force and motion–i chapter 5 copyright © 2014 john wiley & sons, inc. all rights reserved

Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.

Force and Motion–I

Chapter 5

© 2014 John Wiley & Sons, Inc. All rights reserved.

Goals for this chapter

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 THREE Laws of Motion

(Aside – what is a physical LAW?)


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!


Forces

• A force is a push or a pull• A force is an interaction between two objects, or between an

object and its environment• A force is a VECTOR quantity, with magnitude and direction.• A net force causes an acceleration!


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



Superposition Examples



Superposition Examples

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

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.

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.

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

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

tends to stay in uniform motion.”

Newton’s First Law

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.

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 + …

F1F2

F3



F1F2

F3

F3x

F1x

Rx



F1F2y

F3

F1y

F3y

Ry



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

identical magnitude and directiono So we can restate 1st law even more correctly:



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.



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!



Example 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 → mass: 4 kg



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!

Newton’s Second Law

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

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 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/m

a

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 + 2aDx

• So…

• a = (vf2 - vi

2)/2Dx

• a = - 3.9 m/s2

• NOTE a points in the direction of f !

+x



Units of force:


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



A system consists of one or more bodies Any force on the bodies inside a system exerted by

bodies outside the system is an external force Net force on a system = sum of external forces

Forces between bodies in a system: internal forceso Not included in a Free Body Diagram of the system since

internal forces cannot accelerate the system


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):



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

We can write it as a vector:



Eq. (5-12)

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….



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



Example Normal force for a block resting on a horizontal surface that is:o Accelerating vertically at a

y:

o Vertically at rest:



Example Normal force for a block resting on a horizontal surface that is:o Accelerating vertically at a

y:

o Vertically at rest:

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

(b) greater than mg (see 5-13, with positive acceleration)



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

Figure 5-8



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!



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



Figure 5-9



Figure 5-9

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


5-3 Applying Newton's Laws

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

Eq. (5-15)

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.

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?



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



Sample 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

Figure 5-12 Figure 5-13


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

Figure 5-14


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 = ∞)



Sample Problem A block being pulled up a ramp:


Figure 5-15


Newtonian Mechanics Forces are pushes or pulls

Forces cause acceleration

Force Vector quantities

1 N = 1 kg m/s2

Net force is the sum of all forces on a body

Inertial Reference Frames Frames in which Newtonian

mechanics holds

Newton's First Law If there is no net force on a

body, the body remains at rest if it is initially at rest, or moves in a straight line at constant speed if it is in motion.

5 Summary


Mass The characteristic that relates

the body's acceleration to the net force

Scalar quantity

Newton's Second Law

Free-body diagram represents the forces on one object

Newton's Third Law Law of force-pairs

If there is a force by B on C, then there is a force by C on B:

Some Particular Forces Weight:

Normal force from a surface

Friction along a surface

Tension in a cord

Eq. (5-1)

Eq. (5-12)

5 Summary

Eq. (5-15)

force and motion–i chapter 5 copyright © 2014 john wiley & sons, inc. all rights reserved

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