goals for chapter 4 to understand the meaning of force in physics to view force as a vector and...

Goals for Chapter 4

• To understand the meaning of force in physics

• To view force as a vector and learn how to combine forces

• To understand the behavior of a body on which the forces “balance”: Newton’s First Law of Motion

Goals for Chapter 4

• To learn the relationship between mass, acceleration, and force: Newton’s Second Law of Motion

• To relate mass and weight

• To see the effect of action-reaction pairs: Newton’s Third Law of Motion

• To learn to make free-body diagrams

5-1 Newton's Laws

We will focus on Newton's three laws of motion… buto Newtonian mechanics is valid for “everyday” situations

o It is not valid for speeds which are an appreciable fraction of the speed of lighto Viewed as an approximation of general relativity

o It is not valid for objects on the scale of atomic structureo Viewed as an approximation of quantum mechanics

o And not valid for objects in accelerating frames of reference!

5-1 Newton's First and Second Laws

Before Newtonian mechanics:o Some influence (force) was thought necessary to keep a

body movingo The “natural state” of objects was at rest

This seems intuitively reasonable (due to friction)

But envision a frictionless surfaceo Does not slow an objecto The object would keep moving forever at a constant speedo Friction is a force!

What are some properties of a force?

Forces

Characteristics of forces:o Unit: N, the newton; 1 N = 1 kg m/s2

o Forces are vectors

Net force is the vector sum of all forces on an object

Principle of superposition for forces:o A net force has the same impact as a single force with

identical magnitude and direction

There are four common types of forces

• #1: The normal force:

When an object pushes on a surface, the surface pushes back on the object perpendicular to the surface.

• This is a contact force.

5-2 Some Particular Forces The normal force:

o If you are standing on a surface, the push back on you from the surface (due to deformation) is the normal force

o Normal means perpendicular

There are four common types of forces

• #2: Friction force: This force occurs when a surface resists sliding of an object and is parallel to the surface.

• Friction is a contact force.

There are four common types of forces II

• #3: Tension force:

A pulling force exerted on an object by a rope or cord.

• This is a contact force.

There are four common types of forces II

• #4: Weight:

The pull of gravity on an object.

• This is a long-range force, not a contact force, and is also a “field” force.

5-2 Some Particular Forces

The gravitational force:o A pull that acts on a body, directed toward a second bodyo Generally we consider situations where the second body is

Earth

In free fall (+y direction chosen as UP, with no drag from the air):

5-2 Some Particular Forces In free fall (+y direction chosen as UP, with no drag

from the air):

We can write it as a vector:


Weight :

The name given to the gravitational force that one body (like the Earth) exerts on an object o It is a force measured in newtons (N)o It is directed downward towards the center

Example To relate weight to mass, consider an apple in free fall. The only force on the apple is the gravitational force which results in an acceleration of g. Applying Newton’s 2nd Law

Fnet = ma where Fnet = Fg = W and a =g

Example To relate weight to mass, consider an apple in free fall. The only force on the apple is the gravitational force which results in an acceleration of g. Applying Newton’s 2nd Law

Fnet = ma where Fnet = Fg = W and a =g

Fg = W = mg

Thus,

W = mg (mass – weight relationship)

Don’t confuse mass and weight!

• Keep in mind that the Newton is a unit of force, not mass.

• A 2.49 x 104 N Rolls-Royce Phantom traveling in +x direction makes an emergency stop; the x component of the net force acting on its -1.83 x 104 N. What is its acceleration?


Measuring weight:o Use a balance to compare a body to

known masses, find its mass, and compute its weight

o Use a spring scale that measures weight on a calibrated scale

o Weight is not the same as mass: a pan balance will read the same for different values of g, a scale will read differently for different values of g


Weight must be measured when the body is not accelerating vertically

o E.g., in your bathroom, or on a train

o But not in an elevator

o Or on DROP ZONE

o Or in orbit….


Example Normal force for a block resting on a horizontal surface that is:

o Accelerating vertically at ay:

o Vertically at rest:


Example Normal force for a block resting on a horizontal surface that is:

o Accelerating vertically at ay:

o Vertically at rest:

Answer: (a) equal to mg (no acceleration)

(b) greater than mg (with positive acceleration)

5-2 Some Particular Forces Frictional force or friction:

o Occurs when one object slides or attempts to slide over another

o Directed along the surface, opposite to the direction of intended motion

5-2 Some Particular Forces Tension force:

o A cord (or rope, etc.) is attached to a body and pulled tauto Cord pulls on the body with force T directed along the cord,

AWAY from the body!o The cord is said to be under tensiono The tension in the cord is T and is assumed to be the same

EVERYWHERE along the cord!

What are the magnitudes of common forces?

Forces!

A force:o Is a “push or pull” acting on an objecto A net force causes accelerationo Newton’s First Law of Motion:

If there is no NET force on an object it will remain in the same state of motion

Newton’s First Law

• Simply stated:

“An object at rest tends to stay at rest, an object in motion tends to stay in uniform motion.”


• More properly,

“A body acted on by zero net force moves with constant velocity and zero acceleration.”


• In part (a) a net force acts, causing acceleration.

• In part (b) the net force is zero, resulting in no acceleration.


Newton's first law is true in a non-accelerating frame

Inertial frames:

5-1 Newton's First Newton's first law is not true in all frames

You go around a corner in a car, and seem to slide SIDEWAYS!?

Air moving south from the north pole towards the equator seems to move to the WEST!!!??

You are on a bus moving away from a stop sign, and your physics books slides backwards with no one touching it!!!!???


Newton's first law is not true in accelerating frames

(a): a frictionless puck, pushed from the north pole, viewed from space

(b): the same situation, viewed from the ground

Over long distances, the ground is a non-inertial frame

In (b), a fictitious force would be needed to explain deflection

When is Newton’s first law valid?

• You are on roller skates in a stopped BART car…

• The car starts to accelerate forwards.

• What happens to you?

• If no net force acts on you, you maintain a constant velocity (0!)


• You are on roller skates in a moving BART car…

• The car starts to slow - accelerate backwards.

• What happens to you?

• If no net force acts on you, you maintain a constant velocity


• If no net force acts on you, you maintains a constant velocity (a vector!)

• But as seen in the non-inertial frame of the accelerating vehicle, it appears that you are being pushed to the outside!

• Newton’s first law is valid only in non-accelerating inertial frames.

When is Newton’s first law valid?• If NO net force acts on rider, rider maintains a constant

velocity. But as seen in non-inertial frame of the accelerating vehicle, it appears that rider is being pushed.

• Newton’s first law is valid only in non-accelerating inertial frames.


•“An object at rest tends to stay at rest, an object in motion tends

to stay in uniform motion.”


•Mathematically,

“A body acted on by zero net force moves with constant velocity

and zero acceleration.”

Newton’s First Law II

• In part (a) net force acts, causing acceleration.

• In part (b) net force = 0 resulting in no acceleration.

Drawing force vectors

• Use a vector arrow to indicate the magnitude and direction of the force.

Superposition of forces

• Several forces acting at a point on an object have the same effect as their vector sum acting at the same point.

Decomposing a force into its component vectors

• Choose coordinate system with perpendicular x and y axes.

• Fx and Fy are components of force along axes.

• Use trigonometry to find force components.

5-1 Newton's First and Second Laws Principle of superposition for forces:

o A net force has the same impact as a single force with identical magnitude and direction

o Restate 1st law even more correctly:

Notation for the vector sum

• Vector sum of all forces on an object is resultant of forces

• The net force.

1 2 3

R=F +F +F + = F


• Force vectors are added using components: Rx = F1x + F2x + F3x + … Ry = F1y + F2y + F3y + …

F1 F2

F3



F1 F2

F3

F3x

F1x

Rx



F1 F2y

F3

F1y

F3y

Ry

5-1 Newton's First and Second Laws Superposition Examples

5-1 Newton's First and Second Laws Superposition Examples

Answer: (c), (d), (e)

Decomposing a force into its component vectors

• Choose perpendicular x and y axes.

• Fx and Fy are the components of a force along these axes.

• Use trigonometry to find these force components.

Notation for the vector sum—Figure 4.7

• The vector sum of all the forces on an object is called the resultant of the forces or the net forces.

1 2 3

R=F +F +F + = F


• Force vectors are most easily added using components: Rx = F1x + F2x + F3x + … , Ry = F1y + F2y + F3y + … . See Example 4.1 (which has three forces).

Newton’s Second Law

• If the net force on an object is not zero, it causes the object to accelerate.

An object undergoing uniform circular motion

• An object in uniform circular motion is accelerated toward center of circle.

• THEREFORE

Net force on object must point toward the center of the circle.

Force and acceleration

• The acceleration of an object is directly proportional to the net force on the object.

F

a

Mass and acceleration

• The acceleration of an object is inversely proportional to the object’s mass if the net force remains fixed.

Newton’s second law of motion

• The acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to the mass of the object.

• The SI unit for force is the newton (N).

1 N = 1 kg·m/s2

m F a

Using Newton’s Second Law—Example

5-1 Newton's First and Second Laws What is mass?

o “the mass of a body is the characteristic that relates a force on the body to the resulting acceleration”

o Mass is a measure of a body’s resistance to change in motion

o Literally, # of protons, neutrons, & electrons (regardless of how they are grouped into kinds of elements!)

o It is not the same as weight, density, size etc.

o Same force on LARGER mass produces smaller acceleration!

5-1 Newton's First and Second LawsExample Apply identical 8.0 N force to various bodies:

o Mass: 1kg → acceleration: 8 m/s2

o Mass: 2kg → acceleration: 4 m/s2

o Mass: 0.5kg → acceleration: 16 m/s2

And if you saw an acceleration from the application of this force…

o Acceleration: 2 m/s2

Then you can deduce…o Mass = 4 kg

5-1 Newton's First and Second Laws Summarize these behaviors as:

As an equation, we write:

Identify the body in question, and only include forces that act on that body!


As an equation, we write:

Separate applied forces into components along coordinate system axes


If the net force on a body is zero:

o Its acceleration is zero

o The forces and the body are in equilibrium

o But there may still be forces! They just must cancel!


• If the net force on an object is zero, the object will not accelerate.


• If the net force on an object is not zero, it causes the object to accelerate.

An object undergoing uniform circular motion

• An object in uniform circular motion is accelerated toward the center of the circle.

• So net force on object must point toward the center of the circle.

Force and acceleration

• Magnitude of acceleration of an object is directly proportional to the net force on the object.

• | | = F/m

F

a

Mass and acceleration

• Magnitude of acceleration of an object is inversely proportional to object’s mass if net force remains fixed.

• | | = F/ma

Newton’s second law of motion

• The acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to the mass of the object.

• The SI unit for force is the newton (N).

1 N = 1 kg·m/s2

m F a

Using Newton’s Second Law

• Worker pushes box of mass 40 kg, with constant force of 20N. What is acceleration?

Using Newton’s Second Law

Using Newton’s Second Law II—Example

• Shove bottle of mass 0.45 kg at initial speed of 2.8 m/s a distance of 1 m before it stops. What is the magnitude and direction of force on bottle?


• We know:

• Displacement in x = +1.0 m

• Initial x velocity = +2.8 m/s

• Final x velocity = 0 m/s

• THREE THINGS!


• vf2 = vi

2 + 2ax

• So…

• a = (vf2 - vi

2)/2x

• a = - 3.9 m/s2

• NOTE a points in the direction of f !

+x

Systems of units

• SI: force in Newtons, distance in meters, mass in kg.

• British: force in pounds, distance in feet, & mass in slugs.

• CGS: force in dynes, distance in centimeters, & mass in grams

Mass and weight

• The weight of an object (on the earth) is the gravitational force that the earth exerts on it.

• The weight W of an object of mass m is

W = mg

• The value of g depends on altitude.

• On other planets, g will have an entirely different value than on the earth.

A euro in free fall

Don’t confuse mass and weight!

• Keep in mind that the newton is a unit of force, not mass.

Newton’s Third Law

• If you exert a force on a body, the body always exerts a force (the “reaction”) back upon you.

• A force and its reaction force have the same

magnitude but opposite directions.

• These forces act on different bodies.

Objects interact when they push or pull on each other:

We can write this law as a scalar or vector relation:

We call these two forces a third-law force pair

Any time any two objects interact, there is a third-law force pair

We call these two forces a third-law force pair

Any time any two objects interact, there is a third-law force pair

Applying Newton’s Third Law: Objects at rest

• An apple rests on a table. Identify the forces that act on it and the action-reaction pairs.

Applying Newton’s Third Law: Objects in motion

• A person pulls on a block across the floor. Identify the action-reaction pairs.

A paradox?

• If an object pulls back on you just as hard as you pull on it, how can it ever accelerate?

5-3 Applying Newton's Laws


Third-law force pairs:


Answer: (a) they increase (b) yes(c) they begin to decrease slowly (the gravitational force of Earth decreases with height—negligible in an

actual elevator)

(d) yes

A paradox?

• If an object pulls back on you just as hard as you pull on it, how can it ever accelerate?

Free-body diagrams

• A free-body diagram is a sketch showing all the forces acting on an object.

Free Body Diagrams To solve problems with forces, we often draw a free body diagram

The only body shown is the one we are solving for

Forces are drawn as vector arrows with their tails on the body

Coordinate system shown

Acceleration is NEVER part of a free body diagram – only forces on a body are present.

Free Body Diagrams

Answer: F3 = 2 N to the left in

both cases

5-2 Some Particular Forces Real world – Tension will distort a cord/cable/string!

They stretch!

Ideal world - A massless and unstretchable cord exists only as a connection between two bodieso It pulls on both with the same force, T

o True even if the bodies and cord are accelerating, and even if the cord runs around a massless, frictionless pulley

o These are useful simplifying assumptions


Answer: (a) equal to 75 N (b) greater than 75 N (c) less than 75 N

5-3 Applying Newton's LawsSample Problem A block of mass M = 3.3 kg, connected by a cord and pulley to a hanging block of mass m = 2.1 kg, slides across a frictionless surface

Draw the forces involved

Treat the string as unstretchable, the pulley as massless and frictionless, and each block as a particle


Draw a free-body diagram for each mass

Apply Newton's 2nd law (F = ma) to each block → 2 simultaneous eqs.

Eliminate unknowns (T) that are the same, and solve for the acceleration

For the sliding block:

For the hanging block:

Note the direction of “a” for the hanging block!!!

Combining we get:


For the sliding block:

For the hanging block:

Combining we get:

Plugging in we find a = 3.8 m/s2 and T = 13 N

Does this make sense? Check dimensions are correct, check that a < g, check that T < mg (otherwise acceleration would be upward), check limiting cases (e.g., g = 0, M = 0, m = ∞)


goals for chapter 4 to understand the meaning of force in physics to view force as a vector and...

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