Final Review, Day 1. Announcements: Web page:

Transcription

1 Announcements: Final Review, Day 1 Final exam next Wednesday (5/9) at 7:30am in the Coors Event Center. Recitation tomorrow is a review. Please feel free to ask the TA any questions on the course material. Lecture on Friday is a free-for-all review. Please bring any questions you have. I will plan to ask clicker question after clicker question. Web page:

2 Grading for the Course 3 midterm exams, 15% each: 45% Posted Final exam: 23% CAPA homework 15% Tutorial participation (3%) and TA-graded written homeworks 7% In-lecture clicker participation: 4% Posted SmartPhysics Prelecture participation 3%

3 Total Clicker Scores for Course

4 Tutorial Scores Reminder to check your scores

5 Final information Final is on 5/9 from 7:30 10:00am in the Coors Event Center You are allowed a calculator and two double sided sheets of paper with any handwritten notes you like. We may ask you to show your student ID if your TA doesn t recognize you so make sure you bring it with you. The same moments-of-inertia as in exam 3 are provided The following constants are provided: G, g (10 m/s 2 ), density of water, mass of Earth, radius of Earth, and atmospheric pressure.

6 Final information We are using the south end of Coors Event Center. Sit every other seat and as close to the court as possible.

7 4-5 fluid questions Exam questions ~8 questions on oscillations and waves Problems on collisions, work, force and motion, conservation of energy, static equilibrium, angular momentum, torque and angular motion, simple harmonic motion, and 2D kinematics. 4-5 questions of the type from the first midterm 4-5 questions of the type from the second midterm 4-5 questions of the type from the third midterm Carefully review the clicker concept questions from lectures

8 Studying materials Old exams can be found from the content section of the D2L link. Obviously you can ignore the thermodynamics questions. Tutorial Homework Solutions can also be found in the content section of the D2L link.

9 Study suggestions (approximately in order) 1. CAPA: Make sure you understand and can do the CAPA problems. Think about possible modifications to the problems and what you would need to do different to solve it. 2. Lecture: Make sure you understand the lecture notes and can do all of the clicker questions. Think about modifications to the clicker questions that could be asked. 3. Tutorials: Review the tutorials including the homework to make sure you understand. For questions you got wrong, analyze where your thinking went astray and make sure you have corrected that aspect. For ungraded questions, go over them again to see if you have changed your mind about your answers.

10 Study suggestions (approximately in order) 4. End of chapter problems: These problems should help you setup and solve problems. Try to solve the problems using only your formula sheet and calculator as in the exam. 5. Old exams: The exams this time are not as representative because we have not covered gravity yet. Nevertheless, some problems are quite good. Try to solve the problems using only your formula sheet and calculator and impose a time limit. 6. Read the text: There are examples worked out, suggestions for how to solve problems, and interesting relationships explored that might help you remember the essential bits.

11 Material covered since last exam We covered three subjects since the last exam Chapter 14: Fluid mechanics Sections covered on proof of Bernouli s Equation was not covered Chapter 15: Oscillatiions Sections covered. Sections on damped harmonic motion and on forced oscillations were not covered. Chapter 16: Waves Sections , covered on phasors was not covered

12 Chapter 14: Fluid mechanics Defined density: Intrinsic quantity but equal to mass/volume. Defined pressure: A location has a pressure (which is a scalar quantity) and a pressure can exert a force on an area. The force is perpendicular to the area. Pressure is force/area and has units of N/m 2 or pascals (Pa). Atmospheric pressure at sea level is 10 5 Pa. A fluid doesn t hold its shape and can flow. Liquids are (approximately) incompressible fluids Gases are compressible fluids

13 Fluid mechanics: Hydrostatics Statics is when fluids are not moving The weight of a fluid above causes an increase in pressure as the depth increases. At a depth d below the surface which has pressure p 0 in a liquid of density ρ the hydrostatic pressure is This also shows that for a connected liquid, the pressure at a given depth is the same no matter where you are. Since force is pressure times area, one can use a small area to increase the pressure which can exert a large force over a small area. Hydraulic lifts and hydraulic brakes use this principle.

14 Fluid mechanics: Buoyant force The buoyant force is always equal to the weight of liquid displaced by an object. An object completely submerged in a liquid displaces an amount of liquid equal to the volume of the object so and therefore An object floating on the surface must have an upward (buoyant) force which exactly cancels the downward force of gravity (weight) so the buoyant force is The buoyant force is still So which means the floating object displaces an amount of water equal to the weight of the object. but

15 Fluid mechanics: Fluid dynamics The continuity equation simply says the amount of liquid entering a pipe is the same as the amount of liquid exiting the pipe. The equation is which has units of volume/time Bernoulli s equation is a restatement of conservation of energy. Forces associated with pressure can do work which adds a component to the normal potential and kinetic energy terms. Bernoulli s equation: or

16 Chapter 15: Oscillations is the angular frequency (rad/s). It sets how quickly the system oscillates. The time it takes to increase by (a complete cycle) is the period The frequency is how many cycles are completed per second.. Frequency SI unit is hertz (Hz). Units of cycles/second or s -1 are also used. Note that.

17 Mass on end of spring The force exerted by the spring is. 0 The potential energy stored in a spring is. Acceleration is not constant, but can use Newton s second law Dividing both sides by m: We guess a solution

18 Position versus time: Initial conditions Velocity versus time: The initial (t=0) position and velocity are Since is set by, we can solve these two equations for A and (assuming we have the initial position & velocity). Thus, initial conditions (position & velocity) give us A & φ.

19 Energy considerations Our analysis of the mass on a spring tells us about the position, velocity, acceleration as a function of time. Without friction, energy is conserved: Remember that when, so the total energy is The maximum velocity occurs at energy we see so using conservation of so

20 Springs in Parallel x = x 1 + x 2 Force is same on both springs so F = k eff x = k 1 x 1 = k 2 x 2 k eff x = k eff (x 1 + x 2 ) = k 2 x 2 x 1 = k 2 k 1 x 2 k eff = k 2x 2 x 1 + x 2 = If k 1 = k 2, then k eff = k 2 2 k 2 x 2 = k 2 x 2 + x 2 k 1 k 2 k 2 k 1 +1 = k 1k 2 k 1 + k 2

21 Pendulum We found the force on a pendulum along the θ direction is. Using the small angle approximation gives us. L But note also that so The angular frequency is The period of a physical pendulum is T = 2π I mgl

22 Torsion Pendulum Consider a disk suspended from a torsion wire attached to its centre. Called a torsion pendulum. A torsion wire is essentially inextensible, but is free to twist about its axis. As the wire twists it also causes the disk attached to it to rotate in the horizontal plane. Let be the angle of rotation of the disk, and equilibrium position is when the wire is untwisted. k is the torsion constant > 0 Note: I thru center of wire T = 2π I κ

23 Chapter 16: Waves y(x,t) = y m sin(kx ωt + φ) Transverse Displacement v = τ µ τ is the tension in the string µ is the mass per unit length P avg = 2 dk dt avg = 1 2 µv ω 2 y m 2

24 Interference (Standing Waves) y(x,t) = [2y m sinkx] cosωt Amplitude at position x Oscillating Term In a standing wave the amplitude varies with position. The place where amplitude is zero is when k = 2π λ kx = nπ for n = 0, 1, 2,... so x = nλ for n = 0, 1, 2,... (nodes) 2 λ = 2L n for n =1, 2, 3,... f = v λ = n v 2L for n =1, 2, 3,... n is called the harmonic number

25 Clicker question 1 Set frequency to BA A long, taut string is shaken up and down at one end by a machine. This creates a right-going traveling wave on the string that is described by the equation y(x,t)=asin(kx-ωt). The wavelength λ of the wave is 0.50 m, the period T of the wave is 0.10 seconds, and the amplitude of the wave is m (1.0 mm). What is the maximum vertical speed v y of a piece of the string? (A) m/s (B) m/s (C) 5.0 m/s (D) 31.4 m/s (E) m/s

26 Clicker question 1 Set frequency to BA A long, taut string is shaken up and down at one end by a machine. This creates a right-going traveling wave on the string that is described by the equation y(x,t)=asin(kx-ωt). The wavelength λ of the wave is 0.50 m, the period T of the wave is 0.10 seconds, and the amplitude of the wave is m (1.0 mm). What is the maximum vertical speed v y of a piece of the string? (A) m/s (B) m/s (C) 5.0 m/s (D) 31.4 m/s (E) m/s v y = ωacos(kx ωt) v y = 2πfA = 2πA T = 2π(0.001) /.1 =.0628m /s

27 Clicker question 2 Set frequency to BA Referring to the previous problem above, if the frequency with which the machine shakes the string is doubled, what happens to the speed of the wave, v wave? (A) the wave speed increases (B) decreases (C) remains unchanged

28 Clicker question 2 Set frequency to BA Referring to the previous problem above, if the frequency with which the machine shakes the string is doubled, what happens to the speed of the wave, v wave? (A) the wave speed increases (B) decreases (C) remains unchanged In a traveling wave the speed is determined by Tension and mass per unit length not frequency

29 Clicker question 3 Set frequency to BA A mass m is thrown downward with an initial speed v o a height h i above a table top on which sits a spring with spring constant k. The mass compresses the spring by a maximum amount x and stops for an instant at a height h f-. There is no friction in this problem. Which of the following equations correctly expresses conservation of energy and allows one to solve for the compression x of the spring? (A) mgh i + ½ mv 2 = ½ kx 2 (B) mgh i + ½ mv 2 =1/2 kx 2 +mgh f (C) mgh i +1/2 kx 2 =mgh f (D) mgh(h f -h i ) = ½ kx 2 +1/2 mv 2 (E) None of these equations is correct.

30 Clicker question 3 Set frequency to BA A mass m is thrown downward with an initial speed v o a height h i above a table top on which sits a spring with spring constant k. The mass compresses the spring by a maximum amount x and stops for an instant at a height h f-. There is no friction in this problem. Which of the following equations correctly expresses conservation of energy and allows one to solve for the compression x of the spring? (A) mgh i + ½ mv 2 = ½ kx 2 (B) mgh i + ½ mv 2 =1/2 kx 2 +mgh f (C) mgh i +1/2 kx 2 =mgh f (D) mgh(h f -h i ) = ½ kx 2 +1/2 mv 2 (E) None of these equations is correct.

31 Clicker question 4 Set frequency to BA A solid piece of plastic of volume V, and density ρ p is floating in a cup of water. (The density of water is ρ w, and ρ p < ρ w. ) What is the magnitude of the buoyant force on the plastic? (A) Zero (B) ρ p V g (C) ρ w V (D) ρ w V g (E) ρ p V

32 Clicker question 4 Set frequency to BA A solid piece of plastic of volume V, and density ρ p is floating in a cup of water. (The density of water is ρ w, and ρ p < ρ w. ) What is the magnitude of the buoyant force on the plastic? (A) Zero (B) ρ p V g (C) ρ w V (D) ρ w V g (E) ρ p V

33 Clicker question 5 Set frequency to BA The diagram shows a snapshot of a traveling wave at some given time. The frequency of this wave is 120 cycles/sec. What is the speed of the wave? A) 540 m/s B) 120 m/s C) 1.2 m/s D) 360 m/s E) not enough information given

34 Clicker question 5 Set frequency to BA The diagram shows a snapshot of a traveling wave at some given time. The frequency of this wave is 120 cycles/sec. What is the speed of the wave? A) 540 m/s B) 120 m/s C) 1.2 m/s D) 360 m/s E) not enough information given v = fλ = (120s 1 )(3m) = 360m /s

35 Clicker question 6 Set frequency to BA Suppose the "A" string on a piano is one meter long, and has a mass of kg. The frequency of the fundamental of this string is 440 Hz. What tension in the wire is needed? HINT: Piano strings are fixed at both ends. What is the wavelength of the fundamental vibration? (It is NOT 1 m!) A).03 N B) 3.5 N C) 390 N D) 6200 N E) 2.4E7 N

36 Clicker question 6 Set frequency to BA Suppose the "A" string on a piano is one meter long, and has a mass of kg. The frequency of the fundamental of this string is 440 Hz. What tension in the wire is needed? HINT: Piano strings are fixed at both ends. What is the wavelength of the fundamental vibration? (It is NOT 1 m!) A).03 N B) 3.5 N C) 390 N D) 6200 N E) 2.4E7 N v = τ µ v = fλ = (440Hz)2m = 880m /s µ = 0.008kg /1m τ = v 2 µ = (880) 2 (0.008)N

37 Clicker question 7 Set frequency to BA A pendulum consists of a mass m at the end of a string of length L=1.00 m. The mass is released from rest at an angle of θ = 60.0 o What is the distance d through which the mass travels in moving from the initial position to the lowest point? L θ d (A) 0.17 m (B) 0.71 m (C) 1.05 m (D) 30 m (E) None of these.

38 Clicker question 7 Set frequency to BA A pendulum consists of a mass m at the end of a string of length L=1.00 m. The mass is released from rest at an angle of θ = 60.0 o What is the distance d through which the mass travels in moving from the initial position to the lowest point? L θ d (A) 0.17 m (B) 0.71 m (C) 1.05 m (D) 30 m (E) None of these. (1m)(60 π 180 ) =1.05m