Chapter 4 Part 2 Newton's Third Law, Free Body Diagrams, Inclined Plane

Prof. Clements Notes/Hints for Physics Courses

Chapter 4 Part 2 Open Stax College Physics or Most Introductory Physics Courses

Newton's Third Law; Normal Force, Tension

Terms you should know at the end of this unit: free body diagram, tension, normal force, inertial reference frames

Newton’s Third Law of Motion

Can you think of a force in the universe that does not have another force associated with it that is equal in magnitude but opposite in direction? The correct answer is, “NO.”

There are no isolated (unpaired) forces in the universe.

Newton's Third Law: Whenever one object exerts a force on a second object, the second object exerts an equal and oppositely directed force on the first object. F₂₁ = - F₁₂

Examples:

1. As you stand stationary on the floor:

a) There is a downward force on the floor due to your weight.

b) There is an upward force of the floor acting on your feet.

These two forces have the same magnitude (measured in Newtons, or pounds), but the forces act on different objects (floor, you) and are in opposite directions. Your weight pushes down. The floor pushes up.

2. As you walk North across a floor:

a) There is a horizontal force on the floor as you foot pushes on the floor.

b) There is a horizontal force on your foot as the floor pushes on your foot.

These two forces have the same magnitude (measured in Newtons, or pounds), but the forces act on different objects (floor, foot) and are in opposite directions. Your foot pushes in the South direction on the floor. The floor pushes in the North direction on your foot. (Notes: The forces of example 1 are still active in the vertical direction. The force of friction allows walking to take place. Friction will be discussed later.)

3. For the case of a baseball being hit by a bat:

a) There is a force on the ball towards the outfield due to the bat.

b) There is a force on the bat towards home plate due to the ball.

These two forces have the same magnitude (measured in Newtons, or pounds), but the forces act on different objects (ball, bat) and are in opposite directions. The bat pushes the ball towards the outfield. The ball tries to push the bat towards home plate.

The name “Action Reaction” for Newton’s Third Law is misleading in that there is no time delay for the appearance of the “reaction” force. The two forces have equal magnitudes at all times.

The “action reaction” forces always act on different objects.

Make a force diagram for a student who is sitting in a chair. Draw all of the forces that act on the student and all of the Third Law paired forces. You should have an upward force arrow on the person due to the force of contact with the chair. You should have a downward force arrow on the person due to the gravitational force on the person by to the Earth. You should have a downward force arrow on the chair due to the weight of the person. You should have an upward force arrow at the center of the Earth due to the gravitational attraction on the Earth by the person.

It is very important to note that only the forces acting on the object of interest are used in the calculations. You will not use all of the forces drawn in your first sketch.

Systems of Objects and Free-Body Diagram

In systems where there is more than one object you should start your analysis by finding the net EXTERNAL force acting on the system and the total mass of the system. Then you will be able to calculate the acceleration of the system. After this step you will draw the force diagram for a section of the system and calculate the tension in the connection between the masses.

When a force diagram only shows the forces acting on the object of interest it is called a free-body diagram.

Example: A massless rope is attached to a block of wood that has a mass of 8 kg. A student is pulling to the right on the horizontal rope with a force of 9 Newtons. A 3 kg block of aluminum is attached by a massless string to the left side of the block of wood. The wood and aluminum are on a horizontal table. Ignore friction. Calculate the force the string applies to the aluminum object.

Start: Find the acceleration of the system. Only use external forces. There is only one external force, the 9 Newton force acting to the right. Use F = ma The total mass of the system is 11 kg.

9 Newtons = 11 kg * a

a = 0.818 m/s²

Next: Find the tension in the string connecting the aluminum and wood objects. The object of interest is now the aluminum object. The only external force acting on just the aluminum object is the tension in the string. The 9 Newton force does not act directly on the aluminum object so it is not considered.

Use F = ma

The force is the tension in the string. The mass for the object of interest is 3 kg. The acceleration values for the aluminum and wood are both 0.818 m/s² .

Tension in string = 3 kg ( 0.818 m/s² )

Tension in string = 2.45 Newtons

The apparent weight (measured by a bathroom scale) of a person riding in an elevator which is accelerating upward at 1 m/s² is larger than the weight of the same person standing in the elevator as it moves upward at a constant rate of 3 m/s.

Use F = ma

The net force is the addition of the upward force of the bathroom scale on the person and the downward weight of the person.

Force_scale – mg = ma

Force_scale = mg + ma mg is the weight of the person. Force_scale is larger than mg.

Note: When the elevator reaches a constant speed the acceleration = 0 and Force_scale = mg.

Note: When the elevator nears the top of its motion the acceleration becomes negative and

Force_scale = mg + m(-a) and Force_scale < mg .

In September 2017 this YouTube video was available to show this effect.

https://www.youtube.com/watch?v=D-GzuZjawNI

You might want to search YouTube for other videos about scales and elevators. It is a popular topic for physics classes.

Normal, Tension, and Other Examples of Forces

Contact force: a force that occurs between two objects when they are touching

There are always pairs of forces in contact force situations but only the force acting on the object of interest is useful for the calculations.

Normal force: a contact force that is perpendicular to the surface

Tension in a rope is a force.

Inclined Plane Problems

When an object is on an inclined plane the weight vector is not parallel to the plane and is not perpendicular to the plane. Remember that when we write equations we must only use quantities that are parallel to each other. We will be interested in the velocity along the plane and the displacement along the plane. The components of the weight vector can be used in the equations.

Suppose a 7 kg object is resting on a frictionless inclined plane that has an angle of 34 degrees. The object is 1.4 meters up from the bottom of the plane. It is released from rest at a certain instant of time. Draw a free-body diagram at this time with the goal that you will calculate the time for the object to reach the bottom of the plane.

Steps to be done:

1. Calculate the component of the weight acting down the plane. Calculate the component of weight perpendicular to the plane.

2. Calculate the acceleration of the object down the plane using F = ma.

3. Calculate the time required for the object to reach the bottom of the plane using a kinematic equation.

The force of gravity perpendicular to the plane is not important in the calculation because it is not parallel to the plane. When friction is present this force will be used to determine the force due to friction.

A person is standing for 10 minutes, stationary, on a tight rope that is tied between two poles. It impossible for a real rope to not sag when a person is standing on the rope. The net force on the person in the vertical direction must be zero for the person to be stationary. As the rope sags the tension of the rope now has an upward component that balances the downward weight of the person.

An inertial reference frame is a frame in which Newton's Laws are valid. An inertial reference frame may be moving, but it is not accelerating. If the observer is in a non-inertial reference frame the observer will think there are forces present that don't actually exist.

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.

Chapter 4 Part 1 Force, Inertia, Mass, Newton's First Two Laus

Prof. Clements Notes/Hints for Physics Courses

Chapter 4 Open Stax College Physics or Most Introductory Physics Courses

DYNAMICS: FORCE AND NEWTON'S LAWS OF MOTION

Terms you should know at the end of this unit: dynamics, force, mass, inertia, inertial reference frame, internal force, external force, net force, system, weight

This chapter starts our discussion of Dynamics of motion. Dynamics is the study of what causes changes to take place in the motion of an object. The central concept in the chapter is Force. Force is a vector quantity. Newton developed a coherent set of laws that describe motion and describe how force causes a change in the motion of an object. These laws are concise statements that have a very wide range of applications for objects moving at slow speeds (< 5% of the speed of light). You should know that Newton made his contributions to physics in the last half of the 1600’s.

The development of a sun-centered solar system model and the detailed orbits of the planets influenced the development of Newton’s laws and Newton’s Law of Gravitation. I will not delve too deeply into the study of planet orbits. You should know that astronomy had a major influence on the development of physics theories. This relationship between astronomy and physics continues to this day.

The material in this chapter is foundational for future chapters. You should work through example problems in each section and work problems at the end of the chapter in your textbook that have answers in the back of your text on in the student solution guide. If you have trouble understanding this chapter you may have trouble understanding future chapters. See your instructor, or a class mate, when questions arise.

Force Concepts

force … a push or a pull

Think about circumstances where you experienced a force. You experiencing forces at this moment (force of gravity, upward force of chair on your body, etc.)

net force … The vector sum of all forces which act on one object. The vector sum will be done using the methods of chapter 3. You will need to find the X and Y components of the net force. If necessary, you should review right triangles and trig functions.

There are two categories of force: internal and external. Examples of internal force are forces between atoms or molecules in a solid or the force at the link between two railroad cars. Internal forces do not affect the motion of an object and we will be able to ignore them. External forces do affect the motion of the object. You will need to be able to sketch "free-body" diagrams that show the object and all of the external forces that act on the object. You must ignore any forces that do not act on the object of interest.

Newton’s First Law of Motion Inertia, Mass, Weight

The Greek natural philosophers believed that a constant push was required in order for an object to maintain a constant velocity. This was due to their lack of understanding of the force due to friction. Galileo and Newton understood friction and did have a correct understanding of the law of inertia.

Newton’s First Law … An object has a constant velocity unless there is a non-zero net force acting on the object.

Inertia is a measure of the resistance of an object to a change in its velocity.

The velocity an inertial reference frame (coordinate system) must be constant but the velocity does not have to be zero.

Mass is a measure of the inertia of a body. In the metric system the fundamental unit of mass is the kilogram.

Mass and Weight are different types of physics quantities. Mass is a measure of the inertia of an object. Weight is a measure of the gravitational force on an object. Mass and weight are proportional to each other but the quantities have different units and different numerical values (on Earth).

Newton’s Second Law of Motion

You may wish to do a search on YouTube for:

newtons second law demonstration

The word "system" is used often in physics. The system will often have more than one object in the system. Forces that exists between members of the system are internal forces and will be ignored because they do not affect the motion of the overall system. Forces that act on the system from outside the boundaries of the system are the external forces that must be considered. People riding in a car are inside the system of (car + people). The forces the people may apply to the seat or floor do not change the motion of the system.

Imagine what happens to the acceleration of a system if the external force becomes larger while the mass is constant.

The acceleration increases.

Imagine what happens to the acceleration of a system if the mass (inertia) becomes larger while the net force is constant.

The acceleration decreases.

This leads to the following form of the second law: F_NET = ma F_NET is the net external force (the sum of all of the external forces that act on the object of interest). A force of 1 Newton will give a 1 kg object an acceleration of 1 m/s² . The unit of force in the English system is the pound.

The direction of the acceleration is always the same as the direction of the net force.

Newton's Second Law has been verified by experiments.

Suppose two cars, with different masses, have the same velocity before the brakes are applied. Why is a larger braking force required to stop a larger car in the same distance that a smaller car is stopped?

You should think about the value of the acceleration for both cars. The fourth kinematic equation V² = V_o² + 2 * a * (X – X_o) , shows us that the acceleration value is the same for both cars since the velocities and displacement values are equal. F_NET = ma tells us that the Force will be larger for the car that has the larger mass.

Weight The downward force that accelerates objects towards the Earth is called weight. Weight is a vector that points toward the center of the Earth.

Objects that have more mass have more weight near the surface of the Earth.

Objects that have greater weight do not have greater acceleration values towards the Earth. They have the same acceleration values.

One way to calculate weight is to use W = m * g where g is the value of the acceleration due to gravity. You will see another way to calculate weight when Newton's Law of Gravity is discussed. Try these algebra steps: 1) Write F = ma, 2) replace F with W, 3) replace W with m*g , 4) solve for the acceleration, a.

F = ma

W = ma

mg = ma Cancel the mass values, m.

g = a Objects of any mass accelerate towards the Earth with the same value, -9.8 m/s² .

(Note: there is a subtle, but very important, concept regarding mass. There are two types of mass: gravitational mass and inertial mass. Experiments have verified that the two mass values are identical to the full number of decimal places that can be detected.)

Objects that have greater weight do not greater acceleration values towards the Earth.

An astronaut who has a mass of 70 kg will have different weights on the Earth, the Moon, and Mars. Remember, W = mg. The value of g is different for the Earth, Moon, and Mars.

Astronauts on the International Space Station (in orbit about 240 miles ( above the surface of the Earth) do not have zero weight . They are not weightless because “g” is not zero at this location. They are apparently weightless. This may be discussed more in the chapter that discusses gravity.

Calculate the weight in Newtons, and in pounds, for an astronaut on Earth who has a mass of 70 kg. W = mg W = 70 kg * 9.8 m/s² W = 686 Newtons 1 pound/4.448 N → 154 lb

Calculate the weight in Newtons, and in pounds, for an astronaut in the International Space Station who has a mass of 70 kg. 'g” is about 8.7 m/s² at this location.

W = mg W = 70 kg * 8.7 m/s² W = 609 Newtons 1 pound/4.448 N → 137 lb

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.

Chapter 3 Part 3 Addition of Velocities

Prof. Clements Notes/Hints for Physics Courses

Chapter 3 Open Stax College Physics or Most Introductory Physics Courses

Addition of Velocities

I highly recommend that you go to the web site https://openstax.org/details/books/college-physics to access the free OpenStax College Physics textbook and resources.

YouTube videos of my (usually short) lectures and example problems are indexed at http://www.physics.gpclements.com/ . There are also a few videos that give review of basic math tools that are used in an introductory physics course.

Terms you should know at the end of this unit: vector addition

Addition of Velocities

In these situations there is usually motion of an object in a medium that is also moving. The motion of the object is a vector. The motion of the medium is a vector. To analyze the problem you will add the two vectors.

Imagine a small airplane flying at 120 miles/hour at an angle of 30 degrees North of East. At least those are the numbers the pilot programmed into the computer that controls the flight of the plane. After one hour of flying the pilot discovers that the plane is 140 miles from the starting point at an angle of 35 degrees North of East. What could cause the airplane to end up at this position after one hour?

One answer would be that there was a “tailwind” pushing the plane with the tailwind having a little sideways component (relative to the direction of the plane) that changed the angle for the displacement. That is, the wind direction was more than 30 degrees North of East.

You can solve problems that involve motion in a moving medium (airplane in windy air; boat in a river, etc.) by adding vectors. You need to add the vector of the motion to the vector for the motion of the medium. The resultant is the actual motion of the object.

If you are going to study for your private pilot’s license or your captain of a vessel license you should work example problems in this section.

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.

Chapter 3 Part 2 Right Triangles, Analytic Addition of Vectors, Projectile Motion

Prof. Clements Notes/Hints for Physics Courses

Chapter 3 Open Stax College Physics or Most Introductory Physics Courses

Right Triangles, Analytic Addition of Vectors, Projectile Motion

I highly recommend that you go to the web site https://openstax.org/details/books/college-physics to access the free OpenStax College Physics textbook and resources.

YouTube videos of my (usually short) lectures and example problems are indexed at http://www.physics.gpclements.com/ . There are also a few videos that give review of basic math tools that are used in an introductory physics course.

Terms you should know at the end of this unit: analytic addition, component, hypotenuse, projectile motion

Vector Addition and Subtraction Using Analytical Methods

Analytic addition of vectors produces accurate results. This is the preferred method for adding vectors. Having said that, it is worth your time to still make a quick graphical solution to all vector addition problems before you apply analytical methods. This will help you catch errors in your calculations.

In the typical problem you have two vectors that are specified with their magnitude (length) and angle to the positive X axis given. In the analytical method you will break the vectors into components using sine and cosine, add the X components, add the Y components, and then find the resultant vector's magnitude and angle.

Be careful in your thinking if you see Ax + Ay = A in a textbook. The bold font should remind you that these quantities are vectors and must be added using the rules for adding vectors, not the rules of algebra you are used to using with numbers. It is never allowed to do a simple addition of the X and Y component values so it might be better to call Ax + Ay = A vector reconstruction rather than vector addition. You will use the X and Y components to find the resultant vector, A. The magnitude of A is found using the Pythagorean theorem and the angle of the vector is found by using the inverse tangent function and your brain. (i.e. The inverse tangent function does not always give the right result.

Analytical (mathematical) addition of vectors ( C = A + B ) is performed by

1) finding the components of each vector A and B,

2) adding “like” components i.e. C_X = A_X + B_X , C_y = A_y + B_y

3) reconstructing C.

The image below gives a review of Right Triangles, sine, cosine, tangent, and the Pythagorean Theorem.

To perform the mathematical addition of vectors we must be able to resolve vectors into their components. This is accomplished by the trigonometric calculations for a right triangle. The vector is always the hypotenuse in the right triangle. The X and Y components are the sides of the right triangle. If the angle is given in standard form (i.e. measured counterclockwise from the positive X axis) then you can always use the relationships below to find the components of the vector. If A is the magnitude of the vector then

A_X = A cos (theta) and A_Y = A sin (theta) .

Suppose a certain vector has a magnitude of 12 meters and a direction of 37 degrees from the positive X axis. What are the values of the components of the vector?

A_X = 12 cos(37^o) → 9.58 meters A_y = 12 sin(37^o) → 7.22 meters

If you are given the components of a vector you can reconstruct the vector using the Pythagorean Theorem and the tangent function (plus thinking).

( ) ^½ represents square root

A = ( A_X² + A_Y² ) ^½      tan(theta) = A_Y /A_X    (theta) is found by applying tan^-1 to both sides of this relationship. The angle found will only be correct if the vector is in the first or fourth quadrant (X component is positive). If the X component is negative you must adjust the angle by 180^o . Your sketch of the graphical addition of the two vectors should help you find the correct angle.

Perform the reconstruction of A using the A_X and A_Y values given above.

A = ( (9.58)² + (7.22)² )^½ → 11.996 or 12 meters

tan(theta) = A_Y /A_X      tan^-1 ( tan(theta) ) = tan^-1 (A_Y /A_X) →  (theta) = tan^-1 (7.22 / 9.58) →  (theta) = 37^o

If your calculator does not produce this result for the angle check that the mode of the calculator is set for degrees, not radians.

Suppose A has a length of 3 meters and an angle of 30 degrees, B has a length of 7 meters and an

angle of 160 degrees. Find C = A + B.

A_X = 3 cos(30^o) → 2.598 meters A_y = 3 sin(30^o) → 1.5 meters

B_X = 7 cos(160^o) → -6.578 meters B_y = 7 sin(160^o) → 2.394 meters

C_X = -3.9798 C_y = 3.894

C = ( (-3.9798)² + (3.894)² )^½ → 5.57 meters

tan(theta) = A_Y /A_X     (theta) = tan^-1 (3.894 / -3.9798) → (theta) = -44.4^o but this angle is incorrect.  The result vector, C, is in the second quadrant.

In this situation you must add 180^o to -44.4^o to produce the correct value 

for (theta) of 135.6^o .

Vector addition can be extended to any number of vectors. The following image shows how to add three vectors.

When a problem has two dimensions you must work the problem using vector techniques. Make a vector diagram before you attempt to solve the problem.

You should work some example problems and let your instructor know if you have questions.

Projectile Motion

Displacement, velocity, and acceleration are vectors. When motion occurs in a direction not aligned with either the X or Y axis you must work two problems simultaneously. It is very important that you keep all X quantities (horizontal motion) in one equation and all Y quantities (vertical motion) in a separate equation. The only quantity which will be common to both equations is the time value, t.

Your result will have X and Y components of the position, velocity, and acceleration. You will need to reconstruct the vector to determine the magnitude and angle of the displacement or velocity or acceleration.

Projectile motion refers to motion which takes place near the surface of a large mass (e.g. the earth).

Carefully toss a soft object to another person and have the person throw the object back to you. Watch the object carefully as it flies through the air. The object is undergoing projectile motion. In projectile motion the vertical acceleration is -9.80 m/s² (assuming the object is moving near the earth, not Mars etc.) and the value of the horizontal acceleration is 0! (As long as we ignore air resistance.)

Recall that acceleration determines how the velocity changes. Because the horizontal acceleration is 0 the horizontal velocity is constant!

Solving Projectile Motion Problems

Because the acceleration in the X (horizontal) direction is zero the kinematic equations for the X motion simplify to distance = (initial X component of velocity) * time. The Y (vertical) equations will contain -9.8 m/s² for the acceleration value.

It is very important that you sketch the problem in projectile motion problems. You must choose a positive direction for the horizontal and vertical motion. I will choose upward to be the positive direction. You must choose the location for 0 for both the horizontal and vertical coordinates. I will usually choose ground level to be the 0 location for the X and Y values.

Here is the link for the PhET Cannon projectile motion simulation. Click the “Play” button arrow on the graphic to run the simulation. Your computer will need to have JAVA installed.

PhET Cannon

http://phet.colorado.edu/en/simulation/projectile-motion

The following projectile motion example shows how to calculate the 1) maximum height, 2) time in the air, 3) landing location.

The following projectile motion example analyzes the motion of a basketball.

The following projectile motion example analyzes a situation in which two balls collide.

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.

Chapter 3 Part 1 Vectors, Graphical Method of Vector Addition, Subtraction, Multiplication

Prof. Clements Notes/Hints for Physics Courses

Chapter 3 Open Stax College Physics or Most Introductory Physics Courses

Vectors, Vector Addition, Subtraction, Multiplication

I highly recommend that you go to the web site https://openstax.org/details/books/college-physics to access the free OpenStax College Physics textbook and resources.

YouTube videos of my (usually short) lectures and example problems are indexed at http://www.physics.gpclements.com/ . There are also a few videos that give review of basic math tools that are used in an introductory physics course.

Terms you should know at the end of this unit: vector, vector addition and subtraction, negative vector, scalar multiplication

This chapter will move you toward understanding and solving problems where objects move in two dimensions. This chapter will also help your understanding of vectors and vector related calculations. This chapter is not a major step up in difficulty compared to the previous chapter. You will still use the kinematic equations and your understanding of position, velocity, and acceleration. You will need to understand how to work with right triangles and trigonometric functions. Be sure to get help (your instructor, class mates, YouTube videos) if you have lingering questions after you study this material.

Introduction

In the early chapters of physics textbooks you will usually only study problems in which the motion is over a small area and you can consider the Earth to be flat. I will not discuss problems in three dimensions.

The following is a very important concept. You must NEVER directly add numbers that are not in the same (straight line) direction. i.e. For displacement, you must NEVER add 4 km in the North direction to 3 km in the East direction to produce a result of 7 km. This result is wrong.

I will talk about X-type numbers and Y-type numbers. We will have two sets of equations in our problems. One set of equations will apply to motion in the X direction. A separate set of equations will apply in the Y direction. Another way to state this concept is that motion in the X direction has no effect on motion in the Y direction. i.e. An acceleration in the X direction has no effect on velocity in the Y direction. This point will be covered again later in the chapter.

It is common for textbooks to show illustrations that verify that motion in the X direction has no effect on motion in the Y direction.

Graphical Vector Addition and Subtraction

You should review the description of vectors in your textbook. A vector quantity has some magnitude and a direction. e.g. A value of velocity might be 4 m/s at an angle of 30 degrees to the positive X axis. Textbooks often show vector symbols in boldface or with an arrow above the symbol of the quantity.

The two methods of adding vectors are 1) graphical and 2) analytical (mathematical). This post will discuss the graphical method. The following post will discuss the analytical addition of vectors.

The tail of a vector is the starting part of the arrow. The head of the vector is the arrow part. For the graphical method you:

1) draw the first vector with an approximately correct length and the approximately correct angle to the X axis.

2) draw the second vector where the first vector ended. This creates the name of the method: "head-to-tail." Draw the second vector with its approximately correct angle relative to a line that is parallel to the X axis.

3) draw the resultant vector as a straight line starting at the beginning (tail) of the first vector that was drawn and ending at the head (arrow) of the last vector that was drawn.

You should review the examples of vector addition in your textbook. The graphical method is always approximate.

A minus sign in front of a vector symbol indicates that the direction of the vector is changed by 180 degrees while the magnitude is unchanged.

Algebra allows performing subtraction through the addition process. Suppose we want to subtract 3 from 8. We could write 8 - 3 or 8 + (-3) and produce the result of 5 either way. Vector subtraction is done by adding a negative vector. The method is similar to the calculation 8 + (-3) . Review the examples of vector subtraction in your textbook. The image below shows the addition and subtraction of two vectors:

C = A + B               C = A + (- B )

Multiplication of a vector by a scalar:

What is the meaning of 3 * 5? Answer: 3 * 5 is 5 + 5 + 5.

What is the meaning of 3X? Answer: 3X is X + X + X.

What is the meaning of 3A ? Answer: 3A is A + A + A .

Multiplication of a vector by a scalar just makes the vector longer. If the scalar is positive the direction is unchanged. The image above shows the vector multiplication 2A. If the scalar is negative the vector switches direction by 180 degrees.

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.

Chapter 2 Part 4 Vertical Motion and Motion Graphs

Prof. Clements Notes/Hints for Physics Courses

Chapter 2 Open Stax College Physics or Most Introductory Physics Courses

Vertical Motion and Graphs of Motion

I highly recommend that you go to the web site https://openstax.org/details/books/college-physics to access the free OpenStax College Physics textbook and resources.

YouTube videos of my (usually short) lectures and example problems are indexed at http://www.physics.gpclements.com/ . There are also a few videos that give review of basic math tools that are used in an introductory physics course.

Terms you should know at the end of this unit: acceleration due to gravity, free-fall, tangent line, slope

Falling Objects

Free fall is vertical motion with only the force of gravity acting on the object. (i.e. ignore air resistance and don’t allow any propulsion)

“g” is the symbol for the acceleration due to gravity Near the earth’s surface the magnitude of g is about 9.80 m/s² and the direction is towards the center of the Earth. (The value of g changes slightly based on various conditions of the location that we will ignore.) g tells us how much the velocity of an object in free fall will change every second (9.8 meters/second). g is not gravity. Gravity is a force, not an acceleration. The force of gravity produces an acceleration.

For vertical motion I will usually use the symbol Y for the position of the object. I will try to usually use X for position when the motion of the object is in the horizontal direction. We will assume that the motion is occurring near the surface of the Earth. In this case, the value of “g” will be constant. We will usually ignore air resistance. The four kinematic equations may be used for vertical motion because the acceleration of the object is constant.

Write the four kinematic equation with these changes. Wherever X appears use Y. Wherever “a” appears use “g.”

Equation 1 V = V_o + g t

Equation 2 Y = Y_o + [ 0.5 (V + V_o) ] * t or S = [ 0.5 (V + V_o) ] * t

Equation 3 Y = Y_o + V_o * t + 0.5 * a * t² or S = V_o * t + 0.5 * a * t²

Equation 4 V² = V_o² + 2 * a * (Y – Y_o) or V² = V_o² + 2 * a * S

with S = Y - Y_o

Note: I will work problems with the upward direction as the positive direction. This means that I will use g = - 9.8 m/s² .

Note: Some people have a misconception that the value of the free-fall acceleration depends on the weight of the object. In reality, all objects near the earth have a free-fall acceleration of g (9.8 m/s² ) if air resistance is ignored. This fact will proven when force and Newton’s laws are discussed.

Suppose that an object is dropped from rest and is in free fall from a height of 400 meters. Calculate the position of the object for every second before it hits the ground.

Here are some questions for you to consider: Where will you put Y = 0 meters? Is Y positive in the upward direction or downward? (These questions do not have one right answer!) You are in control of labeling your coordinate system. The analysis will give identical physical results for an infinite number of choices for the 0 point and + or – direction on the axes.

Solution:

1. Make a sketch and write the values for quantities that are known. You should make your own sketch right now. I will use upward as the positive direction.

I will label the ground as Y = 0 meters and assign the upward direction as +.

Y_o = 400 m V_o = 0 m/s g = - 9.8 m/s²

Y = Y_o + V_o * t + 0.5 * a * t²

Y = 400 + 0 * t + 0.5 * (-9.8) * t²

for t = 0 seconds Y_o= 400m for t = 1 second Y₁ = 395m

For the rest of the list the subscript for Y is the time value in seconds:

Y₂ = 380m Y₃ = 356m Y₄ = 322m Y₅ = 278m Y₆ = 224m Y₇ = 160m Y₈ = 86.4m Y₉ = 3.1m

Y₁₀ = -90m ...This value will not be used since it is below ground level.

If you would plot the Y value vs. time the graph would be a parabola.

The velocity at any particular time can be calculated with

V = V_o + at

For this problem V = -9.8 (m/s) * t V_o = 0 m/s V₁ = -9.8 m/s

The graph of velocity vs. time is a straight line with a slope of -9.8 m/s².

The value of the slope on the V vs time graph is equal to the acceleration value.

The graph of acceleration vs time is a horizontal line that has a value of -9.8 m/s² .

You should workother sample problems until you are comfortable with free-fall problems. Let your instructor know if you have questions on the example problems.

Graphical Analysis of Linear Motion

A straight line can be represented by the equation Y = m * X + b. The English language (and other languages) does not have enough symbols for all the variables that might be discussed. The Y and X symbols here are NOT vertical and horizontal position. They are just the coordinates of a Cartesian coordinate system.

“m” is the slope of the line. It is common to calculate the slope by dividing the “rise” by the “run.” The “rise” is the change in the vertical variable value. The “run” is the change in the horizontal variable value.

“b” is the Y-intercept. For the graph represented by Y = 5X + 3, the value of the Y intercept is 3.

When X = 0 the value of Y is 3. When X = 1 the value of Y is 8.

The slope of the line between these two points is (8 – 3) / (1-0) or 5. Note that 5 is the coefficient of X in the equation above.

The slope of the graph at a particular time can be found by drawing a straight line tangent to the graph. The tangent line touches the graph at the time of interest and has equal spacing between the tangent line and the graph on both sides of the touch point.  The graph below is a parabola.  The tangent lines for the times of 1 and 3 seconds and their slopes are shown.  The slopes are calculated by selecting two widely separated points on the line and dividing the change in the vertical (on the graph axis) coordinate by the change in the horizontal coordinate.

The slope of the tangent line on a graph of X vs. time gives information about the instantaneous velocity at that time. The slope of the tangent line on a graph of V vs. time gives information about the instantaneous acceleration at that time. You should be able to draw a tangent line on a graph and find the slope value.

The area “under the graph” of velocity vs. time gives information on the change in position of the object. The area “under the graph” of acceleration vs. time gives information on the change in velocity of the object. “Under the graph” means find the area between the line and the time axis of the graph (where “V” or “a” is zero).

Copyright© 2017 by Greg Clements Permission is granted to reproduce this document as long as 1) this copyright notice is included, 2) no charge of any kind is made, and, 3) the use is for an educational purpose. Editing of the document to suit your own class style and purposes is allowed.