Chapter 4

Dynamics: Newton’s Laws of Motion

Units of Chapter 4

• Force

• Newton’s First Law of Motion

• Mass

• Newton’s Second Law of Motion

• Newton’s Third Law of Motion

• Weight – the Force of Gravity; and the Normal Force

• Solving Problems with Newton’s Laws: Free-Body Diagrams

• Applications Involving Friction, Inclines

• Problem Solving – A General Approach

Units of Chapter 4

4-1 Force

A force is a push or pull. An object at rest needs a force to get it moving; a moving object needs a force to change its velocity.

The magnitude of a force can be measured using a spring scale.

4-2 Newton’s First Law of Motion

Newton’s first law is often called the law of inertia.

Every object continues in its state of rest, or of uniform velocity in a straight line, as long as no

net force acts on it.

4-2 Newton’s First Law of Motion

Inertial reference frames:

An inertial reference frame is one in which Newton’s first law is valid.

This excludes rotating and accelerating frames.

4-3 Mass

Mass is the measure of inertia of an object. In the SI system, mass is measured in kilograms.

Inertia The tendency of a body to maintain its state of rest or motion.

Mass is not weight:

Mass is a property of an object. Weight is the force exerted on that object by gravity.

If you go to the moon, whose gravitational acceleration is about 1/6 g, you will weigh much less. Your mass, however, will be the same.

4-4 Newton’s Second Law of Motion

• 1st Law: If no net force acts, object remains at rest or in uniform motion in straight line.

• What if a net force acts? Do Experiments.• Find, if the net force F 0 The velocity v changes

(in magnitude or direction or both).• A change in the velocity v (Δv)

There is an acceleration a = (Δv/Δt)

OR

A net force acting on a body produces an acceleration! F a

4-4 Newton’s Second Law of Motion

• From experiments: The net force F on a body and the acceleration a of that body are related.

• HOW? Answer by EXPERIMENTS! – The outcome of thousands of experiments over

hundreds of years:

a F/m (proportionality)

• We choose the units of force so that this is not just a proportionality but an equation:

a F/m

OR: (total!) F = ma

4-4 Newton’s Second Law of Motion

• Newton’s 2nd Law: F = maF = the net (TOTAL!) force acting on mass m

m = the mass (inertia) of the object.

a = acceleration of object. Description of the effect of F

F is the cause of a.

• To emphasize that the F in Newton’s 2nd Law is the TOTAL (net) force on the mass m, text writes:

∑F = ma ∑ = a math symbol meaning sum (capital sigma)

4-4 Newton’s Second Law of Motion• Newton’s 2nd Law:

∑F = ma

Force is a vector, so ∑F = ma is true along each coordinate axis.

∑Fx = max ∑Fy = may ∑Fz = maz

ONE OF THE MOSTFUNDAMENTAL & IMPORTANT

LAWS OF CLASSICAL PHYSICS!!!

Based on experiment! Not derivable

mathematically!!

4-4 Newton’s Second Law of Motion

Newton’s second law is the relation between acceleration and force. Acceleration is proportional to NET force and inversely proportional to mass.

(4-1)

4-4 Newton’s Second Law of Motion

The unit of force in the SI system is the newton (N).

Note that the pound is a unit of force, not of mass, and can therefore be equated to newtons but not to kilograms.

4-5 Newton’s Third Law of Motion

Any time a force is exerted on an object, that force is caused by another object.

Newton’s third law:

Whenever one object exerts a force on a second object, the second exerts an equal force in the

opposite direction on the first.

Law of action-reaction: “Every action has an equal & opposite reaction”.

(Action-reaction forces act on DIFFERENT objects!)

4-5 Newton’s Third Law of Motion

A key to the correct application of the third law is that the forces are exerted on different objects. Make sure you don’t use them as if they were acting on the same object.

4-5 Newton’s Third Law of MotionRocket propulsion can also be explained using Newton’s third law: hot gases from combustion spew out of the tail of the rocket at high speeds. The reaction force is what propels the rocket.

Note that the rocket does not need anything to “push” against.

4-5 Newton’s Third Law of MotionHelpful notation: the first subscript is the object that the force is being exerted on; the second is the source.

This need not be done indefinitely, but is a good idea until you get used to dealing with these forces.

(4-2)

Example 4-5

4-6 Weight – the Force of Gravity; and the Normal Force

Weight is the force exerted on an object by gravity.

Close to the surface of the Earth, where the gravitational force is nearly constant, the weight is:

4-6 Weight – the Force of Gravity; and the Normal Force

An object at rest must have no net force on it. If it is sitting on a table, the force of gravity is still there; what other force is there?

The force exerted perpendicular to a surface iscalled the normal force. It is exactly as large as needed to balance the force from the object (if the required force gets too big, something breaks!)

Example 4-6

m = 10 kg

The normal

force is NOTalways equal to the weight!!

m = 10 kg∑F = ma

FP – mg = ma

Example 4-7

4-7 Solving Problems with Newton’s Laws – Free-Body Diagrams

1. Draw a sketch.

2. For one object, draw a free-body diagram (force diagram), showing all the forces acting on the object. Make the magnitudes and directions as accurate as you can. Label each force. If there are multiple objects, draw a separate diagram for each one.

3. Resolve vectors into components.

4. Apply Newton’s second law to each component.

5. Solve.

4-7 Solving Problems with Newton’s Laws – Free-Body Diagrams

When a cord or rope pulls on an object, it is said to be under tension, and the force it exerts is called a tension force.

Example 4-11

NF

aF

mamgFFmaF

sm

Kg

N

m

FamaF

NNF

NNF

N

yN

ypyNyy

pxxxpx

py

px

0.78

0;00.980.20

/46.3

0.10

6.34

0.20)0.30)(sin0.40(

6.34)0.30)(cos0.40(

2

0

0

Example 4-13

NagmamgmF

smgmm

mma

ammgmm

fromsubtract

amamgmF

amamgmF

EEET

CE

CE

CECE

CCCCT

EEEET

10500)(

/68.0

)()(

)2()1(

)2(

)1(

2

Example 4-14

2

0.

2

mgF

aspeedconst

mamgF

T

T

The advantage of a pulley

Example 4-15

NF

F

FFF

FF

aFFF

pT

Tpy

RCRB

RBRCx

1700sin2

0sin2

0;0coscos

Problem 25 Take up as positive!

m1 = m2 = 3.2 kg

m1g = m2g = 31.4 N

Bucket 1: FT1 - FT2 - m1g = m1a

Bucket 2: FT2 -m2g = m2a

∑F = ma (y direction), for EACH bucket separately!!!

FT1

FT2

FT2

m2g

m1g

a

a

Problem 29 Take up positive!

m = 65 kg

mg = 637 N

FT + FT - mg = ma

2FT -mg = ma

FP = - FT (3rd Law!)

∑F = ma (y direction) on woman + bucket!

FTFT

a

FP

mg

4-8 Applications Involving Friction, Inclines

On a microscopic scale, most surfaces are rough. The exact details are not yet known, but the force can be modeled in a simple way.

We must account for Friction to be realistic!

– Exists between any 2 sliding surfaces.– Two types of friction:

Static (no motion) friction

Kinetic (motion) friction– The size of the friction force: Depends on the

microscopic details of 2 sliding surfaces.• The materials they are made of

• Are the surfaces smooth or rough?

• Are they wet or dry?

• Etc., etc., etc.

4-8 Applications Involving Friction, Inclines

• Kinetic Friction: Experiments determine the relation used to compute friction forces.

• Friction force Ffr is proportional to the magnitude of the normal force FN between two sliding surfaces.

DIRECTIONS of Ffr & FN are each other!!

Ffr FN

FaFfr

FN

mg

a

For kinetic (sliding) friction, we write:

k is the coefficient of kinetic friction, and is different for every pair of surfaces.

k depends on the surfaces & their conditionsk is dimensionless & < 1

4-8 Applications Involving Friction, Inclines

4-8 Applications Involving Friction, Inclines

Static friction is the frictional force between two surfaces that are not moving along each other. Static friction keeps objects on inclines from sliding, and keeps objects from moving when a force is first applied.

• Static Friction: Experiments are used again.

• The friction force Ffr exists || two surfaces, even if there is no motion. Consider the applied force Fa:

There must be a friction force Ffr to oppose Fa Fa – Ffr = 0 or Ffr = Fa

FaFfr

FN

mg

The object is not moving

∑F = ma = 0 & also v = 0

• Experiments find that the maximum static friction force Ffr

(max) is proportional to the magnitude (size) of the normal force FN between the two surfaces.

DIRECTIONS of Ffr & FN are each other!! Ffr FN

• Write the relation as Ffr (max) = sFN

s Coefficient of static friction

– Depends on the surfaces & their conditions– Dimensionless & < 1

– Always find s > k

Static friction force: Ffr sFN

4-8 Applications Involving Friction, Inclines

The static frictional force increases as the applied force increases, until it reaches its maximum.

Then the object starts to move, and the kinetic frictional force takes over.

4-8 Applications Involving Friction, Inclines

Example 4-16

Example 4-18

Example 4-19

Example 4-20

NamFF

smmm

Fgma

amamFgm

amFF

amFFF

amFgmF

NFF

NgmF

AfrT

BA

frB

BAfrB

AfrT

AfrTAx

BTBBy

Nkfr

AN

17

/4.10.20.5

8.96.19

8.94920.0

498.90.5

2

4-8 Applications Involving Friction, Inclines

An object sliding down an incline has three forces acting on it: the normal force, gravity, and the frictional force.

• The normal force is always perpendicular to the surface.

• The friction force is parallel to it.

• The gravitational force points down.

If the object is at rest, the forces are the same except that we use the static frictional force, and the sum of the forces is zero.

4-9 Problem Solving – A General Approach

1. Read the problem carefully; then read it again.

2. Draw a sketch, and then a free-body diagram.

3. Choose a convenient coordinate system.

4. List the known and unknown quantities; find relationships between the knowns and the unknowns.

5. Estimate the answer.

6. Solve the problem without putting in any numbers (algebraically); once you are satisfied, put the numbers in.

7. Keep track of dimensions.

8. Make sure your answer is reasonable.

xkxNkx

NNy

Gy

Gx

mamgmgmaFmgF

mgFmgFF

mgF

mgF

)cos(sinsin

cos0cos

cos

sin

Example 4-21

N 1 2F m m g fr N 1 2k kF F m m g

FpFfr

FN

(m1+m2) g

a

P fr 1 2 xF F F m m a a = 1.9 m/s2

Problem 48

For contact forces analyze forces on the two blocks separately.

Summary of Chapter 4

• Newton’s first law: If the net force on an object is zero, it will remain either at rest or moving in a straight line at constant speed.

• Newton’s second law:

• Newton’s third law:

• Weight is the gravitational force on an object.

• The frictional force can be written:

(kinetic friction) or (static friction)

• Free-body diagrams are essential for problem-solving