PHY 2048 C - Lecture notes All of them PDF

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Summary

This is from the lecture and lab separate class (not studio physics) taught by Stephen Hill. He goes really fast in lecture, so these should help on stuff that you missed. He will post power points but not stuff he does on the projector, and these notes contain the projector work....

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1/9 Ch 1 & Ch 2 (start) Tuesday, January 9, 2018

9:33 AM

A Short Introduction: • We are first going to learn about classical laws of mechanics that govern the motion of moving bodies. ○ As formulated by Galilei, Kepler, and Newton • This will be followed by an introduction to wave motion and thermodynamics. • Newton's First Law (Law of Inertia): an object in motion stays in motion unless acted upon by an external force. • Relativity: concepts of space and time change at large relative velocities. • Quantum Mechanics: concept of matter changes on small length scales. • Classical laws of mechanics break down in these limits, and much remains to be discovered. Ch.1 International System of Units (SI Units) • Scientists measure quantities through comparisons with standards. • Every measured quantity has an associated unit. • Even though there exist an essentially infinite number of different physical quantities, we need no more than seven base units/quantities from which all others can be derived. • In mechanics, we really only need three bas unites: meters, seconds, and kilograms. • In thermodynamics, we need two more: temperature in Kelvin (K) and quantity in moles (mol). • Need to know from nano- (10-9) giga- (109). • Example: How many inches in 1 light-year?

- SI Unit Standards • 1792: French established a new system of weights and measures. ○ 1 m= distance from N. pole to equator/ 10-million • In 1870s: 1 m = distance between fine lines on Pt-Ir bar ○ Accurate copies sent around the world • In 1960s: 1 m = 1650763.73 * wavelength 86Kr (orange) • 1983 until now: ○ 1 m= distance light travels in 1/(299 792 458) sec • Standard of time: ○ Length of a day ○ Period of vibration of a quartz crystal ○ Now we use atomic clocks. • Current kilogram standard is a Pt-10Ir cylinder (IPK) in Paris (likely to change) • Accurate copies around the world, US version at NIST ○ Note: We measure "mass" in kilograms. Weight is something completely different, which we measure in Newtons (=kg*m/s2)

Ch. 2 • We will define the position of an object using the variable x, which measures the position of the object relative to some reference point (origin) along a straight line (x-axis). • When graphing, position is usually the y-axis and time is the x-axis. • Displacement: the difference in position. ○ DeltaX = X2 - X1 • Average velocity is the change in position over the change in time.

1/11 Ch 2 cont Thursday, January 11, 2018

9:22 AM

• • • • •

Reading: up to page 27 (Ch 2) Displacement ^x: ^x=x2-x1 -> final-initial position Like x, the sign of ^x is crucial. Its magnitude represents a distance The sign of ^x specifies a direction.

• • • • • • • • • •

Vavg or v = ^x/^t => (x2-x1)/(t2-t1) Like displacement, the sign of Vavg indicates direction. Average speed Savg: S= total distance/^t Savg doesn’t specify a direction; it is a scalar as opposed to a vector, and thus lacks an algebraic sign (it's always positive). Using the strict definition of average velocity, if the displacement is zero, the average velocity is also zero. v=(-10m + 10m)/6s-0 = 0 s=total distance/^t -> (18m+18m)/6s = 6ms-1 Vector=includes direction Scalar= doesn't include direction, only magnitude. The instantaneous velocity is v=lim as ^t->0 (^x/^t) => dx/dt = local slope ○ Just taking the tangent at a point. ○ Instantaneous speed = magnitude of v An object is accelerating if its velocity is changing Average acceleration aavg= a = (^v/^t) = (v2-v1)/(t2-t1) Instantaneous acceleration a: a=lim as ^t-> 0 is a=dv/dt When a position function is concave down, it is decelerating. When a position function is concave up, it is accelerating.

• • • •

What is the average velocity between t=0 and 4s?

What is the avg vel betwee t = 0s and 8s?

Based on 2, what is the average veloicty between t=4 and t=8? Zero because their position at both times was 17, which means their displacement is zero.

So it is a parabola.

What is the average speed between t=0 and t=4s?

What is the average speed between t=0 and t=8?

• Constant Acceleration: a special case ○ a= dv/dt = ^v/^t = (v-v0)/(t-0) • Projectiles/gravity problems indicate constant acceleration. Polynomials indicate not a constant acceleration. • If acceleration is constant, then velocity should be linear. Meaning, the acceleration can be solved by not using calculus. ○ v-v0 = at or ○ v= v0 + at

• It is rigorously true: ○ v-v0 = area under a(t) curve = at x-x0 = Area under v(t) curve = v0t + 1/2 at^2 • What we have discovered through integration. ○ a(t) = dv/dt -> ^v = x0->x{ dx = 0->t{v(t)dt = 0->t{(v0+at)dt = Area ○ x-x0 = v0t + 1/2 at^2 • One can easily eliminate a, t, or v0 by solving Eqs 2-7 and 2-10 simultaneously. Eq # 2.7 • 2.10 2.11

Eq

Missing quantity

v=v0+at x-x0 x-x0=v0t+1/2at^2 V v^2=v0^2

1/16 Ch 2 cont & start Ch 3 Tuesday, January 16, 2018

9:31 AM

Ex. From previous mini exam #1

• If one eliminates the effects if air resistance, one finds that ALL objects accelerate downwards at the same constant rate at the earth's surface, regardless of their mass. • The rate is called the free-fall acceleration g • The value of g varies slightly with latitude, but for this course g is taken to be 9.81 m/s^2 at the earth's surface • It is common to consider y as increasing in the upward direction. Therefore, the acceleration a due to gravity is in the negative y direction, i.e ay= -g= -9.81m/s^2 • In the free fall equations, y is substituted for x from the constant acceleration problems. AND ay is substituted for g. ○ Not on the exam equation sheet.

• In free fall problems, we will use v0, vtop, and vend ○ Vtop=0

In this problem, V0 = 10m/s^2 * 10s = 100 m/s

What if you didn’t know t ?

Ex. From the old mini exam: 1. You are driving home from school at 17.2 m/s along a straight road when you suddenly hit the brakes, causing your car to decelerate at 3.70 m/s^2 a. What is your stopping distance, i.e. the distance traveled from the instant you hit the brakes to when you come to a halt.

Part 1: During a 100m race, runner 1 starts perfectly at the gun and accelerates to the finish at 1 m/s^2. Runner 2 gets an awful start, only 1 second after the gun. What acceleration is required for runner 2 to win the race? Assume acceleration is constant throughout race.

CH 3 START • A vector is a quantity that has both a magnitude and a direction, e.g. displacement, velocity, acceleration… • Consider displacement as an example, if you travel from point A to B: ○ It doesn't matter how you get from A to B, the displacement is simply the straight arrow from A to B. ○ All arrows that have the same length ad direction represent the same vectors, i.e. • Note: overhead arrow is used to denote a vector quantity. • If you travel from point A to B, and then from point B to point C, your resultant displacement is the vector from point A to Point C. • In spite of the fact that vectors must be handled mathematically quite differently from scalar, the rules for addition are the same.

Resolving vector components:

They have length equal to unity (1), and the point respctively along the x,y,z axes of a right -handed Cartesian coordinate system.

1/18 Ch.3 cont Thursday, January 18, 2018

10:00 AM

• Resolving vector components: ○ Ax = A*cos(theta) ○ Ay= A*sin(theta) • The inverse process ○ Use Pythagorean theorem ○ Or tan(theta)= Ay/Ax • Theta is often (but not always) measured from x to y (in a right-handed sense around the z-axis).

a=5m, b=10m Theta(a) = 53.13 deg Theta(b) = 36.87 deg

• Average velocity, v-> = displacement/time interval = (r2-r1)/delta T = delta R/ delta T • Instantaneous velocity is then the derivative at that time.

a) A swimmer heads directly across a river, swimming at 4.10 m/s relative to still water. She arrives at a point 39.0 m downstre am from the point directly across the river, which is 78.0 m wide. At what angle is the resultant velocity of the swimmer, as measured relative to the line drawn directly across the river. b) What is the speed of the river current? c) What is the swimmer's speed relative to the shore? d) In what direction should the swimmer head so as to arrive at the point directly opposite his starting point> Express your ans wer in terms of the angle with respect to a line drawn directly across the river.

1/23 Ch. 3 cont Tuesday, January 23, 2018

9:31 AM

• Vertical motion is unaffected by horizontal motion. Meaning, if one ball is thrown horizontally from a high point, and another is dropped, they will fall at the same constant rate, irrespective of their horizontal component of motion. • a=-g acts in the -y direction. -Projectile Motion • Motion in a vertical plane where the only influence is the constant acceleration due to gravity. • In projectile motion, the horizontal motion and vertical motion are independent of each other, i.e. they do not affect each other. • This feature allows us to break the motion in two: the x and y components. ○ Vox = Vo * cos thetao ○ Voy = Vo * sin thetao • Vx = Vox + AxT ○ Ax = 0 ○ Vx = Vo * cos thetao

This only works if on horizontal ground.

A baseball player hits a home run high into the stands, to a height of 45.0 m above the level playing field. The ball remains in flight for 4 s and travels a horizontal distance of 130 m. Determine the initial speed and the initial angle of the trajectory (assume no air resistance)

• Although the speed, v, does not change, the direction of the motion does, i.e. the velocity, which is a vector, does change. • Thus, there is an acceleration associated with the motion. • We call this centripetal acceleration.

1/25 Ch. 4 Thursday, January 25, 2018

9:31 AM

• Linguistic arguments: ○ Some sort of interaction - loosely speaking, a push or a pull on an object. ○ We call this a force, which can be said to act on a body. ○ Examples of forces: ▪ Normal or "contact force" ▪ Gravitational force ▪ Electromagnetic force ▪ Weak and strong nuclear forces • "A body in uniform motion (constant velocity) remains in uniform motion, and a body at rest remains at rest, unless acted upon by a nonzero net force." • "If no net force acts on a body, then the body's velocity cannot change; that is, it cannot accelerate. • "Forces cause changes in motion (acceleration)" • Warning Labels: ○ Friction and air resistance have a tendency to distort our comprehension of the nature of forces. ○ Acceleration applies to velocity, not speed: there are situations in which your speed remains constant, yet you are accelerating. ○ "Net Force" implies that the sum of all forces is non-zero. • Free Body Diagrams: ○ The forces shown below are what we call "external forces" ○ They act on the "system" S.

○ S may represent a single object, or a system of rigidly connected objects. We do not include internal forces which makes the system rigid in our free body diagram. • Newton's Definition: "The rate at which a body's momentum changes is equal to the net force acting on the body."

• Note that Newton's 2nd law includes the 1st law as a special case F=0. • We may treat the components separately.

• The mass, m, is a scalar quantity. • 1N = (1 kg)(1 m*s^-2) = 1 kg*m.s^-2 • During free fall |a| = -g jhat ○ F=ma then F= -mg jhat • This is always true at the surface of the earth, and will usually* be the case for problems worked in this class. • Even when a mass is stationary, e.g. • Weight is measured on newtons, because kg is mass and weight is your mass times the pull of gravity. • In outer space, everything is weightless (cause no gravity) but not massless! • Mass is simply the characteristic of a body that relates the ent force

• Normal Force: ○ The internal forces within the table supply a normal force which is directed normal to the surface of the table, I,e, up. ○ If body remains stationary, then the normal force must be equal in magnitude (opposite in direction) to the weight. ○ N = W = Fg = mg newtons (N)

• Newton's laws apply only in "inertial reference frames" The scale reading is equal to the normal force in the passenger from the

• Newton's laws apply only in "inertial reference frames" ○ The scale reading is equal to the normal force in the passenger from the scale. ○ If we want to use Newton's laws (specifically 2nd law) we must analyze the problem from an inertial frame, i.e., from a reference frame that is not accelerating. We can do this by stepping outside of the elevator and analyzing the motion from there.

What is the normal force on the block shown below, with mass m=1,200 kg, if Flift = 5,000 N

What is the normal force on the block shown below, with mass m=1,200 kg, if Fpull = 5,000 N and theta = 28 degrees?

• Newton's 3rd Law: ○ If object A exerts force on object B, then object B exerts an oppositely directed force of equal magnitude on object A. ○ For every "action" force, there is always an equal and opposite "reaction" force; we call these a "third-law force pair"

1/31 Ch. 5 Wednesday, January 31, 2018

• • • • • • •

10:02 PM

A taut cord is said to be in a state of tension. If the body pulling on the cord does so with a force of 50 N, then the tension in the cord is 50 N. A taut cord pulls on objects at either end with equal and opposite force equal to the tension (Newton's 3rd law). Cords are usually massless, pulleys are usually massless and frictionless. F= ma = T a= mg/m + M where if M has a very large mass it is the inertia and mg is the force. Hooke's Law: F=-kd ○ D is the displacement of the free end of the spring from its position when in a relaxed, or equilibrium state. ○ K is the spring constant, or force constant, and is a measure of the stiffness of the spring with dimensions of N.m^-1 ○ Force is always in opposite direction to the displacement ▪ Hooke's Law (scalar version): F = -kx

02/01 Ch. 5 Thursday, February 1, 2018

10:05 AM

1. What is the tension, T, if the mass is static? 2. What is the acceleration if the tension is T=0?

1. If mass is static, a=0 by definition; therefore, T = mgsintheta 2. If T=0, there must be an acceleration --> a= -g sintheta

What happens if m = M/2? What happens if m > M/2? What happens if m < M/2?

• In a circle, the speed does not change but the velocity does because the vector changes.

When a plane turns, it banks to give the wing's lifting force, Fw, a horizontal component that provides a centripetal force. If a plane is flying level at 950 km/h and the banking angle theta is not to exceed 40 degrees, whats the minimum radius of curvature for the turn?

The bigger the angle of curvature, the smaller the radius.

2/6 Friction Tuesday, February 6, 2018

9:42 AM

• Characteristics of Friction: a. When you set an object in motion on a typical surface, it slows down and stops if you do not continue to push. b. Even if you do continue to push a moving object with a constant force, it does not guarantee acceleration. c. If you try to push an extremely heavy object, it does not move at all, no matter whether you push hard or gently. d. If you really push with all your might, it may eventually give way and begin to slide. • C and D tell us that the frictional force is greater for heavier objects. • Fs is the static friction force. • Fk is kinetic friction force. ○ Fk is always less than Fs and that is cause the object is sliding in Fk. • Static Friction: 1. In static friction, the static frictional force exactly cancels the component of the applied force parallel to the surface. 2. The heavier the object, the more difficult it is to make it slide. Evidently, the maximum frictional force depends on the normal force between the surface and the object, i.e. a. Fs,max = UsN i. This equation is only for max static friction; however, static friction can be anywhere from 0 to Fs,max ii. Where Us is the coefficient of static friction and N is the magnitude of the normal force. Us is a parameter that depends on both surfaces. Once the force component parallel to the surface exceeds Fs,max then the body begins to slide along the surface. 3. If a body begins to slide along the surface, the magnitude of the frictional force instantly decreases to a value of Fk given by a. Fk = UkN b. Where Uk is the coefficient of kinetic friction and N is the magnitude of the normal force. Therefore, during the sliding, a kinetic frictional force of magnitude Fk opposes the motion. 4. When several agents push in different directions on an object, the frictional force opposes the component of the net force on the object which is parallel to the surface. Practice Problem: a. What is the minimum force, F required to move the block? b. What is the acceleration once it moves [assume force from part a]?

1. At what angle does it begin to slide? 2. What is the acceleration at this angle?

2/8 Work Thursday, February 8, 2018

9:35 AM

• Energy is scalar* quantity (a number) that we associate with a system of objects eg plants orbiting the sun, masses attach to springs, electrons bound to nuclei, etc. • Forms of energy: kinetic, chemical, nuclear, thermal, electrostatic, gravitational… • It turns out that energy possesses a fundamental characteristics which makes it very useful for solving problems in phys (energy is always conserved) • Kinetic energy K is energy associated with the state of motion of an object. The faster an object moves, the grater its kine energy • Potential energy U represents stored energy in a spring. It can be later released as kinetic energy. • Work W is the energy transferred to or from an object by means of a force acting on the object. Energy transferred to th object is positive, and energy transferred from the object is negative work. • If you accelerate an object to a greater speed by applying a force on the object, you increase its kinetic energy K; you performed work on the object. • Similarly, if you decelerate an object, you decrease its kinetic energy. The object did work on you. • If an object moves in response to your application of a force, you have performed work. • The further it moves under the influence of your force, the more work you perform. • There are only two relevant variables in one dimension: the force, Fx, and the displacement, deltaX • Even in one dimension, Fx and deltaX are still vectors, i.e., we must take into account of their directions.

• Fx is the component of the force parallel to the objects motion, and deltaX is its displacement. ○ Ex. Pushing furniture across a room. ○ Ex. Carrying boxes up to your attic.

Work - Examples:

Friction is a non-conservative force. Meaning, once I lose energy due to friction into thermal energy, I can never get that back. Friction will always be negative work.

PHY 2048C Page 38

PHY 2048C Page 39

To calculate the work, done on an object by a force during the displacement, we use only the force component along the object's displacement. The force component perpendicular to the displacement does zero work.

• Power is defined as "the rate at which work is done" • If an amount of work W is done in a time interval deltaT by a force, the average power due to force during the time interval is defined as:

PHY 2048C Page 40

A 5,000 kg truck traveling at 15.7m/s ( 35 mph) applies the brakes, which are capable of producing a 2 kN braking force. a) What is the initial power dissipated in the breaks? b) What is the stopping distance?

The spring was able to store the kinetic energy of the block and eventually give this energy back to the block.

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