Chapter 3: The Causes of Motion: Dynamics

Chapter 3: The Causes of Motion: Dynamics Chapter 3 Goals: • To introduce the concept of inertia and its relationship to changes in motion • To show how the net force is the physical cause of changes in motion • To present and understand Newton’s First and Second laws of motion • To introduce the idea of momentum • As important examples: to discuss uniformly accelerated motion, and simple harmonic motion • To introduce the ideas of work and energy

Newton’s First Law Also called the law of inertia “An object in motion remains in motion, and an object at rest remains at rest, unless a net force acts on the object” Not a quantitatively useful result, but it captures an incredibly insightful observation about motion Until Isaac Newton, most people assumed that motion would cease unless the force continued to act, but this was because ‘friction’ is ubiquitous If an observer sees motion that obeys N1 (again, with no forces present), the observer is said to be in an ‘inertial frame of reference’ Example of non-inertial reference frame: the bed of an accelerating pickup truck

Some of the Forces in Nature

Relationship of Mass to Weight Weight force W is the gravity force on a body of mass m Mass m is the amount of matter in a body [m] = kg, slug g = 9.81 m/s2, so a 1 kg mass has a weight of 9.8 N g = 32.2 ft/s2, so a 1 slug mass has a weight of 32.2 lb • If an object ONLY feels the gravity force, then it is in free-fall, which means that it is accelerating downward at g (whether moving up or down!). • If the object is YOU, you would say you were ‘weightLESS’ but in fact what you lack is a force to counteract W.

Weight Force and Tension Force • Lamp is object • There are two forces acting on the object: • Tension force (from chain) is UP and therefore POSITIVE • Gravity force (from Earth) is DOWN and therefore NEGATIVE • By N1, if the two forces add to zero, the lamp is either motionless or moving at constant velocity (its acceleration is zero!) W

Weight Force and Normal Force • Monitor is object • There are two forces acting on the object: • Normal force (from desk) is UP and therefore POSITIVE • Gravity force (from Earth) is DOWN and therefore NEGATIVE • By N1, if the two forces add to zero, the monitor is either motionless or moving at constant velocity (its acceleration is zero!) N W

Newton’s Second Law Serves to define the mass of the object “When a net force acts on a body, the body accelerates in the direction of the net force. The acceleration is directly proportional to the force, and inversely proportional to the body’s mass” SI: [a] = m/s2 [m] = kg  [F] = kg-m/s2 := N(ewton) A 1 N force causes a 1 kg mass to accelerate at 1 m/s2 USA: [a] = ft/s2 [m] = slug  [F] = slug-ft/s2 := lb (‘pound’) A 1 lb force causes a 1 slug mass to accelerate at 1 ft/s2

What is a force? “When a net force acts on a body, the body accelerates in the direction of the net force. The acceleration is directly proportional to the force, and inversely proportional to the body’s mass” Often you literally feel them: when an object presses on you, your biology can sense it Usually: but from the moment you came to life you have felt gravity, and it acts at all points in your body, so you are NOT aware of it until it goes away What is a force? A force is a push or a pull. Enuf said? NO, but that definition has survived and works really well. It is a tautology: anything that is a push or a pull must be a force

An Alternative Formulation of N2, and a Mention of Newton’s Third Law Momentum of a moving object is p := mv Assume the mass is constant: then Therefore N2 takes the elegant form The real power of the momentum concept will emerge once we consider systems of bodies, and Newton’s Third Law: “To every action (force) on a body there is an equal but opposite reaction (force) (on a different body!)”

Remarks on T and N • The direction of N is by definition always ‘normal’ to the surface • If a surface exerts a non-normally directed force on a body, it is usually treated as some kind of friction—or GLUE! • Often, the normal force magically adjusts itself to suit what the kinematics is doing! • If a weight is supported by a surface, the normal force is equal to the weight in magnitude—but ONLY if the weight is not accelerating!

Remarks on T and N • T lives without change in all parts of the string or cable that is ‘under’ tension • T can go around corners by the use of pulleys • If a weight is tied to a string and it hangs, the tension is the weight—but ONLY if the weight is not accelerating! • Often, the tension magically adjusts itself to suit what the kinematics is doing! Example: a 2 kg hockey skate is accelerating sideways at 4 m/s2 on an ice rink because a child is pulling on its lace. Find the tension.

Some Scenarios Combining T, N and W Let the weight W of the body be 12 N (so what is mass?) At the beginning, the body is on a horizontal surface Tension T is provided by a hand pulling up on the rope  The body may or may not be in equilibrium (that is, a may or may not be zero), and there may or may not be a third force N T=12 N T=10 N T=15 N

The Spring Force There does NOT need to be a block, yet!!!

The Wonders of the Spring Force • It is much like tension or compression, depending whether the spring is stretched longer than equilibrium position, or squeezed shorter • There is a very elegant expression for the force exerted BY the spring: Hooke’s Law F = – k (x-x0) • Here, k is the spring constant • x is the location of the movable end of the spring • x0 is a constant too: the location of the movable end when the spring is neither stretched nor squeezed • Often, one takes x0to be zero

A graph of Hooke’s law F(x) Stiff spring F exerted BY spring Limp spring x x0 • Slope of the graph is –k • [k] = N/m

What kind of acceleration occurs to a mass? • At this instant, the spring is squeezed (x < 0) so its force is to the right (F > 0) • Therefore, acceleration of mass is to the right • If it is moving to the right, it will be speeding up and fly right through equilibrium, since a = 0 there • If it is moving to the left, it will be slowing down and eventually stop, since a is growing and opposed to the motion {show Active Figure AF_1502}

What kind of acceleration occurs to a mass? • The only horizontal force on the mass is the spring (ignore up/down forces) • Fnet = – kx = ma a = – kx/m • So the acceleration is in the opposite direction to the position, and proportional to it!! {show Active Figure AF_1501}

What kind of acceleration occurs to a mass? • What functions of time look an awful lot like their own second derivatives, to the point where if the function is added to its second derivative the answer could be zero? • NOT powers of t . . . Thinking . . . • AHA! Trig functions!! • What are the meanings of A, w and f? Let’s plug in and see what happens…

Plugging the trial solution into the DE Inserting this stuff into the differential equation, we get

Plugging the trial solution into the DE • It works great, if we take w = √(k/m) • No information about A or f at this stage Cosine works just as well!! • A is amplitude • T is the period and T = 2p/w= 2p √(m/k) • f is the phase constant – f {show Active Figure AF_1506} Here, f is roughly +150° = 2.6 rad

Relationship of Simple Harmonic Motion to Circular Motion • This kind of motion: x = A sin wt or A cos wt , with perhaps a phase constant f in there, is called Simple Harmonic Motion • One can think of it as a ‘projection’ of circular motion at the same angular speed/frequency {show Active Figure AF_1513} y A wt + f A sin (wt + f) x A cos (wt + f)

Relationship of v(t) and a(t) to x(t) A phase constant f = + p/2 –A wA – wA w2A – w2A

Problem 3.7 • A body is constrained to move on a straight path so that its acceleration is proportional to its displacement from equilibrium, that is a = -g x where g = 2.5 s-2 • Determine the period of the oscillations. • If the amplitude of the oscillations is 0.40 m, determine the speed of the body as it passes through the centre of oscillation. • c) [added] If the body is at 0.2m and moving to the right at t = 0.0 s, sketch x(t) and v(t) [hint: you must deal with the phase constant].

The definition of Work • Consider a force that varies with a body’s position F(x) • If the body moves a smalldisplacement dx, so small that F(x) doesn’t change much, in the same (or opposite) direction as the force, the small bit of work done by the force is dW := F(x) dx • This is of course the (signed) area of the thin strip of width dx and height F(x) on a graph of F versus x • For a non-small displacement (from xA to xB), the work done by the force is the integral of dW (area under F(x)): NOTE: this is the work done by any force!! If it is the net force, see the next slide!!

The definition of Kinetic Energy, and the Work-Energy Theorem • If the force is the net force we get • Since Fnet = ma = mdv/dt we have (watch closely!!) • Define Kinetic Energy of a body • Thus we have the Work-Energy Theorem: the work done on a body by the net force that acts in some process is equal to the change in kinetic energy of the body during that process, or WAB,net= DK

Units of Work and Energy • [W] = N-m = kg-m2/s2 = Joule (J) • If a force of 1 N acts on a body, and the body moves in the direction of the force a distance of 1 m, the force does 1 J of work (pretty small…) • If a force of 1 dyne (= 1 g-cm/s2) acts, and the body moves1 cm, the force does 1 erg of work (really small…) • If a force of 1 lb… moves 1 ft… 1 foot-pound of work • 1 calorie (cal) = 4.186 J [mechanical equivalent of heat] • 1 British Thermal Unit (BTU) = amount of heat needed to raise 1 lb of water 1 °F = 1055 J • 1 kcal = 1 Cal = 1 food calorie • 1 electron-Volt (ev) = 1.602x10-19 J • 1 kilowatt-hr = another common energy unit…

The Potential Energy associated with an x-dependent force, and the Mechanical Energy • Work • Here, f(x) is called the antiderivative of F(x) • If F is not a function of x, [“F ≠ F(x)”] you can’t do this! • Potential Energy Change of the body-force combination is DUAB := – DfAB • So we argue that when the K of a body grows, due to forces acting, those forces lose U (and vice versa) • Therefore, in such a process, the total mechanical energy does not change: DE=DK + DU = 0

Mechanical Energy and Energy conservation • Mechanical Energy of a situation E:=K+U • Mechanical energy change in a process is DE=DK + DU +ELOST • K is owned by the body (in m and v) • U is owned by the body’s location relative to the source (s) of the force(s), in some way • There are many forces in nature for which one cannot even define a potential energy function!! • Examples include FRICTION, TENSION, NORMAL… • If all the forces are ‘derivable’ from potentials, and if any non-potential-owning forces do no work, Mechanical Energy is Conserved: DE = 0

Potential energy facts • Suppose the process is in the presence of F(x) • Let us move from a starting point x0 to a variable endpoint x. Introduce the ‘dummy’ variable x’: • U(x0) is called the reference value of the potential energy and if you are clever you can make it 0 • Since U is the integral of F, F is the derivative of U

Potential energy example: Gravity • Almost trivial example: the weight force W = – mg • Reference potential energy U(y0) is evidently mgy0 • So just take your initial height to BE zero and the reference potential energy to be zero too. Freedom!! • Conclusion: potential energy of gravity is • Check: what is the force of gravity, from the potential?

Potential energy example: Spring • Important example: the spring force F = – k(x-x0) • Reference potential energy U(x0) is buried in there • But you can just take potential energy to be zero if x = x0 and so U(x0) is taken to be zero: thus one says

Potential energy example: Spring • Checking the integral as an area • For simplicity, take x0 = 0 • The end of the spring moves from xmax < 0 to 0 • The force of the spring is positive, and the area under the curve is too • Area = ½ (base)(height) = ½ (–xmax)((– k xmax)

Potential energy functions U(x) • What information is encoded in U(x)? • For x > 0, slope of U is positive, so F is to left • For x < 0, slope of U is negative, so F is to right • A restoring force • For x = 0, slope of U is zero, so F is zero • Stable equilibrium point • Unstable equilibrium point: object tends to ‘fly away’

Power: the rate of energy consumption • Power is the timerate of energy consumption, or production, or the time rate of doing work Formal definition and fairly useless Cool!! • [P] = J/s = Watt (W) • so 1 J = 1 W-s and 1 kW-hr = 3.6x106 J • a kW-hr of energy costs about 12 cents • 1 horse-power (hp) = 746 W

Energetics of Simple Harmonic Motion sin and sin2

Chapter 3: The Causes of Motion: Dynamics