Mechanics
Newton’s Laws of Motion

 Objectives Introduction to Newton's Laws of Motion

·               State Newton’s three laws of motion and display an understanding of their applications.

·               Demonstrate an understanding of the meaning of net force.

·               Distinguish between weight and mass and show how they are related.

·               Define inertia and its relationship to mass.

·               Demonstrate an understanding of the nature of frictional forces.

·               Understand the definition of free fall and the causes of air resistance and terminal velocity.

·               Describe the relationship between acceleration and force.

·               Describe the relationship between acceleration and mass.

·               Explain why the acceleration of objects in free fall do not depend upon the object’s mass.

·               Identify action-reaction forces when given an interaction.

Notes

So far we have studied linear motion, free fall, projectiles, vectors, and other ways to talk about things that move. In a car you step on the gas; we can calculate the car’s acceleration, how long it takes to get somewhere, or determine its direction.  A projectile is fired; we can calculate its landing position, time of flight, or velocity.  So far we have only focused on how things move.

kinematics – the study of how objects move; Galileo’s focus of study.

There are other things to think about in terms of moving objects.  In fact, it is the most asked question of all two year olds:  Why?  The next several lessons answer the question:  Why do things move the way they do?

dynamics – the study of why objects move the way they do; Newton’s focus of study.

Newton’s Laws of Motion are all intertwined.  To fully understand any one of them, you must know a little bit about each one.

Newton’s First Law of Motion – an object at rest will stay at rest unless acted on by an outside force and an object in constant motion will continue its motion unless acted on by an outside force; when no forces act on an object, there is no acceleration; SF = 0, a = 0.

Newton’s Second Law of Motion – acceleration is proportional to force, ; acceleration is inversely proportional to mass, ; therefore acceleration is proportional to the ratio of force to mass, ; .

Newton’s Third Law of Motion – for every action there is an equal and opposite reaction; the force in is equal to the opposite of the force out; Fin = -Fout.

Demonstrations and Notes

µ            Place a coin on the desk.  Move the coin by pushing it sideways, observing the cause of the motion.  Stop pushing the coin, observing the motion of the coin.  Determine what is required to put the coin in motion.  Place a ball on the desk.  Gently push the ball sideways, observing the motion of the ball.  Determine what caused the ball to move.  Determine what caused the ball to continue to roll.

force (F) – a push or a pull; vector; represented by vector arrows; all of the rules involving vectors apply to force vectors.

newton (N) – units of force; 1 .

F = m a  = N

µ            Place a book on the table and push it to the right with a force of 10 newtons.  Determine the force being applied to the book due to pushing.  Push the book with a force of 10 newtons to the right and 10 newtons to the left.  Determine the total force applied to the book due to pushing.  Push the book with a force of 8 newtons to the right and 10 newtons to the left.  Determine the total force applied to the book due to pushing.

net force (Fnet or SF) – the combination or sum of all forces acting on an object; calculated using vector addition.

equilibrium – state of balance; occurs when SF = 0 or net force acting on an object is zero.

Using Newton’s Second Law, Fnet = m a

If Fnet = 0, then either a = 0 or m = 0

Since mass never changes,

Therefore when Fnet = 0 then a = 0

Another way to state Newton’s Second Law of Motion is when SF = 0, a = 0; in other words, when no forces act on an object, there is no acceleration.

Imagine a ball rolling across the table at a constant velocity.  Since the ball is moving at a constant velocity, there is no change in the velocity or no acceleration.  Ignoring friction, there are no horizontal forces acting on the ball.  The obvious vertical force acting on the ball is gravity pulling it down.  This is the force due to its weight.

weight (W) – the effect of gravity on an object, always acts down toward the Earth (negative).

mass (m) – the quantity of matter in an object.

volume (V) – the amount of space taken up by an object.

Using Newton’s Second Law, Fnet = m a.  Since weight is a force, F = W.

Since the acceleration caused by the force is due to gravity, a = g

Since mass never changes, m = m

Therefore W = m g

Since the ball is traveling at a constant velocity, there is no acceleration; a = 0.  According to Newton’s Second Law of Motion, when there is no acceleration, the net force acting on the ball must also be zero, Fnet = 0.  If the weight of the ball is acting down, there must be another force acting on the ball in the up direction to balance out the weight, producing a net force of zero.  This force is known as the normal force.

normal force (FN) –force perpendicular to the surface of contact.

When the surface of contact is horizontal, the normal force is equal and opposite to the weight of the object on that surface, Fw = -FN,

Summary Review – Introduction to Newton’s Laws of Motion and Forces

ü             Newton’s First Law of Motion states that an object at rest will stay at rest unless acted on by an outside force and an object in constant motion will continue its constant motion unless acted on by an outside force.

ü             Newton’s Second Law of Motion states that acceleration is proportional to the ratio of force to mass, Fnet = m a.

ü             Newton’s Third Law of Motion states that for every action there is an equal and opposite reaction.

ü             A force is a push or a pull measured in newtons (N).

ü             Net force is the combination or sum of all forces acting on an object.

ü             When in a state of equilibrium, the net force acting on an object is zero.

ü             Weight is the force of gravity acting down on all objects with mass.

ü             The normal force is a perpendicular force that pushes surfaces together.

Demonstrations and Notes

µ            Cover the top of a glass with a note card.  Place a coin on top of the note card.  Flick the note card sideways and observe what happens to the coin.  Repeat with different objects of various mass.

µ            Place a large hoop on a narrow flask.  Balance a coin on top of the hoop.  Hit the hoop sideways very quickly and observe what happens to the coin.

Newton’s First Law of Motion – Law of Inertia; an object at rest will stay at rest unless acted on by an outside force and an object in motion will stay in motion unless acted on by an outside force; when no forces acts on an object, there is no acceleration; SF = 0, a = 0.

inertia – the reluctance of any body to change its state of motion; measured by the mass of an object; the larger an object’s mass, the more inertia it has.

The affects of inertia are easily seen when looking at a floater in a drink.  When the glass is rotated, the floater remains in the liquid at rest.  To see the floater, the glass must be turned slowly so the inertia of the glass and liquid move together.

Things tend to keep doing what they are already doing.  When outside forces (like friction, a push or pull, and gravity) act on an object, the motion often changes.  If friction is ignored, like in the vacuum of outer space, an object at rest will never move and an object in motion will continue its motion at a constant velocity forever!

Although a force is required to get an object moving, a force is not necessary to sustain the motion of the object.  However, frictional forces oppose motion causing objects in motion to eventually stop moving.

µ            Place a tablecloth on the table and set up dishes on the tablecloth.  Quickly remove the tablecloth by pull it horizontally.  Observe the dishes.

µ            Hang a ball of large mass from a string.  Connect another string to the bottom of the ball.  Slowly pull the string and observe the strings.  Using the concept of inertia, determine which string will break.  Connect another string to hang the ball.  Pull the lower string quickly.  Using the concept of inertia, determine which string will break.

Summary Review – Inertia

ü             Newtons First Law of Motion is often called the Law of Inertia.

ü             Inertia is the reluctance of any body to change its state of motion and is determined by the mass of an object.

ü             Objects tend to keep doing what they are doing.

ü             A force is required to get an object moving but a force is not necessary to sustain motion.

Demonstrations and Notes

µ            Place a ball on the desk.  Gently push the ball sideways, observing the motion of the ball.  Determine why the ball stopped rolling.

friction – force that acts between materials that touch; always opposes motion and shown parallel to the surface of contact; caused by the irregularities in the surfaces of the objects touching.

The smoother the surface, the less friction that is present.  Surfaces include the table, floor, air, or anything composed of atoms.  All objects interact differently with various materials.  The way they interact depends upon the composition of the material.  The more rough a material, the more friction that is present.

coefficient of friction (m) – constant that depends on the two surfaces in contact; no units; varies for different materials; represents the percentage of force lost due to friction.

µ            Place a book on the table and connect a spring scale to the book.  Pull the book sideways and observe the scale reading focusing on the reading when the motion begins.  Observe the scale reading after the motion has started.

static friction – the force that opposes the start of motion.

sliding friction – the opposing force between surfaces in motion; also called kinetic friction.

These types of friction are often experienced when moving large objects like a car.  Initially, it is difficult to get the car to move but once the motion starts, it is easier to maintain the motion.

Demonstration and Notes

There is another form of friction we have been ignoring up until now:  air resistance.

µ            Drop a ball and determine what forces act on the ball as it falls.

air resistance (Fdrag) – force of friction acting on an object moving through air; often called drag.

Since the ball is in free fall it is accelerating or increasing in velocity due to gravity.  As the ball increases speed the amount of air resistance also increases.  This is observed when sticking your arm out the window of a moving car.  If the car is going slow (5 m/s) you can feel the air hitting your arm when you stick it out the window.  When the speed is increased (40 m/s) the air resistance is also increased as noted by the way your hand jerks back when placed out the window.  The same holds true for the ball in free fall.  As the velocity increase due to the acceleration due to gravity, the force due to air resistance also increases, which decreases the acceleration of the ball.

Notice how the force of air resistance compares to the weight of the ball after a few seconds of free fall.  Eventually, the force of air resistance equals the weight of the ball.  When this occurs, the net force acting on the ball becomes zero.  According to Newton’s First Law, when there is no force acting on an object, there is no acceleration.  This can only occur in two ways:  no motion or constant velocity.  Since we know the ball is moving, the velocity must be constant.

terminal velocity – speed at which the acceleration of a falling object terminates because friction balances the weight.

Fdrag = W

Fnet = 0, a = 0

The ball is now moving at a constant velocity

Summary Review – Friction and Air Resistance

ü             Friction is a force that acts between materials that touch.

ü             Static friction opposes the start of motion.

ü             Sliding or kinetic friction opposes the motion of an object.

ü             Air resistance, or drag, is the force of friction acting on an object moving through air.

ü             Terminal velocity occurs when the force due to air resistance of a falling object is equal to the force of the object’s weight, creating a net force of zero to act on the object.

Demonstration and Notes

µ            Drop two balls of varying mass from the same height.  Determine which ball will hit the ground first.

Often it is said that gravity affects all objects the same.  This is true, but to explain why gravity affects all objects the same requires Newton’s Second Law of Motion.

Newton’s Second Law of Motion tells us that acceleration and force are proportional.  If we only take that into consideration, we are led to believe that the object that produces the larger force will accelerate faster.  The object with the larger weight will hit the ground first because it has a larger force due to gravity and the larger the force, the greater the acceleration.

Newton’s Second Law of Motion also tells us that acceleration and mass are inversely proportional.  If we only take that into consideration, we are led to believe that the object with the smaller mass will accelerate faster.  Applying the concept of inertia, the less mass an object has, the easier it is to move.  A more massive object will resist motion and take longer to begin free fall.  Therefore, since the smaller object is easier to move due to its smaller mass, it will hit the ground first.

What we often fail to realize is that both force (weight) and mass affect an object in free fall.  To determine the acceleration of an object in free fall, use all of Newton’s Second Law of Motion.

F = m a

Compare the ratios of force and mass for each falling object:

 small ball large ball

The small ball has a small force of weight and a small mass.  The large ball has a large force of weight and a large mass.  However, the ratios of force to mass for each object is equal!  Since the ratios are equal, each object has the same acceleration.  This is also proven mathematically.

Small Ball:  -9.8 m/s2 = g

Large Ball:  -9.8 m/s2 = g

Summary Review – Freefall Using Newton’s Second Law of Motion

ü             Both force and mass affect an object in free fall.

ü             All objects fall at the same rate because the ratio of weight to mass is always constant.

Demonstration and Notes

µ            Push on a large object like the wall, a desk, or a closed door.  Observe the forces acting on the door and the person pushing.

Newton’s Third Law of Motion – for every action there is an equal and opposite reaction; the force in is equal to the opposite of the force out; Fin = -Fout often called action-reaction.

When we think of a force, we usually speak of it as a push or a pull.  Although that is true, a more precise definition of a force would include the term interaction.

interaction – a mutual action between objects where each object exerts an equal and opposite force on the other.

Interactions always happen in pairs.  For example, you interact with the ground when you walk.  You push against the ground and the ground pushes against you.  To determine the action and reaction forces in an interaction, a simple formula is used.

Action:  Object A exerts a force on object B.

Reaction:  Object B exerts a force on object A.

To use the formula, simply determine what the object is acting on and reverse the roles.  If you are pushing on the wall, the wall is pushing on you, too!

Determine the action-reaction forces of the following interactions.

 A car is cruising down the street. A rocket is lifting off toward space. A ball is dropped off a very tall building. The car’s tires push down on the road. The rocket pushes down on the gas (rocket fuel exhaust). The Earth is pulling the ball down. The road is pushing up on the car’s tires. The gas pushes up on the rocket. The ball is pulling up the Earth!

To understand how action-reaction forces work, look at the situation where a ball is falling toward the Earth.  If the action is the Earth pulls the ball down, then the reaction must be the ball pulls the Earth up.  It is true to say the ball falls down to the Earth.  It is also true to say the Earth falls up to the ball!  How much the Earth moves is another story.  In terms of force, the Earth and the ball apply an equal and opposite force on one another.  Comparing the mass of the ball and the Earth, it is obvious that the Earth has a lot more mass.  Using Newton’s Second Law of Motion, the acceleration of both the ball and the Earth can be calculated.

The effects of Newton’s Third Law of Motion can be experienced and calculated using a less dramatic example:  firing a gun.  The gun applies a force to the bullet and the bullet applies an equal and opposite force to the gun.  Since these forces are equal, they can be labeled as F.  The mass of the bullet is small, labeled small m, and the mass of the gun is large, labeled large m.  To determine the acceleration of the two objects, use Newton’s Second Law of Motion.

F = m a

 Bullet Gun

The bullet has a much larger acceleration than the gun.  In other words, the bullet is fired at a high acceleration and the gun has a recoil of a small acceleration.  The same thing applies to the falling ball and the falling Earth.  The difference is that the Earth has an enormous mass compared to the ball so the acceleration of the Earth is so small we don’t even notice it!

Action-reaction forces are always equal in magnitude and opposite in direction.  However, they do not cancel one another out.  This can be confusing, but by looking at the systems involved, the key to Newton’s Third Law of Motion is revealed.  (Even Newton admitted this concept was difficult to fully understand!)

When looking at the systems of an interaction, the effect of the action-reaction forces in question are dependent upon how the systems are defined.  For example, when kicking a football, it is true to say that your foot hits the ball and the ball hits your foot.  The force you apply to the ball is equal and opposite to the force the ball hits your foot.  When viewing your foot and the ball as one system, it is true to say the forces cancel out and produce no net force.  However, when viewed independently, your foot hits the ball with a force that will accelerate the ball.  This holds true with Newton’s Second Law of Motion.  The force the ball applies to your foot is independent to the motion of the ball and does not cancel out.

Whenever action-reaction forces are internal to a system, they cancel each other out.  They do not cancel each other when either one is external to the system being considered.

An excellent example of this concept is seen through the explanation of the horse-cart problem.

Summary Review – Action-Reaction

ü             An interaction is a mutual action between objects where each object exerts an equal and opposite force on the other.

ü             Action-reaction forces do not always cancel each other out.  They are dependent upon the system.

Introduction – What’s Going On?

Newton’s Laws of Motion are based upon the concepts of inertia, acceleration, mass, force, and weight.  To see how these concepts affect one another, various experiments can be performed that help to show the relationship that exists between mass, acceleration, and force.  This lab contains three parts that apply the concepts of inertia, acceleration, mass, force, and weight in an effort to develop a conceptual understanding of Newton’s Laws of Motion.

This activity is based on Newton’s Second Law of Motion, which states:

Ř      Acceleration is proportional to force,

Ř      Acceleration is inversely proportional to mass,

Ř      Acceleration is proportional to the ratio of force to mass,

Ř

Part I – Setting Up

• Each group consists of three members:  a skater, a pusher, and a holder.
• The skater sits on the skateboard.  The pusher holds a bathroom scale to the skater’s back.  The holder holds the skateboard still and prevents anyone from hitting the walls.
• Select a starting point and finish line in the hall.
• Zero the scale while it is on the skater’s back.
• Determine the force to be applied to the scale.
• While the holder is holding the skater still, the pusher should apply the determined force.

Part II – Collecting Data

• On a predetermined signal, the helper releases the skater.
• Take notes, observing what occurs.  The skater should focus on what they feel, the pusher should focus on the scale, and the helper should focus on the big picture.
• Repeat the pushing of the skater with various forces, skaters, and distances pushed.  When changing skaters, use group members of varying weight.
• Each group member should try all three positions multiple times.

Part III – Wrap Up

• Write a few paragraphs explaining what was observed and how changing the variables affected the results.  (10 points)
• Answer the following questions.  (2 points each)

Ř      What effect did the increased force have on the acceleration of the skater?

Ř      How is the acceleration of the skater related to the mass of the skater?

Ř      How did the increased distance affect the acceleration of the skater?

Ř      When the force is constant, how does the final velocity relate to the distance traveled?

Ř      Explain applying a force to the skater and no change in motion occurring?

 While pushing the skater down the hall, maintain a constant force on the scale.