Chapter 4Chapter 4

Forces and Mass

Classical MechanicsClassical Mechanics

does not apply for• very tiny objects (< atomic sizes)• objects moving near the speed of light

Newton’s First LawNewton’s First Law

• If the net force F exerted on an object is zero the object continues in its original state of motion. That is, if F = 0, an object at rest remains at rest and an object moving with some velocity continues with the same velocity.

• Contrast with Aristotle!

ForcesForces

• Usually a push or pull• Vector• Either contact or field force

Contact and Field ForcesContact and Field Forces

Fundamental (Field) ForcesFundamental (Field) Forces

Types• Strong nuclear force• Electromagnetic force• Weak nuclear force• Gravity

Strong Nuclear ForceStrong Nuclear Force

• QCD (Quantum chromodynamics) confines quarksby exchaning gluons

• Nuclear force: binds protons and neutronsby exchanging pions

Electromagnetic ForcesElectromagnetic Forces

• Opposites attract, like-signs repel• Electric forces bind electrons in atoms• Magnetic forces arise from moving charges

Weak Nuclear ForceWeak Nuclear Force

• Involves exchange of heavy W or Z particle

• Responsible for decay of neutrons

GravityGravity

• Attractive force between any two bodies• Proportional to both masses• Inversely proportional to square of

distance

F =Gm1m2

r 2

Inertia (Newton’s First Law)Inertia (Newton’s First Law)

• Tendency of an object to continue in its original motion

MassMass

• A measure of the resistance of an object to changes in its motion due to a force

• Scalar• SI units are kg

Newton’s Second LawNewton’s Second Law

• Acceleration is proportional to net force and inversely proportional to mass.

rF∑ =m

ra

Units of ForceUnits of Force

• SI unit is Newton (N)

• US Customary unit is pound (lb)• 1 N = 0.225 lb

F =ma

1 N =1kg⋅m

s2

WeightWeight

Weight is magnitude of gravitational force

w=mg

w=GMearthm

r2

g=GMearth

Rearth2

weight

mass

Weight vs. MassWeight vs. Mass

• Mass is inherent property• Weight depends on location

Newton’s Third LawNewton’s Third Law

• Single isolated force cannot exist• For every action there is an equal and

opposite reaction

Force on “1” due to “2”

rF12 =−

rF21

Newton’s Third Law cont.Newton’s Third Law cont.

• F12 is action force F21 is reaction force• You can switch

action <-> reaction

• Action & reaction forces act on different objects

Action-Reaction PairsAction-Reaction Pairs

rn =− ′

rn

Fg =−Fg'

Define the Define the OBJECT OBJECT (free body)(free body)

• Newton’s Law uses the forces acting ON object

• n and Fg act on object

• n’ and Fg’ act on other objects

Assumptions for F=maAssumptions for F=ma

• Objects behave as particles• ignore rotational motion (for now)

• Consider only forces acting ON object• neglect reaction forces

Definition of EquilibriumDefinition of Equilibrium

rF∑ =0

Example 4.1aExample 4.1a

A Ford Pinto is parked in a parking lot

There is no net force on the Pinto

A) TrueB) False

Example 4.1bExample 4.1b

A Ford Pinto is parked in a parking lot

The contact force acting on the Pinto from the parking lot surface ______________ .

A) Points upwardsB) Is zeroC) Points downward

Example 4.1cExample 4.1c

A Ford Pinto drives down a highway on the moon at constant velocity (where there is no air resistance)

The Pinto’s acceleration is __________

A) Less than zeroB) Equal to zeroC) Greater than zero

Example 4.1dExample 4.1d

A Ford Pinto drives down a highway on the moon at constant velocity (where there is no air resistance)

The force acting on the Pinto from the contact with the highway is vertical.

A) TrueB) False

Mechanical ForcesMechanical Forces

• Strings, ropes and Pulleys• Gravity• Normal forces• Friction• Springs (later)

Some Rules for Ropes and PulleysSome Rules for Ropes and Pulleys

• Force from rope points AWAY from object• Magnitude of the force is tension• Tension does not change when going

over frictionless pulley

Example 4.2Example 4.2

a) Find accelerationb) Find T, the tension above the bowling ballc) Find T3, the tension in the rope between the pailsd) Find force ceiling must exert on pulley

a) a = g/6 = 1.635 m/s2

b) T = 57.2 Nc) T3=24.5 Nd) Fpulley=2T = 114.5 N

Example 4.3aExample 4.3a

2) Which statements are correct?Assume the objects are static.

T1 is _____ T2

cos(10o)=0.985 sin(10o)=0.173

A) Less thanB) Equal toC) Greater than

Example 4.3bExample 4.3b

2) Which statements are correct?Assume the objects are static.

T2 is ______ T3

cos(10o)=0.985 sin(10o)=0.173

A) Less thanB) Equal toC) Greater than

Example 4.3cExample 4.3c

2) Which statements are correct?Assume the objects are static.

cos(10o)=0.985 sin(10o)=0.173

A) Less thanB) Equal toC) Greater than

T1 is _____ Mg

Example 4.3dExample 4.3d

2) Which statements are correct?Assume the objects are static.

T1+T2 is ______ Mg

cos(10o)=0.985 sin(10o)=0.173

A) Less thanB) Equal toC) Greater than

Example 4.4Example 4.4

Given that Mlight = 25 kg, find all three tensions

T3 = 245.3 N, T1 = 147.4 N, T2 = 195.7 N

Cable Pull DemoCable Pull Demo

Inclined PlanesInclined Planes

• Choose x along the incline and y perpendicular to incline

• Replace force of gravity with its componentsFg,x =mgsinθ

Fg,y =mgcosθ

Example 4.5Example 4.5

Find the acceleration and the tension

a = 4.43 m/s2, T= 53.7 N

Example 4.6Example 4.6

Find M such that the box slides at constant v

M=15.6 kg

M

Forces of FrictionForces of Friction

• RESISTIVE force between object and neighbors or the medium

• Examples:• Sliding a box• Air resistance• Rolling resistance

Sliding FrictionSliding Friction

• Parallel to surface, opposite toother forces

• ~ independent of the area of contact

• Depends on the surfaces in contact

f ≤μsNf =μkN

μs > μk

Coefficients Coefficients of Frictionof Friction

f ≤μsNf =μkN

μs > μk

Static Friction, ƒStatic Friction, ƒss

• μs is coefficient of static friction

• N is the normal forcef

F

fs ≤μsN

Kinetic Kinetic Friction, ƒFriction, ƒkk

• μk is coefficient of kinetic friction

• Friction force opposes F• n is the normal force

F

f

f =μkn

Friction DemoFriction Demo

Example 4.7Example 4.7

The man pushes/pulls with a force of 200 N. Thechild and sled combo has a mass of 30 kg and the coefficient of kinetic friction is 0.15. For each case:What is the frictional force opposing his efforts?What is the acceleration of the child?

f=59 N, a=3.80 m/s2 / f=29.1 N, a=4.8 m/s2

Example 4.8Example 4.8

Given m1 = 10 kg and m2 = 5 kg:a) What value of μs would stop the block from sliding?b) If the box is sliding and μk = 0.2, what is the acceleration?c) What is the tension of the rope?a) μs = 0.5 b) a=1.96 m/s2 c) 39.25 N

Example 4.9Example 4.9

What is the minimum μs required to prevent a sled from slipping down a hill of slope 30 degrees?

μs = 0.577

Other kinds of frictionOther kinds of friction

• Air resistance, F ~ Area v2

• Rolling resistance, F ~ v

Terminal velocity:

Fresistance =CAv2

=mgat terminal velocity

Coffee Filter DemoCoffee Filter Demo

Example 4.9Example 4.9

An elevator falls with acceleration a = 8.0 m/s2. If a 200-lb person stood on a bathroom scale during the fall, what would the scale read?

36.9 lbs

Accelerating Reference FramesAccelerating Reference Frames

• Equivalent to “Fictitious” gravitational force

g fictitious =−aframe

Fictitious Force: DerivationFictitious Force: Derivation

Eq. of motion in fixed frame

x =v0t+12

at2

=v0t+12

Fm

t2

F-maf looks like force in new frame, maf acts like fake gravitational force!

x0 (t)=12

aft2

x−x0 (t) =v0t+12(F −maf )

mt2

Example 4.10Example 4.10

You are calibrating an accelerometer so that you can measure the steady horizontal acceleration of a car by measuring the angle a ball swings backwards.If M = 2.5 kg and the acceleration, a = 3.0 m/s2:a) At what angle does the ball swing backwards?b) What is the tension in the string?

θ = 17 degT= 25.6 N

θ

Example 4.11aExample 4.11a

A fisherman catches a 20 lb trout (mass=9.072 kg), and takes the trout in an elevator to the 78th floor to impress his girl friend, who is the CEO of a large accounting firm. The fish is hanging on a scale, which reads 20 lb.s while the fisherman is stationary. Later, he returns via the elevator to the ground floor with the fish still hanging from the scale.

In the instant just after the elevator begins to move upward, the reading on the scale will be ______________ 20 lbs.

a) Greater thanb) Less thanc) Equal to

Example 4.11bExample 4.11bA fisherman catches a 20 lb trout (mass=9.072 kg), and takes the trout in an elevator to the 78th floor to impress his girl friend, who is the CEO of a large accounting firm. The fish is hanging on a scale, which reads 20 lb.s while the fisherman is stationary. Later, he returns via the elevator to the ground floor with the fish still hanging from the scale.

On the way back down, while descending at constant velocity, the reading on the scale will be ________________ 20 lbs.

a) Greater thanb) Less thanc) Equal to

Example 4.11cExample 4.11cA fisherman catches a 20 lb trout (mass=9.072 kg), and takes the trout in an elevator to the 78th floor to impress his girl friend, who is the CEO of a large accounting firm. The fish is hanging on a scale, which reads 20 lb.s while the fisherman is stationary. Later, he returns via the elevator to the ground floor with the fish still hanging from the scale.

In the instant just before the elevator comes to a stop on the 78th floor, the mass of the fish will be ______________ 9.072 kg.

a) Greater thanb) Less thanc) Equal to