Forces

1. Forces Chapter 2

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?

3. Summary of Important Equations to understand for the HW: • F = m · a • Weight = w = Fg = mg • Fcentripetal = m v2/r

4. 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.

5. Net Force = mass x acceleration Any push or pull that causes a change in the motion of (i.e., accelerates) a particle. • 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 netF = 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.

6. 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 = Dv/Dt, a = 0  Dv=0 … So velocity cannot change unless a force acts on an object. a = F / m

7. What is a Force ? (Vector) Force is simply: A PUSH or A PULL Forces are vectors they have both magnitude and direction

8. F2 = 20N +x If we now add a 2nd force, F2, pushing to the left, what happens? The minus sign tells us the direction! Total force = Ftot = 10 N – 20 N = -10 N Total Force F1 = 10 N M = 2kg -x 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 We have to consider both forces.. That is, the total force actingon the object. a = Ftot/M = -10 N / 2 kg = -5 m/s2

9. Total Force Force #1 Force #2 Force #1 Force #2 Forces are Vectors so Directions are Important Total Force = 0 Forces Add Forces Cancel!

10. If m is large  a is small F a = ----- m If m is small  a is large Inertia and Mass Inertia: Tendency for a body to maintain its state of motion, whether moving or at rest. Large inertia  it’s “hard” to accelerate the object Small inertia  it’s “easy” to accelerate the object Mass is the way we quantify “inertia”. A common unit of mass is kilograms [kg]

11. 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.

12. Weight Weight is defined to be the forceon an object due to gravity. w = weight in Newtons [N]m = mass in [kg]g = acceleration due to gravity = 9.8 [m/s2] w = mg (F = ma) • 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 !

13. 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

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

15. 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

16. 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

17. Counter-intuitive! Is there a natural state of motion? • 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

18. 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))

19. 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

20. 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 In Class Exercise #1: m = 1000kg F = ? N vi = 0m/s vf = 30m/s ti = 0s tf = 10s

21. 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)

22. Chair pushingup on man So, there must be another force presentpushing up on the man with equal magnitudebut opposite in direction to his weight. Reaction Force I 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 Weight(800 N) In this case, the chair exerts an upward force of800 N on the man  Fnet = 0

23. Newton's Third Law of motion Can’t touch someone without them touching you right back • 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?

24. 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!

25. 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)

26. 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…

27. The earth! Gravitational Force Everyone 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?

28. Gravity • The 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 downwardIt is given the special letter “g” (for gravity) g = 9.8 m/s2 • All objectswhich are allowed to fall near the surface of the earth will experience this same acceleration (neglectingair resistance) (feather vs billiard ball demo)

29. Law of Universal Gravitation • Fgµ 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

30. 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.

31. 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!

32. Forces Chapter 2

33. 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

34. 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