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Lecture Notes for Chapter 2
- Reading Memo Insights:
- Where did Newton base his laws? Is it through inventions, accidents, or natural occurance?
- 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 Force → paid with Energy
- Motion is the key to life (e.g., strange effects of
relativity occur with speeds close to speed of light)
- 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.
- Space & Time are the very fabric of physical reality. And space & time are inexorably linked via velocity: v = d/t... i.e., motion!
- Motion is initiated by Forces which arise from the interaction of matter
- Forces come in pairs
- What is a Force? 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 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 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.
- 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).
- Forces are "paid" for with energy
- But another useful concept is to think of these changes occuring by virtue of a field, with the force simply being a response to this field.
- Motion is the key to life (e.g., strange effects of
relativity occur with speeds close to speed of light)
- 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)
- A measure of inertia
- 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
- Mass is an intrinsic property of matter
- 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
- Keeps block on inclined plane (increases with tilt)
- Let's you 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
- Four fundamental forces
- Newton's first law
- Object at rest remains at rest
- Unless acted upon by a net, external force
- Object in uniform motion (v=const) remains in motion
- Unless acted upon by a net, external force
- Thus: no motion and uniform motion are equivalent as far as forces concerned
- 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
- 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
- Different from Aristotle/intuition
- Force also needed to change direction (velocity is a vector)
- 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 "pseduo-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))
- Object at rest remains at rest
- 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: compute force required to accelerate a 1,000kg car from 0 to 30m/s in 10s
- Solution Process Outline: Create known/unknown table first
Known Unknown m = 1000kg F = ?N vi = 0m/s vf = 30m/s ti = 0s tf = 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
- Solution Process Outline: Create known/unknown table first
- 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 Force = constant Acceleration
- Zero Velocity (draw graph 1 p. 54) or Constant Velocity (Uniform Motion; graph 2)
- 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
- Composite of horizontal and vertical motion
- 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)
- We'll only deal with the simple situation of constant Forces
- 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 goes 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?
- 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)
- 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
- Law of Universal Gravitation
- F ∝ m1 m2/r2
- Really is univeral (acts between people)
- Different from Aristotlean cosmology
- In orbit, gravity gives rise to centripetal acceleration
- 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 = F
- Governs orbits of all stellar objects
- Gravitational Field
- VERY important concept
- Action at a distance very odd (what mediates)
- 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 mediated by the atoms and molecules that make up the rope itself (the field is like the array of atoms/molecules that make up the rope).
- Concept of field
- How something placed in space is affected by force -- i.e., map out what the forces on it will be at various points in space
- 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!
- 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