Static Forces

Chapter 4

The Golden Gate bridge does not move. Well, actually, it does. Vibrations due to vehicle

traffic, wind, and even earthquakes are all considered in the design. To build a bridge that

does not fall down takes an analysis of forces that would like to make it move. On Earth,

gravity is a significant force on structures. In space, forces associated with thrusting rocket

engines and other subtle vibration effects are important to consider in designing spacecraft and

space stations. If you want to design something that does not move, it is critical to understand

the forces that want to make it move.

Forces Beginning with Newton – Lesson 1

Lesson Objectives

Describe the four known forces in nature and their order of strengths

Understand the law of inertia

Be able to visualize the concept of force and its connection to motion

State examples of the law of action-reaction

Vocabulary

Inertia

Force

Check Your Understanding

1. In the study of motion, you learn how to predict the future motion of a body. What are

some examples of what makes an object move in the first place?

2. Questions about projectiles sometimes ask for a final velocity of an object just before it

hits the ground. Describe the motion of that object from the last moment it was a

projectile to when it is at rest on the ground.

Introduction

In the history of science, forces are at the center of the scientific revolution. It is the

understanding of forces that answers questions about why objects move. The study of motion

allows us to predict the position, velocity, acceleration, and higher order changes in motion.

Analyzing motion does not address why or how something can move? This lesson will describe

the historical origins of the concept of force largely through one man, Isaac Newton. By the

end of the lesson, you will be able to describe why things move and address how objects

interact with each in ways you may have never considered.

Lesson Content

History of Forces

Throughout history, there are many great figures who get credited for the work of many they

represent. Isaac Newton worked mostly alone and revolutionized science in work mostly done

in a little over one year. There is a quote from Newton that says, “if I have seen further, it is

because I have stood on the shoulders of giants.” This acknowledgement is to the like of

Galileo, Rene Descartes, and others for setting the stage for what Newton was to do. Newton

took an interest in the natural philosophy sometime in college through his curiosity about

astrology. Newton always enjoyed puzzles and the challenges associated with mechanical

systems. At a young age and even in college, Newton was considered a bright pupil but not the

outstanding figure he was to become. This fascination with astrology came from the possibility

of uncovering secrets about predicting what happens in nature. Astrology and astronomy were

indiscernible during this time. The science of astronomy and chemistry were still connected

with hints of the mysticism.

Figure 1. Isaac Newton, 1642-1727.

Once Newton realized that the predictions of astrology were mathematically based, he began to

dedicate himself to mastering his math skills. When a plague broke out in London, Newton took

the only preventative measures available at the time to avoid possibly dying – he left town. At

his family’s farm house, Newton spent the next 15 months using mathematics to explain the

universe. When the mathematics stopped being useful, Newton invented new mathematics.

Newton’s annus mirabilis or miracle year led to several major discoveries. Isaac Newton

wanted to explain how things moved in the universe. For instance, the motion of the Moon and

the planets was not explained. Rene Descartes, whose work Newton studied intensely, claimed

that all circular motions were the result of remnant vortices left over from the original creation

by God the creator. Newton believed God’s work could be uncovered with mathematics.

Newton needed to calculate things where the variables themselves varied. Newton came up

with what we now call calculus to describe his three laws of motion and also his universal law of

gravitation. He also established that white light was made up an equal combination of colors,

and invented the reflecting telescope.

Video of Dr. Neil deGrasse-Tyson talking about the accomplishments of Isaac Newton

http://video.answers.com/the-genius-of-sir-isaac-newton-516946383

Before exploring the laws of motion, you should understand the concept of force and how it

applies to the big picture; the universe itself.

What is a Force?

A force simply is a push or a pull. A force is the answer to why an object changes its motion.

The current understanding of forces has categorized all types of forces into four distinct types.

The most obvious force is the weakest. The four known types of forces in reverse order of

strength are discussed below in table 1. Each force is shown by name, the recipe for that force

to happen, and the relative strength.

Fundamental Force Recipe Relative Strength

Gravity Masses attract other masses 3610

Weak nuclear Reason Neutrons decay when isolated from protons,

root cause of radioactivity

1

10,000

Electromagnetic Like charges repel, opposite charges attract 1

Strong nuclear Protons that are around each other in close

proximity with neutrons, reason atomic nuclei exist

and source of energy of stars, fission and fusion

reactors and weapons

137

Table 1. Fundamental Forces in the Universe

The information in the table of fundamental forces tells an amazing story. The most noticeable

force in the universe, gravity, is the weakest of all the known reasons why anything would

change its motion. In fact, it is so small it should not be noticeable. Gravity, compared to the

only other force most people know about, is a fraction:

36110

1000000000000000000000000000000000000

times the force of electricity and

magnetism. The reason gravity is the force we notice the most and the one we will apply most

is that the other forces end up either canceling out or fading out very quickly with distance.

The nuclear forces are only so strong over distances like those in the interiors of atoms.

Electromagnetic forces depend on charges, and there are two kinds – positive and negative.

The universe has had a long time to let the balance of charges to be about the same. So, only

the weak and meager force of gravity is noticeable on the large scale.

Newton’s First Law of Motion: the Law of Inertia

Inertia is a concept many people mix up with the mass of an object. However, it is not quite

the same thing. Inertia is defined as a tendency of an object to resist changes in its motion.

On the surface, this seems to just describe mass. The bigger the mass of an object suggests

that it would be harder to make it change its motion. But, consider the case of a 10 foot pole in

the conceptual example.

Inertia depends on the shape and mass of an object. That brings us to the:

1st law of motion

An object at rest will remain at rest unless acted on by an outside force. An object in linear

motion will remain in linear motion unless acted on by an outside force.

Objects like to keep doing what they are doing unless forced to do otherwise. In this sense,

inertia is the measure of the laziness of an object as it follows this first law.

Examples of the first law:

Tablecloth trick

Conceptual Example – Understanding the difference between mass and inertia

You and a friend are each asked to carry something from one room to another room downstairs. The

two objects are a 1.0kg ball that fits in the palm of your hand, and a skinny 10 foot fiberglass pole.

Do the two objects have the same inertia? Is the 1.0kg (or 2.2 pound) ball as easy or harder to carry

through the building downstairs to another room compared to the 10 foot pole?

Solution:

Even though the two objects have the same mass, they would not take an equal effort to

transport from one room to another. The distribution of mass of the 10 foot pole makes it

harder to control and thus takes more force to move around corners and change orientation to

get in and out of doorways.

Second Law of Motion

The second law of motion can be stated as an equation describing the net force on an object.

2nd law of motion

netF ma

In Newton’s famous book, the Principia, he stated the 2nd law as a relationship to acceleration.

He said neta F : which says the harder you push something, the more acceleration it should

experience. He also said that 1

am

: which says that larger mass objects should acceleration

less than smaller mass objects. If you combine these two relationships and solve for net force,

you get the equation in the second law. Net force is the vector force that is left over after

combining all the forces on a body.

Third Law of Motion

The third law describes how a rocket works.

3rd law of motion

For every action, there is an equal and opposite reaction.

Conceptual Example – How do rockets work in space if there is no atmosphere to push

against?

Robert Goddard was an American scientist who was building and testing rockets in the early 1900’s.

When Dr. Goddard published results on what it would take to send a rocket away from the Earth, like

to the Moon for instance, the New York Times published the following editorial on January 13, 1920.

“That Robert Goddard, with his ‘chair’ in Clark College and the countenancing of the Smithsonian

Institution, does not know the relation of action to reaction, and of the need to have something

better than a vacuum against which to react – to say that would be absurd. Of course he only seems

to lack the knowledge ladled out daily in high schools.”

What is wrong with the editorial statement in the Times back in 1920?

Solution:

A rocket works by containing a chemical reaction that ejects gases from the bottom end

of the rocket. The action is the ejection of the gases at high speed. The reaction is an

equal but opposite force on the rocket upward. The ejected gas does not push against

an atmosphere. The NY Times issued the following statement on July 17, 1969 – one

day after the astronauts of Apollo 11 landed on the Moon:

“Further investigation and experimentation have confirmed the findings of Isaac Newton in the 17th century

and it is now definitely established that a rocket can function in a vacuum as well as in an atmosphere. The

Times regrets the error.”

Lesson Summary

A force is a reason why an object will change its motion and is defined simply as a push

or a pull.

There are four fundamental forces known in the universe and they include strong

nuclear, electromagnetic, weak nuclear and gravity in order of strengths.

Gravity is by far the weakest force in the universe, but most noticeable due to short

range or balanced charges.

Objects like to keep moving or sitting still unless a net force is present.

You cannot touch an object without being touched.

The net force on an object results in a net acceleration according to netF ma.

Review Questions

Conceptual Example – If the action of gravity is pulling you down, what is the reaction?

You jump off of a step. You fall due to the force of gravity. How does Newton’s 3rd law apply?

Solution:

he action is you being pulled toward the center of the Earth. The reaction is the Earth being pulled

toward the center of you. Newton’s 3rd law states that the action and reaction are equal in

magnitude. . This can be combined with Newton’s 2nd law:

The reaction of the Earth is real but so tiny that no instruments can measure it.

Review Problems

Further Reading / Supplemental Links

•

Points to Consider

1. How do the laws of motion connect to kinematics?

2. A book is stationary on a slightly tilted desk. Describe how the laws of motion describe

what is happening.

Practical Forces – Lesson 2

Lesson Objectives

Understand the differences between mass and weight

Describe how surfaces interact with each other through normal force

Understand the macroscopic definition of static friction

Be able to draw free body diagrams

Check Your Understanding

1. What is inertia?

2. How is force related to motion?

3. Is it possible to make something move without being affected yourself?

Introduction

Newton’s laws of motion seem like common sense to many people. However, it took one of the

greatest scientists in history to establish these laws using rules of mathematics. Analyzing

interacting bodies requires an understanding of some basic principles. These basic principles

show up in the first steps of any force analysis. It is these first steps where most of the

thinking has to be done to solve problems, making this a very important lesson.

Lesson Content

Mass and Weight

As a creature living on the surface of the Earth, it would be easy to confuse mass and weight.

By definition, mass is the amount of matter an object contains. Matter is stuff, like the various

subatomic particles. A curious third grader could ask, “what is stuff or what is matter?”

Science has measured mass and there are some interesting theories that relate mass to energy.

In the end, mass is an observable and measurable quantity but it somewhat vague. As much

as science knows about nature, there is still much to be discovered. Weight is defined as the

force due to gravity on an object. On the surface of the earth, weight is the force associated

with an object of mass m being pulled so that it wants to accelerate at –g in the vertical

direction. Weight can be written as: ˆWeight WF F mgy . The magnitude of the weight is just

mg.

Normal Force

If a book is sitting on a desk, the desk will push back on the book just hard enough to keep it

from falling through the desk. The force exerted by a surface is always perpendicular to that

surface. In math conventions, the direction perpendicular to a surface is called the normal

direction. Therefore, the normal force is the force exerted by a surface in a perpendicular

direction to resist applied forces like weight. Figure 2 shows a normal force for a flat surface

and for a slanted surface.

Normal force on a flat surface Normal force on a slanted surface

Figure 2. Normal force and weight force.

Static friction

Friction is a type of force similar to normal force in the sense that reacts to other conditions.

Normal force is as big as it needs to be to stop a force from breaking a material at its surface.

Friction is a force that resists motion. In particular, static friction is a force that resists motion

without there being any motion. If you were to use a microscope to look at what you perceive

as a smooth surface, it may be surprising to find many jagged edges. When one jagged edge is

pushed against another and a force tries to make them slide across each other, there will be a

physical resistance. This microscope model of static friction is visualized in Figure 3.

Figure 3. Microscopic view of static friction.

There is a macroscopic approach to defining static friction that is very useful. There are two big

factors that determine how hard a static friction force can resist a motion. One of those factors

is how rough the surfaces are when pressed together. The roughness factor for a combination

of two surfaces is called the coefficient of static friction, symbolized by the greek lower case

letter mu with the ‘s’ for static, s. The other important factor is how hard you push the two

surfaces together. This one has already been defined. The force that a surfaces pushes back

to prevent its surface from being penetrated is called normal force, or FN. Putting these two

factors together creates the working definition of static friction:

,f static s NF F

The inequality is an important concept that states static friction is only as big as its needs to be

up to some maximum amount. The only time that the static friction equals the product of the

two factors is when the object is just about to move. Friction forces are parallel to the surfaces

creating the resistance.

Free Body Diagrams

A free body diagram, or FBD, is a simple vector diagram that shows all the forces acting on a

body without drawing everything around it. This diagram will be central to applying Newton’s

laws so it is important to be able to draw them correctly.

Conceptual Example – Drawing Free Body Diagrams

Draw a FBD for each of the case described below.

FBD Case Description FBD

object in free fall with no air

resistance

Object accelerating downward

with air resistance

Object falling at its terminal

velocity (constant speed)

Book sitting on a desk

(stationary)

Book sitting on a frictionless

desk being pulled with a force,

F (accelerating)

Book sitting on a desk with

friction being pulled with a

force, F, but not moving.

Calculator on a tilted desk with

no friction (accelerating)

Calculator on a tilted desk with

friction (stationary)

Conceptual Example – Drawing Free Body Diagrams, Solutions

FBD Case Description FBD Explanation

object in free fall with

no air resistance

The only force present is the weight

force. This object has not choice but

to accelerate downward.

Object accelerating

downward with air

resistance

The weight force is larger than the air

resistance force, so the net force is

downward as it the acceleration.

Object falling at its

terminal velocity

(constant speed)

The upward and downward forces are

the same size so they cancel each

other out for a net force of zero. Zero

net force means no acceleration

constant speed.

Book sitting on a desk

(stationary)

The normal force is just as big as the

weight force to prevent the book from

falling through the desk.

Book sitting on a

frictionless desk being

pulled with a force, F

(accelerating)

The lone horizontal force, F, has no

counterpart. Therefore, the book will

accelerate to the right.

Book sitting on a desk

with friction being pulled

with a force, F , but not

moving.

Despite being pulled to the right, the

book does not move. Static friction is

doing what it does – resist motion

within its limits.

Calculator on a tilted

desk with no friction

(accelerating)

In this tug-of-war, the two forces are

not in a straight line and cannot cancel

each other out.

Calculator on a tilted

desk with friction

(stationary)

In this two dimensional force diagram,

the three forces combine in the just

the right way to balance each other.

Lesson Summary

Weight is not always the same, but mass is the same for an object. Weight depends on

the pull of gravity on an object which can change in different places in the Universe.

Mass is a measure of how much matter something contains.

Normal force is the push perpendicular to a surface that resists objects from going

through them.

Static friction resists motion due to the roughness of the two surfaces and amount of

force pushing the two surfaces together.

Free body diagrams show an object in complete isolation with force vectors drawn to

show specific applications of pushes or pulls.

Review Questions

Review Problems

Further Reading / Supplemental Links

Points to Consider

1. In what circumstances can a normal force point downward?

2. If two boxes are attached by a rope, what would be the implications for the free body

diagram?

3. How would you find the net force on an object with three forces on its free body

diagram in various directions?

Applying Newton’s Laws: the Force Method – Lesson 3

Lesson Objectives

Know how to set up force problems using the Force Method

Understand interacting free body diagrams

Determine when a described system of objects is in equilibrium or not

Vocabulary

equilibrium

Check Your Understanding

1. How do you find the weight of a 10kg object?

2. Describe the interaction between a steep roof and a bag of nails on the roof?

3. Explain the equation: ,f static s NF F.

Introduction

Setting up a force problem requires a careful analysis of how objects will interact with each

other. Since forces are vectors, it is critical to keep track of them in terms of components and

apply Newton’s laws in the appropriate way. The end result will be an analysis that says why

an object will move or why it will not. This new skill will be gained by using clearly defined

steps in a “Force Method.”

Lesson Content

To understand Newton’s laws of motions, you should apply them. Applications of Newton’s

laws represent the basis of a lot of engineering. In the case of this lesson, you will only deal

with cases where the forces are in perfect balance. That means the objects in the analysis will

either be stationary or moving at a constant speed.

Applying Newton’s laws of motion can be accomplished with 4 steps collectively called the

“Force Method” which are shown in Table 2.

Step One: Draw a free body diagram. For every object of interest, draw a FBD and

include an arrow off of the body to indicate

the direction it is moving or the direction it

wants to move. Also, show your choice of x

and y axes.

Step Two: Break all forces into

components.

For each object, find the components for all

of the forces in whatever x and y directions

are defined by the chosen axes.

Step Three: Decide if each direction of

each object is in equilibrium or

not.

This step introduces the 2nd law of motion

into the problem solving. The sum of the

forces are equal to the net force, and net

force is netF ma.

If the object is in equilibrium, a = 0 and the

sum of forces equal zero.

Step Four: Solve for unknowns. This step involves following whatever

mathematical steps necessary to solve for

unknown quantities. You should also

introduce any other known relationships like

the equation for friction is applicable.

Table 2. The Force Method used to set up and solve physical situations involving the forces

between interacting objects.

Example

A 20.0kg box is pulled by a force of 225N to the left and an unknown force to the right at an

angle of 30 degrees above the horizontal.

a) What is the magnitude of force F if

the box is in equilibrium?

b) What is the normal force on the box?

Solution:

Use the force method to set up the problem and find the solution.

Step One: Draw a FBD.

Step Two: Break each force into components.

Each force needs to be written in terms of its x and y components. The x and y

axis are the conventional horizontal and vertical. A table can be used to list the

forces. In this example, only one force needs to have its components calcultated.

X y

F F cos30 F sin30

225N -225N 0

mg 0 -mg

FN 0 FN

Step 3: Decide if each direction of each object is in equilibrium or not.

In this set of lessons, the focus is on cases where the forces cancel out to create a

state of equilibrium. For both x and y directions, the sum of forces will equal zero.

X y

F F cos30 F sin30

225N -225N 0

mg 0 -mg

FN 0 FN

0xF 0yF

The upper case Greek letter sigma means to add up a series of terms. The xF is

replaced by the sum of everything in the x-column and the yF does the same for

the y-column. The end result will be a custom set of equations for this problem.

0xF cos30 225 0F N

0yF sin30 0NF mg F

Step 4: Solve for unknowns.

Rearranging the x-equation gives: cos30 225F N . 260F N *Answer to part a*

Using this result in the y-equation gives: 2260 sin30 20.0 9.8 0mNs

N kg F

66.1NF N *Answer to part b*

Lesson Summary

Applying Newton’s laws of motion to situations where objects interact can be done using

the force method, which uses the following four steps:

o Step 1: Draw a FBD

o Step 2: Break each force on each body into components.

o Step 3: Decide if the object is in equilibrium in the x and y axes.

o Step 4: Solve for unknowns.

Review Questions

Review Problems

Further Reading / Supplemental Links

Points to Consider

1. How would the force method be useful for structures that do not move, like a bridge?

2. Can the sum of forces equal to zero for an object that is moving?

Applied Statics – Lesson 4

Lesson Objectives

Be able to solve force problems for objects on inclined planes

Understand how to apply the force method to stationary interacting bodies

Apply the static force method to objects moving at a constant speed

Vocabulary

Check Your Understanding

1. In the use of the force method, when do you draw a FBD?

2. When do you apply the definition of static friction in the force method?

Introduction

The force method suggests that four steps can be applied to any physical situation to complete

an analysis of forces. This analysis is application of Newton’s laws of motion. By the end of

this lesson, you will be prepared to apply what you know in a new detailed method of analysis.

Lesson Content

Static cases are the focus of the applications of the force method and Newton’s laws in this

lesson. In static cases, an object or system of objects do not move. This lesson will prepare

you for using the force method to solve problems by exploring several examples.

Example: Interacting bodies with pulleys

Three boxes are connected to each other by light strings on a table top as shown using pulleys

to redirect the force directions. The masses are given as: m1 = 5.0 kg, m2 = 20.0 kg, and

m3 = 10.0kg. The coefficient of static friction is 0.38.

a) With friction present, draw free body diagrams of all three masses.

b) How much friction force is required on the second mass to keep the system in

equilibrium?

c) Mass is added to the first mass until the whole system is about to move to the left. How

much mass can be added to m1 without the system moving to the left?

Solution:

a) The friction force is to the

left since this system wants to

move to the right. This

assumption is based on the larger

mass hanging to the right.

b) It is tempting to try to use the definition of static friction and ignore the

conditional less-than equal-to symbol. Static friction force is only as big as

it has to be to resist motion up to its maximum holding ability. In this case,

friction force is just a variable that must be solved in step four of the force

method. Part (a) was step one. Step two does not require any right

triangle be constructed since all of the forces are already in the x or y

directions. A table is often useful to keep track of the components anyhow.

Box

1

Box

2

Box

3

x y x y x y

T1 0 T1 FN 0 FN T2 0 T2

m1g 0 -m1g T1 -T1 0 m3g 0 -m3g

Ff -Ff 0

m2g 0 -m2g

T2 T2 0

Step three involves summing up the forces in each column and setting it equal to

zero in the case of equilibrium. In actuality, the sum of forces is by definition the

net force and net force is equal to mass times acceleration. In the case of

equilibrium, acceleration is zero.

Box 1: Box 2: Box 3:

0yF 0xF 0yF 0yF

1 1 0T m g 1 2 0fT F T 2 0NF m g 2 2 0T m g

Step four will use the equations developed in step 3 to solve for unknowns. The

equations from box 1 and 3 can be used to solve for the tensions, and those results

can be plugged into the x-equation for Box 2.

1 1T m g and 2 3T m g 1 2 0fT F T becomes 1 3 0fm g F m g

Solving for the force of friction gives: 3 1fF m m g . To show work as clearly as

possible, numbers can be substituted at the last step. 210.0 5.0 9.8 mf s

F kg kg

49fF N

c) For this part, the free body diagram changes in one place. The force of

friction on block two will now point to the right since the system is about to

move to the left.

Step 1:

Step 2:

Box

1

Box

2

Box

3

x y x y x y

T1 0 T1 FN 0 FN T2 0 T2

m1g 0 -m1g T1 -T1 0 m3g 0 -m3g

Ff Ff 0

m2g 0 -m2g

T2 T2 0

Step 3:

Box 1: Box 2: Box 3:

0yF 0xF 0yF 0yF

1 1 0T m g 1 2 0fT F T 2 0NF m g 2 3 0T m g

Step 4:

This time friction and m1 are unknowns. However, since the system is at the

verge of motion, the equation for static friction becomes its maximum value so you

get an extra equation to use:

fs s NF F 2NF m g

1 1T m g 2 3T m g 1 2 0fT F T becomes

1 2 3 0sm g m g m g . 1 3 2sm m m 1 10.0 0.38 20.0m kg kg 1 2.4m kg

Any amount over 2.4kg will send the system into motion.

Example: Interacting bodies with an incline plane and variable force

A 40.0kg box on an inclined plane is attached to a hanging 8.0kg mass by a pulley connected to

the top of the incline. There is also a horizontal force applied to the box that changes in time

according to: 100.0 5.0 Ns

F N t .

a) At t = 0s, find the force of friction on box 1 to keep the system in equilibrium.

b) How much time before the system is about to move up the incline?

Solution:

(a) Step One: Draw a FBD for each object of interest in the system

Step Two: Break forces into components.

In the case of inclined planes, it is typically to your advantage to rotate the x and y axes so the

x-axis is parallel to the inclined plane. This makes the direction of motion or preferred motion

in one direction only. When you tilt the x-axis, the angle of the incline becomes the angle

between the weight force and the tilted y-axis.

Box 1 Box

2

x y x y

T T 0 m2g 0 -m2g

Ff Ff 0 T 0 T

F cos40F sin40F

m1g 1 sin40m g 1 cos40m g

FN 0 FN

Step 3: Determine equilibrium or not (build the equations)

Box 1: Box 2:

0xF 0yF 0yF

1cos40 sin40 0fT F F m g 1 sin40 0NF m g F

2 0T m g

Step 4: Solve for unknowns

Plugging in the tension from box 2 into the equation for box 1 will create an equation that can

be solved for friction force.

2 1cos40 sin40 0fm g F F m g 1 2sin40 cos40fF m g m g F

2 240.0 9.8 sin40 8.0 9.8 100.0 cos40m mf s s

F kg kg N 97.0fF N

(b) In the case of the box about to move UP the incline, the friction force will change its

direction. Also, since the box is about to move up the incline, the friction force will be

equal to its maximum amount just before slipping. The time that this occurs can be

found by taking the x-equation for box 1 and solving for time.

Since everything is the same from part (a) except the sign of the friction force, this solution

picks up from Step 3.

Box 1: Box 2:

0xF 0yF 0yF

1cos40 sin40 0fT F F m g 1 sin40 0NF m g F 2 0T m g

Step 4:

1 1sin40 100.0 5.0 sin40Nf s N s

F F m g F m g N t

2 1 1100.0 5.0 sin40 100.0 5.0 cos40 sin40 0N Ns s

m g m g N t N t m g

Collecting the time terms on one side gives:

1 25.0 cos40 sin40 1 sin40 100.0 cos40 sin40Ns

t m g m g N

1 21 sin40 100.0 cos40 sin40

5.0 cos40 sin40Ns

m g m g Nt

60.3t s

Lesson Summary

Objects connected by a single rope will feel the same tension in opposite directions from

Newton’s third law.

Pulleys change the direction of a tension force.

Static friction always opposes the way an object wants to move up to a maximum

amount, but only with as little force as necessary to keep it stationary.

Objects on inclines are easier to manage if the x and y axes are tilted so that the

direction of sliding motion is along the x axis.

The angle of an incline is the same as the angle between the weight and the tilted y-

axis.

Review Questions

Review Problems

Further Reading / Supplemental Links

•

Points to Consider

1. Can an object be in equilibrium and still move?

2. In the inclined plane example, once the system starts to accelerate, how will the

acceleration of box 1 compare to that of box 2?