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Lecture Notes for Chapter 3
- Reading Memo Insights:
- Why is there no work done when carrying a box? Is the work done when one lifts the box up?
- I thought that momentum was an object's ability to keep going, or inertia (which is really equal to mass) and since mass doesn't change therefore momentum can't change? Is Newton's 2nd law true? So how can he have a change in momentum in his formula?
- What is energy?
- Summary of Important Equations to understand for the HW:
- Conservation Law: [#]before = [#]after
- p = mv (total mvbefore = total mvafter)
- Work = F · d
- KE = ½ mv2
- PE = mgh
- E = KE + PE (Total energybefore = Total energyafter)
- P= Work/time = Energy/time
- General Conservation Laws -- Blocks Analogy
- In the 1st chapter, we studied kinematics,
the study of motion, which led to → dynamics, the
study of the causes of motion, and now we're finally
led to → Energy (or Energetics) -- which is the
basis for EVERY process/transaction (how forces are
"paid" for) -- it underlies all physical phenomena and,
in fact, existence itself...
- But this idea of energy is strange and elusive...
- Energy is an abstract, mathematical idea -- but, just like
Copernicus' heliocentric "universe" & the idea of fields, it
just happens to be real!
- It's a numerical quantity which doesn't change
- It is NOT a description or a mechanism
- We have no idea of what energy is... it's just a strange fact of nature:
- Calculate some number
- Watch nature go through her tricks
- Calculate number again: same!
- Just like bishop on a red square after a number of moves
- Law of conservation of energy
- Certain quantity (energy) does not change
- Since abstract, illustrate by analogy (thanks to Prof. Feynman)
- Child has 28 blocks (indestructible, indivisible, identical)
- Mother counts the # of blocks → she discovers conservation of blocks ⇒ # of
blocks is always constant
- Number of blocks always constant (i.e.,
doesn't change)
- [# of blocks]before = [# of blocks]after
- Number of blocks always constant (i.e.,
doesn't change)
- One day, some blocks disappear in a 16 oz. box but she can't look inside the box.
- If you didn't know the weight of each block, you could use
the conservation law to figure out exactly how many
blocks were hidden in the box:
[# of blocks]before = [# of blocks]after constant = constant 28 = 28 28 = (# of visible blocks) + (# of hidden blocks) ⇒ (# of hidden blocks) = 28 - (# of visible blocks)
- If you didn't know the weight of each block, you could use
the conservation law to figure out exactly how many
blocks were hidden in the box:
- Suppose she knew, or confirmed, that each block weighed 3 oz:
- CONSTANT = 28 = (# of visible blocks) + ( [ (wt. of box) - 16oz ]/3oz )
- Some disappear in bathtub (originally 6 inches of water); block raises water 0.25 inches
- CONSTANT = 28 = (# of visible blocks) + ( [ (wt. of box) - 16oz ]/3oz ) + ( [(height of water) - 6 inches]/0.25 inches )
- Increasing complexity of her world increases number of terms representing ways of calculating # of blocks
- Complex formula, quantity which has to be
computed and always stays the same:
- 28 = (# of visible blocks) + (# of blocks hidden in box) + (# of blocks hidden in bathtub) + ...
- Complex formula, quantity which has to be
computed and always stays the same:
- Eventually, actual number of visible blocks goes to zero!
- The mother has to infer the existence of the number of blocks
- And, she can use the process in reverse... if she knows she should end up with 28 blocks, she can infer how many blocks should be in the box or tub
- This is what makes conservation laws so useful... if you know what the number should be (either before or after), you can find intermediate values
- Like block example, many different forms in which
energy hides (with a formula for each one) but total energy obeys strict conservation law
- gravitational, kinetic, heat, elastic, electrical, chemical, radiant, nuclear, and mass energy
- But TOTAL energy obeys strict conservation law
- I.e., [#]before = [#]after
- In the 1st chapter, we studied kinematics,
the study of motion, which led to → dynamics, the
study of the causes of motion, and now we're finally
led to → Energy (or Energetics) -- which is the
basis for EVERY process/transaction (how forces are
"paid" for) -- it underlies all physical phenomena and,
in fact, existence itself...
- 1st Conservation Law: Linear Momentum
- p = mv
- units of kg • m/s
- mass = measure of inertia/matter; velocity tells us how
space & time change and are inter-related (combines all
three fundamental quantities/qualities of physical
reality: space, time, & matter)
- In fact, Einstein combined space and time into a single space-time continuum; it turns out, that empty space, or void, has something called a "vacuum energy". In addition, matter is the same thing as energy (with c2 serving as the conversion factor between the two) so it turns out, all three things are, in some sense, energy...
- mass times velocity = new quantity ⇒ momentum
- Basis for a fundamental formulation of physics!
- Since velocity can change, momentum can change
- Reading Memo Answer: change in velocity is acceleration
- dp/dt = d(mv)/dt = m (dv/dt) = ma = Fnet
- F * dt = Impulse = change in momentum
- Law of conservation of linear momentum
- Total linear momentum of an isolated system is constant
- Constant means net, external (since isolated) force is zero
- F = dp/dt = d(Constant)/dt = 0
- Constant means net, external (since isolated) force is zero
- Total linear momentum of an isolated system is constant
- In collision, total mvbefore = total mvafter
- In-Class Exercise (Example 3.2): A 1000kg car
collides with stationary 1500kg car; final v of both is
4m/s; what was the vcar1 before the collision (see p. 87)
Known Unknown mcar1 = 1000kg vcar1, initial = ?m/s mcar2 = 1500kg vcar2, initial = 0m/s vcar1, final = vcar2, final = 4m/s - [Total Initial Momentum] = [Total Final Momentum]
- (Initial Momentum)car1 + (Iniitial Momentum)car2 = (Final Momentum)car1 + (Final Momentum)car2
- [mcar1(initial velocity)car1] + [mcar2(initial velocity)car2] = [mcar1(final velocity)car1] + [mcar2(final velocity)car2]
- m1vi + m2(0) = m1vf + m2vf
- m1vi = (m1 + m2)vf
- In-Class Exercise (Example 3.2): A 1000kg car
collides with stationary 1500kg car; final v of both is
4m/s; what was the vcar1 before the collision (see p. 87)
- Path-independent; don't need any information about how interaction occurred
- Why is p conserved? Because of Newton's 2nd and 3rd laws
- 2nd: dp/dt; 3rd: equal & opposite
- p = mv
- Work is key to energy
- Energy takes many forms (e.g., gravitational, elastic, etc.)
- A good metaphor for energy is financial assets/net worth, which comes in many forms: cash, investments, real estate, etc.
- Every interaction in our universe involves the
transfer or transformation of energy
- "That which is transferred when work is done" (p. 94) is very much like ancient Indo-European mystics
- An exact definition of energy is elusive (just like an exact definition of any of the fundamental quantities was impossible) so we do what we did before: we define it relative to some other quantity (in this case, a strange fact of nature we call Work)
- Look at lever (p. 90)
- We happen to find when we calculate (like blocks analogy):
- Fleft/Fright = dright/dleft
- F * d has same value on both sides == Work
- Task requires same energy whether we:
- move smaller dist with larger F
- move longer dist with smaller F
- Work = F · d (d is in the direction of the vector F) regardless
- We happen to find when we calculate (like blocks analogy):
- Units of Work: 1 J = 1 N-m -- A measure of how much energy is transferred or transformed
- In class Exercise Example 3.4 on p. 91: 30kg barrel raised 1.2m; amount of work done?
Known Unknown m = 30kg W = ?J Δd = 1.2m agravity = g = 9.8m/s2 - No mechanical work done in carrying box
- When lifted, work is done against gravity
- When dropped, by gravity
- Come in pairs: work is done by one force and against the other force in the pair
- Example of catching ball
- Move hand back when catching = smaller Force
- Work is still the same! (The ball has a certain KE which it has to dissipate in order to stop, v=0)
- Ball does work on hand
- Energy takes many forms (e.g., gravitational, elastic, etc.)
- 2nd Conservation Law: Energy
- Energy is the measure of a system's capacity to do work
- Energy is transferred (or transformed) when work is done
- Units of energy = units of work (Joules)
- In mechanics, two main forms of (mechanical) energy:
- Work done on a system results in Kinetic Energy whereas the ability to have work done by a system is called its Potential Energy
- KE = energy due to motion (can be converted to Work)
- Depends only on final speed
- KE = ½ mv2 (2v = 4KE)
- Derivation on p. 95
- W = Fd = (ma) (½ at2) = ½ m (at)2 = ½ mv2
- Derivation on p. 95
- KE is relative since v is relative (ship example)
- PE = energy due to position
- Equal to work done on it to get it to that position (STORED work, or the ability to do work)
- Work done in lifting it is stored as gravitational PE
- Different kinds of PE:
- Gravitational PE
- Elastic PE
- Depends on how strong spring is & how much stretched or compressed
- Internal PE (whenever you see Internal Energy == Think Heat)
- Just KE and PE of atoms (heat)
- Work always results in transfer of energy, transformation of energy, or both
- Work done on a system increases its E; done by system decreases E; within system ⇒ E transformation
- (Mechanical) Energy cannot be created or destroyed
- Total energybefore = Total energyafter
- E = KE + PE = CONSTANT
- In class Exercise (worked out on board) (p. 100): If I drop something from 10 m, how fast will it be going at bottom?
Known Unknown di = 10m v = ?m/s df = 0m agravity = g = 9.8m/s2 - v doesn't depend on mass! (mgh = ½ mv2)
- Energy is the measure of a system's capacity to do work
- Collisions & Power
- Elastic collision: KEbefore = KEafter
- Inelastic collision: KEbefore ≠ KEafter
- Irrespective, total linear momentum is always conserved
- In inelastic collision, like car crash:
- Some energy goes to Internal Energy (heat), sound, etc.
- Whenever you see internal energy, think heat
- Some energy goes to Internal Energy (heat), sound, etc.
- Power = rate of doing work or transferring energy
- P = Work/t = Energy/t
- Units of Power: Watts
- 1 W = 1 J/s
- Power and speed are both rates of change
- 3rd Conservation Law: Angular Momentum
- Angular momentum is rotational analogue of linear momentum
- ω = mvr
- Torque is rotational analogue of Force
- Example of person spinning on a chair with arms out then in:
- (mvr)out = (mvr)in
- Angular momentum is rotational analogue of linear momentum