Dynamics: Why Things Move•A force is a push or a pull on

an object.•Forces cause an object to

accelerate…•to speed up•to slow down•to change direction

•Force is a vector.

• There must be a net force on an object in order to change the magnitude or direction of a velocity.

• Net force is the unbalanced (greater than zero) or resultant force that acts on an object.

• The unit of force is the Newton (N). A Newton is equal to a 2s

mkg

Types of Forces

•Contact forces: involve direct contact between bodies.•normal support force; friction

•Field forces: act without necessity of contact.•gravity; electromagnetic forces

of attraction and repulsion; strong and weak nuclear forces

Forces and Equilibrium• If the net force on a body is zero (add

up all the forces acting on a body and the resultant force is 0 N), the body is in equilibrium.

• The acceleration of the body is 0 m/s2; a = 0 m/s2.

• An object in equilibrium may be moving relative to us in a straight line at the same speed (called dynamic equilibrium).

• An object in equilibrium may appear to be at rest (called static equilibrium).

Balanced Forces•Balanced forces are equal and opposite and produce a net force of zero.•The acceleration of an object will be 0 m/s2 if the forces acting on the object are balanced

(Fnet = 0 N).

•When balanced forces act on an object:

the object is either at rest, orthe object is moving with constant velocity (same speed in the same direction).

Normal Force on Flat surface

Fn = m·g for objects resting on horizontal surfaces.

m·g

Fn

Galileo’s Thought Experiment

This thought experiment lead to Newton’s First Law.

Galileo’s Thought Experiment

Newton’s First Law: Inertia

• If an isolated object is at rest, it will remain at rest; if it is in motion, it will continue moving along a straight line at constant speed.

• The first law is also called the law of inertia. Inertia is the property of an object that resists acceleration (changes in speed and/or direction).

• The first law applies to an object that has balanced forces acting on it (the net force is zero).

Newton’s First Law: Inertia• For example, the horizontal component

of velocity, which is assumed not to change during the flight of a projectile, is an example of Newton’s law of Inertia.

• If ΣF = 0 N, then velocity is constant (acceleration = 0 m/s2).

• An object with a constant velocity is said to be in a state of equilibrium; its direction remains unchanged (it moves along a straight line); and its speed is constant.

• Keep in mind that a speed of zero (object at rest) is a constant speed.

Newton’s First Law: Inertia

•Inertia is why you wear a seatbelt. Without the seatbelt, you would continue to move forward in the car until an unbalanced force acts upon you to slow you down to rest.•Without the air bag and seat belt, when the front end of the car strikes an object, the driver would continue to move forward until hitting the steering wheel or the windshield.•The air bag and seatbelt help to reduce the accelerating force on the driver.

Newton’s First Law: Inertia

•Inertia causes the pendulum to swing forward, pushing up on the locking bar and causing the bar to catch a ratchet on the ratchet wheel. This locks the seat belt in place and prevents the person from continuing forward.

The First Law is Counterintuitive

Implications of Newton’s 1st Law

• If there is zero net force on a body, it cannot accelerate, and therefore must move at constant velocity, which means• it cannot turn,• it cannot speed up,• it cannot slow down.

What is Zero Net Force?

Even though there are forces on the book, they are balanced, therefore, there is no net force on the book.

F = 0

PhysicsA book rests on a table.

Gravity pulls down on the book.

FG

The table pushes up on the book.

FT

Diagrams

• Draw a force diagram and a free body diagram for a book sitting on a table.

N

W

Free Body Diagram

Physics

N

W

Force Diagram

Unbalanced Forces

• Unbalanced forces are not equal and opposite and do produce a net force.

• The net force is 2 N to the right.Fnet = 5 N + -3 N = 2 N

• Acceleration is always in the direction of the net force.

Net ForceNow let’s take a look at what happens

when unbalanced forces do not become completely balanced (or cancelled) by other individual forces.

An unbalanced forces exists when the vertical and horizontal forces do not cancel each other out.

Example 1

Notice the upward force of 1200 Newtons (N) is more than the downward pull of gravity (800 N). The net force is 400 N up.

Example 2Notice that while the vertical normal force and gravitation forces are balanced (each are 50 N) the force of friction results in an unbalanced force on the horizontal axis. The net force is 20 N left.

Another way to look at balanced and unbalanced forces

Newton’s Second Law: F = m·a

• The vector sum of all the forces acting on an object or system whose mass is m produces an acceleration a on the object.

• Mass can be considered a quantitative measure of inertia. If the mass is large the acceleration produced by a given force will be small.

m

ΣFa

Newton’s Second Law: F = m·a

• Mass and acceleration are inversely proportional if the force remains constant. If the same force is applied to a large mass and to a small mass, the large mass will have a small acceleration and the small mass will have a large acceleration.

• Ex. F = 10 N; m1 = 2 kg; m2 = 5 kg

a1 = 10 N/2 kg = 5 m/s2 (smaller m, larger a)

a2 = 10 N/5 kg = 2 m/s2 (larger m, smaller a)

Working 2nd Law Problems

1. Draw a force or free body diagram.2. Set up 2nd Law equations in each

dimension.Fx = m·ax and/or Fy = m·ay

3. Identify numerical data. x-problem and/or y-problem

4. Substitute numbers into equations. “plug-n-chug”

5. Solve the equations.

Newton’s Third Law

• If object A exerts a force on object B, then object B exerts an equal and opposite force on object A

FA on B = FB on A • In other words, you cannot

touch without being touched!• Forces always occur in pairs

acting in opposite directions on two different objects.

• The action and reaction forces NEVER act on the same object!

• For every interaction, the forces always occur in pairs and are equal and opposite.

• The hammer exerts a force on the nail directed to the right (Fhn); the nail exerts an equal force on the hammer directed to the left (Fnh).

• When an automobile accelerates, the force of the tires on the ground is the action force and the ground below the tires provides the reaction force. • The reaction force supplies the force that accelerates the car.

• When you sit on a chair, your weight pushes down on the chair; the chair pushes up with a force equal to your weight.• If the chair pushes up with a force less than your weight, you

would fall through it.• If the chair pushes up with a force greater than your weight, you

would be pushed up above the seat.

• If you were standing in a small boat and tried to jump across to the nearby dock, you would fall into the water. The force you exert on the boat as you jump pushes the boat away, and the equal and opposite force the boat exerts on you does not result in any forward motion for you.

Weight, Mass & Gravity

•The weight of an object is the downward force the object experiences on or near the Earth’s surface due to the pull of gravity.•All freely falling bodies, regardless of their mass, fall with the same acceleration due to the pull of gravity.

Weight, Mass & Gravity

• At a given location, the weight of an object must be proportional to its mass.

• Fw (weight) = mass·gravity

• unit = N• Gravitational acceleration g varies

from one location to another, so the weight of an object will vary according to its location. The mass of the object will remain constant no matter where the object is located.

Commonly Confused Terms

• Inertia is the resistance of an object to being accelerated

• Mass is the same thing as inertia (to a physicist).

• Weight is the gravitational attraction between masses

inertia = mass weight

Systems of Connected Bodies

• Newton’s laws also apply to a group of objects that are connected or in contact with one another. You find the acceleration of the system of connected objects by dividing the net force on the system by the total mass being accelerated.

totalm

ΣFa

Systems of Connected Bodies

• Force acting on m1 has to accelerate m1, m2, and m3 with acceleration a.

• m1, m2, and m3 will all have the same acceleration a.

• Ftotal = (m1 + m2 + m3)·a

Systems of Connected Bodies• Net or resultant force on each mass =

m·a

• Ftotal = F1 net + F2 net + F3 net

1 1

2 2

3 3

F net resultant m a

F net resultant m a

F net resultant m a

Systems of Connected Bodies

• Force between m1 and m2 has to accelerate both m2 and m3 with acceleration a.

• F1 on 2 = (m2 + m3)·a

Systems of Connected Bodies

• Force between m2 and m3 has to accelerate m3 only with acceleration a.

• F2 on 3 = m3·a

Systems of Connected Bodies

• Ftotal + F1 on 2 + F2 on 3 – F2 on 1 – F3 on 2 = (m1 + m2 + m3)·a

• F1 on 2 equal and opposite to F2 on 1.

• F2 on 3 equal and opposite to F3 on 2.

• Resulting equation: Ftotal = (m1 + m2 + m3)·a

Systems of Connected Bodies

• F1 > F2, therefore, the acceleration of all three masses will be to the right in the direction of the greater force

• Apply Newton’s second law: ΣF = m∙a

F1 – F2 = (m1 + m2 + m3)·a

Systems of Connected Bodies

• Remember that all three masses will have the same acceleration.

1 2

1 2 3

F - Fa =

m +m +m

Systems of Connected Bodies

• To determine F1 on 2, isolate m1 and the forces that act only on m1 to produce the acceleration.

• Apply Newton’s second law, with the direction of the acceleration as positive: ΣF = m∙a; F1 – F2 on 1 = m1·a

F2 on 1 = F1 - m1·a

Direction of F2 on 1 is to the left

Systems of Connected Bodies• Newton’s third law explains that F1 on 2

is equal in magnitude, but opposite in direction, to F2 on 1.

• To determine F1 on 2, isolate m2 and the forces that act only on m2 to produce the acceleration.

• Apply Newton’s second law, with the direction of the acceleration as positive: ΣF = m∙a; F1 on 2 – F3 on 2 = m2·a

F3 on 2 = F1 on 2 – m2·a

Direction of F3 on 2 is to the left

Systems of Connected Bodies• Newton’s third law explains that F2 on 3

is equal in magnitude, but opposite in direction, to F3 on 2.

• To determine F2 on 3, isolate m3 and the forces that act only on m3 to produce the acceleration.

• Apply Newton’s second law, with the direction of the acceleration as positive: ΣF = m∙a; F2 on 3 – F2 = m3·a

F2 on 3 = F2 – m3·a

Direction of F2 on 3 is to the right

• The procedure for determining the forces acting between the masses works for any type of contact forces found between masses.

Example: Comparing Tensions

Blocks A and B are connected by massless String 2 and pulled across a frictionless surface by massless String 1. The mass of B is larger than the mass of A. Is the tension in String 2 smaller, equal, or larger than the tension in String 1?

The blocks must be accelerating to the right, because there is a net force in that direction. We use the massless string approximation to directly relate the string tensions on A and B due to String 2: TA on B=TB on A

(FA net)x=T1-TB on A

= T1-T2 = mAaAx

so T1 = T2 + mAaAx

Therefore, T1 > T2.

Free-Body DiagramsFree-Body Diagrams

Drawing a Free-Body DiagramDrawing a Free-Body Diagram Identify all forces acting on the object. Draw a coordinate system. Use the axes defined in your pictorial representation. If those axes are tilted, for motion along an incline, then the axes of the free-body diagram should be similarly tilted. Represent the object as a dot at the origin of the coordinate axes. This is the particle model. Draw vectors representing each of the identified forces. Be sure to label each force vector. Draw and label the net force vector Fnet. Draw this vector beside the diagram, not on the particle. Or, if appropriate, write Fnet = 0.

Then check that Fnet points in the same direction as the acceleration vector on your motion diagram.a

Identifying Forces Identify “the system” and “the environment.” The system is the object whose motion you wish to study; the environment is everything else. Draw a picture of the situation. Show the object—the system—and everything in the environment that touches the system. Ropes, springs, and surfaces are all parts of the environment. Draw a closed curve around the system. Only the object is inside the curve; everything else is outside. Locate every point on the boundary of this curve where the environment touches the system. These are the points where the environment exerts contact forces on the object. Name and label each contact force acting on the object. There is at least one force at each point of contact; there may be more than one. When necessary, use subscripts to distinguish forces of the same type. Name and label each long-range force acting on the object. For now, the only long-range force is weight.

The Forces on a Bungee Jumper

T

w

nfk

The Forces on a Skier

T

w

The Forces on a Rocket

Fthrust

wD

Using Free-Body Diagrams

x nx gx x xF F F F ma 0 0 x xF ma

xx

Fa

m

y ny gy y yF F F F ma 0 0n g yF F ma

n gF F The vector sum of the forces of a free-body diagram is equal to ma.

Example: A Dogsled Race During your winter break, you enter a dogsled race in which students replace the dogs. Wearing cleats for traction, you begin the race by pulling on a rope attached to the sled with a force of 150 N at 25° above the horizontal. The mass of the sled-passenger-rope system is 80 kg, and there is negligible friction between the sled runners and the ice.

(a) Find the acceleration of the sled

(b) Find the magnitude of the normal force exerted on the surface by the sled.

n gF F F ma

cosx xF F ma sin 0y n yF F mg F ma 2cos / (150 N)(cos 25 ) /(80 kg) 1.7 m/sxa F m

2sin (80 kg)(9.81 m/s ) (150 N)(sin 25 ) 720 NnF mg F

Example: Unloading A Truck You are working for a big delivery company, and you must unload a large, fragile package from your truck, using a 1 m high frictionless delivery ramp. If the downward component of velocity of the package is greater than 2.50 m/s when it reaches the bottom of the ramp, the package will break.

What is the greatest angle q at which you can safely unload?

0 sinnx gx xF F mg ma sinxa g / sinx h

2 2( sin )( / sin ) 2x xv a x g h gh

sind xv v max

2

arcsin[ / ]

arcsin (2.5 m) / 2(9.81 m/s )(1.0 m)

34.4

d xv v

x

Holt Visual Concepts

• Force• Contact and Field Forces• Free Body Diagrams• Newton’s First Law• Mass and Inertia• Newton’s Second Law• Action and Reaction Forces• Newton’s Third Law• Comparing Mass and Weight• Normal Force

• The Physics Classroom: Newton’s First Law Animations• The Car and the Wall• The Truck and Ladder

Drag Force & Terminal Speed

• Drag forces slow an object down as it passes through a fluid as the fluid exert a kind of friction as the object moves through them.

• This “friction” is called drag and exerts a drag force on the object.

• Drag forces:• act in opposite direction to velocity.• are functions of velocity.• impose terminal velocity.• We will only look at the simplistic case of a

spherical object moving relatively quickly through air.

Drag Force & Terminal Speed• In this type of situation, the formula for the

drag force is:

where ρ is the density of air, A is the effective cross-sectional area of the body and v is the velocity (really the speed) of the body.

• C is called the drag coefficient and is not truly a constant (it varies in a complex manner as a function of the body’s shape, speed and density).

2D

1F C A v

2

Drag Force & Terminal Speed• The drag force will act

against the force that is accelerating an object through a fluid.

• If we consider the case of free-fall, then we can see that the drag force acts against the force of gravity that is accelerating an object towards the Earth.

• The figure shows the forces acting on a falling object.

No animals were harmed in the making of this lecture…

Drag Force & Terminal Speed• At some point, the

speed of the falling object becomes large enough that the drag force equals the force from gravity.

• At that point the object will no longer accelerate but rather continues to fall at a constant speed.

Drag Force in Free Fall

m·g

m·g

FD

m·g

FD

m·g

FD

When the drag force and the force from gravity become equal, the falling object has reached what is called its terminal speed vt

This speed is given by:

wt

2 Fv

C A

Drag as a Function of Velocity

• FD = b·v + c·v2

• b and c depend upon:• shape and size of object.• properties of fluid.

• b is important at low velocity.• c is important at high velocity.

Drag Force in Free Fall

for fast moving objects

FD = c·v2

c = ½·D··AwhereD = drag coefficient = density of fluidA = cross-sectional area

FD = b·v + c·v2

for slow moving objects

FD = b·v

mg

FD