is the movement in the y direction that determines how long it takes to reach the ground. We can therefore answer questions about flight time and maximum altitude by studying only the one-dimensional motion in the lateral direction.