ForcesChapter 2

Reading Memo Insights:

Where did Newton base his laws? Is it through inventions, accidents, or natural occurence?

Why is a car having zero force if it is speeding? Forces come in pairs; again, how does a wall

show force?

Summary of Important Equations to understand for the HW:

F = m · a Weight = w = Fg = mg

Fcentripetal = m v2/r

Motion → caused by Forces → paid with Energy

Motion is the key to life (e.g., strange effects of relativity occur with speeds close to speed of light) Space & Time are the very fabric of physical reality.

And space & time are inexorably linked via velocity: v = d/t... i.e., motion!

In Ch. 1, we were concerned with what was going on. We observed (recall that observation is one of the keys of the scientific method) what was going on and described it. Kinematics, the study of motion, is what we discovered last chapter.

The rest of this term will concern us with dynamics, or the study of the causes of motion.

Net Force = mass x acceleration

The law, Fnet=ma, is really a program, or a general rule or a recipe, for analyzing nature but the particular application of this program depends on the phenomena we're studying (e.g., gravitation).

E.g., you might know the net acceleration of a certain mass so you find that the net F = m · a. BUT, there might have been many forces contributing to the object's net acceleration (e.g., both gravitational and electrical). In this case, the net force is: Fnet = Fgrav + Felec = m · a.

So we can observe the net acceleration and get the net force (Fnet = m · a) and set this equal to a sum of all the forces that are actually acting on it (we'll of course derive/compute a formula for each of the forces). This is the kind of reasoning you apply to the elevator problem: Fgravity + Fscale = m · a. When the elevator is at rest, the two forces are balanced since the net acceleration is 0 (i.e., Fgravity + Fscale = 0 → Fscale = - Fgravity). But when the elevator moves upward with a certain acceleration, you get: Fscale = - Fgravity + m · a (note: since gravity is defined in the downward direction, the negative of Fgravity is a positive).

For example, when something is subjected to a gravitational field, we observe that its motion changes; that it accelerates. In fact, it accelerates at a rate of g, the acceleration due to gravity. So what we observe when we place a mass, m, in a gravitational field is that it moves with a force Fnet=ma (where the force in this case is its weight, w=mg); that is the end result of the recipe, that is what we observe (the kinematics). But the specific causes (the dynamics) of that end result, the recipe itself, is found by seeing the gravitational force is the (only) one that's causing that motion: Fg = m · a, where Fg=GMm/r2. The m here is the same as the m in the end result, W=mg. But the a in this specific case for this particular phenomenon is a = GM/r2.

Any push or pull that causes a change in the motion of (i.e., accelerates) a particle.

Acceleration and Force

How do you make objects accelerate?

*** You MUST apply a force ***

If there is no force being applied on an object, it cannot accelerate. That is, it must have:

acceleration = 0

Since a = v/t, a = 0 v=0 … So velocity cannot change unless a force acts on an object.

a = F / m

What is a Force ? (Vector)Force is simply:

A PUSH A PULLor

Forces are vectors they have both magnitude

and direction

Forces are vectors they have both magnitude

and direction

Total Force

M = 2kgF1 = 10 N

If F1 is the only force acting on the mass M, we expect the block to accelerate to the right with an acceleration

a = F1/M = 10 N / 2 kg = 5 m/s2

F2 = 20N

If we now add a 2nd force, F2, pushing to the left, what happens?

We have to consider both forces.. That is, the total force actingon the object.

+x

a = Ftot/M = -10 N / 2 kg = -5 m/s2

Total force = Ftot = 10 N – 20 N = -10 NThe minus sign tells us

the direction!

-x

Forces are Vectors so Directions are Important

Force #1

Force #2

Force #1Force #2

Total Force

Total Force = 0

Inertia and Mass

Mass is the way we quantify “inertia”.

Inertia: Tendency for a body to maintain its state of motion, whether moving or at rest.

A common unit of mass is kilograms [kg]

Large inertia it’s “hard” to accelerate the object

Small inertia it’s “easy” to accelerate the object

Fa = ----- m

If m is large a is small

If m is small a is large

Forces and Energy

Motion is initiated by Forces which arise from the interaction of matter Forces come in pairs

Forces are "paid" for with energy But another useful concept is to think of these

changes occurring by virtue of a field, with the force simply being a response to this field.

WeightWeight is defined to be the force on an object due to gravity.

w = mgw = weight in Newtons [N]m = mass in [kg]g = acceleration due to gravity = 9.8 [m/s2]

Notice that weight is NOT the same as mass.

Mass has to do with the amount of matter inside the object

Weight depends on the mass, and also the value of “g”

On the moon, the value of “g” is only about 1/6th of the value on earth! Therefore, objects “weigh” about 1/6th as much !

(F = ma)

Mass vs. Weight Mass is an intrinsic property of matter

A measure of inertia Inertia = property of matter that makes it resist accelerations

Also a measure of the amount of matter Fundamentally, made of protons, neutrons, and electrons

Force on a car vs. same force on a book (less massive)

Weight = force of gravity acting on a body Weight is a "contact force" that acts in response to the force of

gravity

Mass is invariant under displacement; weight is not

Friction Mechanism

Corrugations in the surfaces grind when things slide.Lubricants fill in the gaps and let things slide more easily.

Friction Four fundamental forces

electromagnetic, gravitational, strong nuclear force, weak nuclear force

Gravity is the weakest We live in an electromagnetic world Friction = force of resistance to relative motion between two

bodies in contact Think of it at the atomic level Friction keeps block on inclined plane (increases with tilt) Friction allows you to walk:

Foot is still relative to floor Body adjusts forward If relative motion, foot would slide on floor instead

Static friction = no relative motion between objects Kinetic friction = relative motion between two contacting objects

Weaker than static friction

Newton's first law Object at rest remains at rest

Unless acted upon by a net, external force Object in uniform motion (v=constant) remains in

motion Unless acted upon by a net, external force

Thus: no motion and uniform motion are equivalent as far as forces concerned

Counter-intuitive!

Different from Aristotle/intuition The ancient Greeks had an interesting idea about existence; a complex mix of

science, philosophy, and theology. The changes were from Aristotle → Galileo → Newton

Aristotole thought everything came to a stop Galileo suggested that, barring any forces, something will continue in constant motion Newton extended this to say that no motion and uniform, linear motion were the same Just like when you’re on a train and see another train go by: is the other train moving

(and yours is still) or is your train moving and the other one standing still? Reason: we rarely see an object with NO forces acting on it Reading Memo Answer:

Car traveling at constant velocity has zero net force acting on it

All forces (impulse, gravity, air resistance, friction, etc.) all cancel each other out

If force acts, object accelerates

If force stops, acceleration stops

So car needs net force ONLY while speeding up or down

Is there a natural state of motion?

Force needed to change direction

A net external force must act on object to speed it up, slow it down, or change its direction (the vector nature of Force)

Therefore, a force is required to produce an acceleration!

Centripetal force produces centripetal acceleration

Changes direction only If cut, flies straight off at tangent (linear motion)

Tendency to move in straight line "gives rise" to centrifugal force

"Centrifugal" Force is a "pseudo-force" because it only arises in an accelerating frame of reference but "disappears" in an inertial frame of reference (e.g., if you are in a car going around a bend (accelerating reference frame), you feel a force pushing you towards the outside; but, to an observer on the sidewalk (in an inertial reference frame), it's just centripetal motion (and your tendency to go straight))

Newton's Second Law of motion

Net external force = gives rise to an acceleration F = m a Sideways force = causes centripetal acceleration

Fcentripetal = m v2/r

In Class Exercise #1: Compute force required to accelerate a 1,000kg car from 0

to 30m/s in 10s Find equations to fit known/unknown table

Find acceleration by Δv/Δt = Δv x 1/Δt to get correct units

Δ means change in any quantity. Therefore, Δv means CHANGE in velocity and is computed as Δv = vf - vi (always final - initial)

For m • a, separate numerical measurement into magnitude & units

Δv/Δt = Δv • 1/Δt = 30m/s • 1/10s = 3 m/s2

(1000 kg) • (3 m/s2) = (1000) (kg) • (3) (m/s2) = (3000) • (kg) (m/s2) =

3000 kg m/s2 = 3000 N

Known Unknown

m = 1000kg

vi = 0m/s

vf = 30m/s

ti = 0s

tf = 10s

F = ? N

Different kinds of forces: We'll only deal with the simple situation of constant Forces

Zero Velocity (draw graph 1 p. 54) or Constant Velocity (Uniform Motion; graph 2)

a = 0; net external F = 0; it's in equilibrium

Uniform Acceleration (draw graphs 3 & 4 on p. 54) -- velocity changes at a constant rate

Constant Acceleration = constant Force

Projectile Motion

Composite of horizontal and vertical motion

Example of a ship and relative motion of the ball, in the reference frame of the person on the ship and in the reference frame of the observer on shore

Horizontal motion continues unabated

E.g., Galilean relativity example of a rock falling on a ship as observed from shore

Vertical motion equivalent to throwing straight up and down

Constant g makes v decrease on way up and increase on way down

Simple Harmonic Motion

Example of spring

Restoring force (when displaced from equilibrium position); object oscillates up & down

Cyclical motion with a constant frequency

Recurs frequently in physics

Skip: Air resistance (Terminal speed = equilibrium = net force is zero): depends on speed (not distance, as in SHO)

Transparency #1 (Table 2.2 on p. 60)

Reaction Force I

Weight(800 N)

Suppose you are sitting on a chair. Are all the forces acting on the man shown in the diagram? A) YES B) NO

If the only force was the man’sweight pushing down, then hewould accelerate, right?

Fnet = ma

Chair pushingup on man

So, there must be another force presentpushing up on the man with equal magnitudebut opposite in direction to his weight.

In this case, the chair exerts an upward force of800 N on the man Fnet = 0

Newton's Third Law of motion Forces always come in pairs

Equal and opposite: FB on A = - FA on B

Push on wall, wall pushes back Otherwise, hand would go through!

Roller skate example Forward force (push on wall)

accelerates you in opposite direction

Think of it at the atomic level Like charges repel (sorta like magnets)

Air deflected (forced) downward produces lift on wing and propeller

Question: Can a propeller work in a vacuum?

Can’t touch someone without them touching you right back

A piano on a sidewalk

Piano does not fall through the sidewalk Did gravity disappear? No, stick your foot under it to see gravity’s still

pulling down on it So the piano is pushed downward by its weight

and the sidewalk, in turn, is pushed down by the weight of the piano

So why doesn’t piano fall through sidewalk? The sidewalk pushes back! Newton’s 3rd law!

Other Misc Forces Centrifugal "force" = reaction to net force Being pushed back in seat in accelerating car Skip: Pushing chair to overcome static friction

Push on chair so it doesn't move Forces balanced? Now how about when I push chair and it moves?

Skip: Rocket motion: rocket and exhaust form a system that just spreads out and stays in the same place

Transparency #2 (Concept Map 2.1 on p. 66)

Forces (II) Since two or more objects must be involved, force is intimately tied to the notion of an interaction.

Interactions are now believed to occur through the exchange of “force carriers”. This is a very important point, and we’ll come back to it later in the course…(particle exchange demo)

So far, we know only of four types of fundamental forces in nature:

Gravity, Electromagnetic, Weak, and Strong

All other forces in nature are understood to be the residual effects of these fundamental forces. We’ll come back to these later in thesemester…

Gravitational ForceEveryone knows that when you drop an object from, say 2 [m],it speeds up, and eventually hits the ground.

That is, …it accelerates..

According to Newton’s Second Law, if the object accelerates, thereMUST BE A NET EXTERNAL FORCE ACTING ON THE OBJECT.

The force which gets the credit for this is called the gravitational force.

But, if forces have to occur between 2 objects… What is the “other” object? The earth!

GravityThe acceleration an object (like you) experiences near the earth is due to the gravitational force which the earth exerts on you.

The acceleration near the surface of the earth is equal to 9.8 m/s2 downward

It is given the special letter “g” (for gravity)

g = 9.8 m/s2

All objects which are allowed to fall near the surface of the earth will experience this same acceleration (neglectingair resistance) (feather vs billiard ball demo)

Law of Universal Gravitation Fg m1 m2/r2

More mass means more gravity Less distance separating them means more gravity Distance is inversely proportional to gravity

Really is universal (acts between people and heavenly objects) Different from Aristotlean cosmology

In space, gravity gives rise to centripetal acceleration – e.g., orbits

On Earth's surface, gravity gives rise to a downward force Reaction (or contact) force that counters downward,

gravitational force is weight Gravitational acceleration derived from w = Fg Governs orbits of all stellar objects

Gravitational Field Concept VERY important concept Action at a distance is very odd (what mediates it?) E.g., imagine a really, really, really long rope (perhaps

thousands of miles long, going around the Earth). Suppose you yank on it.

When does other end know it's been pulled?

Is it instantaneous (which is what action at a distance would imply) or is it mediated by the atoms and molecules that make up the rope itself?

The field is analogous to the array of atoms/molecules that make up the rope.

Concept of field

Map out what the forces on an object will be at various points in space Shows how something

placed in space is affected by force

Goes out in all directions (in 3-d) but gets weaker with distance

NOT a wall; rather, a vector field (direction and magnitude)

Field itself causes force to act (Einstein)

Actual warping of spacetime!

ForcesChapter 2

Chaos == non-linear dynamics

xnext = rx (1 - x)

Universe harbors randomness but that randomness is ordered!

Butterfly effect Just can't know things to arbitrary accuracy Quantum Mechanics (HUP) limits to arbitrary

precision

Tides

Differing gravitational attraction on different parts of Earth by Moon

Differing gravity = differing weights Differing weights "flow" down inclined surface to

form tidal bulges