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Quiz 4 Solutions 1. (3 pts) Two charged parallel plates are used in many devices like a mass spectrometer or a cyclotron to speed up ions or elementary particles. The two parallel plates are raised to different voltages, which results in an electric field between the plates that can be treated as uniform (constant and all in the same direction). Charged ions drift through a small hole in one of the plates and are accelerated by the field between the plates. The figure on the right shows a C14+ ion about to drift into the region between the plates. As it is accelerated, it will pass through the points marked a and b. At which point will the magnitudes of the electric potential, the force felt by the ion, and the acceleration of the ion be greater? Check one answer from each column. (The plates are close enough together that the holes through which the ions enter and leave are small enough that their effect on the field may be ignored.) Potential Force Acceleration ____ Va > Vb ____ Fa > Fb ____ aa > ab ____ Va = Vb ____ Fa = Fb ____ aa = ab ____ Va < Vb ____ Fa < Fb ____ aa < ab ____ Not enough info to tell ____ Not enough info to tell ____ Not enough info to tell 4/7/2014 Physics 132 1 Quiz 4 Solutions 2. (3 pts) In the figure at the right are shown four arrangements of charge. Each charge has the same magnitude, but some are + and some are -. All distances are to the same scale. If the same test charge were to be placed at the point P in each of the examples, in the box below rank the order of the electric potentials the test charge would measure from greatest to smallest. Use “>” to mean greater than and “=” to mean equal. Do not use “<” signs. Your answer should be a string of letters that looks something like E = F > G > H meaning E and F are equal and bigger than G and G is bigger than H. You, of course, should use the letters ABCD and the appropriate ranking. C>A>B=D 4/7/2014 Physics 132 2 Quiz 4 Solutions 3. Four charges of equal magnitude are placed on a grid as shown in the figure at the right. 3.1 (2 pts) If a small test charge were placed at the black dot indicated on the x-axis, in what direction would the electric field it detects point? A. In the +x direction E. In the +y direction B. In the –x direction F. In the –y direction C. It would be 0. G. In some other direction D. You can’t tell without knowing the sign of the test charge 3.2 (2 pts) In the box at the right, sketch a graph of the electrostatic potential the test charge would measure as it moves along the x-axis. The positions of the charges are marked on the graph by vertical bars, and the V axis crosses the x-axis at x = 0. 4/7/2014 Physics 132 3 Damped Oscillations & Resonance 4/7/2014 Physics 132 4 The Simple Pendulum Consider a mass m attached to a string of length L which is free to swing back and forth. If it is displaced from its lowest position by an angle θ, Newton’s second law for the tangential component of gravity, parallel to the motion, is: 4/7/2014 Physics 132 5 The Simple Pendulum If we restrict the pendulum’s oscillations to small angles (< 10°), then we may use the small angle approximation sin θ ≈ θ, where θ is measured in radians. and the angular frequency of the motion is found to be: 4/7/2014 Physics 132 6Slide 14-74 A ball on a massless, rigid rod oscillates as a simple pendulum with a period of 2.0 s. If the ball is replaced with another ball having twice the mass, the period will be Physics 132 s 4. 0 s. 2. 8 1. 0 4/7/2014 12% 2% s. E. 15% s. D. 33% 2. 0 C. 38% s. B. 1.0 s. 1.4 s. 2.0 s. 2.8 s. 4.0 s 1. 4 A. 7 On Planet X, a ball on a massless, rigid rod oscillates as a simple pendulum with a period of 2.0 s. If the pendulum is taken to the moon of Planet X, where the free-fall acceleration g is half as big, the period will be Physics 132 s 4. 0 2. 8 1. 0 4/7/2014 s. 4% s. E. 14% s. D. 23% 2. 0 C. 30% 29% s. B. 1.0 s. 1.4 s. 2.0 s. 2.8 s. 4.0 s 1. 4 A. 8 Tactics: Identifying and Analyzing Simple Harmonic Motion 4/7/2014 Physics 132 9 The Physical Pendulum Any solid object that swings back and forth under the influence of gravity can be modeled as a physical pendulum. The gravitational torque for small angles (θ < 10°) is: Plugging this into Newton’s second law for rotational motion, τ = Iα, we find the equation for SHM, with: 4/7/2014 Physics 132 10 A Swinging Leg as a Pendulum Whiteboard, TA & LA 1 𝐼𝐼 = 𝑀𝑀𝐿𝐿2 3 4/7/2014 Physics 132 11 A Swinging Leg as a Pendulum 4/7/2014 Physics 132 12 A solid disk and a circular hoop have the same radius and the same mass. Each can swing back and forth as a pendulum from a pivot at one edge. Which has the larger period of oscillation? 4/7/2014 Physics 132 di sk . ci rc ul bo ar th ho ha op ve . th Th e s er am e is e pe no ... te no ug h in fo rm ... lid Th ey D. 31% 0% so C. 37% 31% Th e B. The solid disk. The circular hoop. They both have the same period. There is not enough information to tell. Th e A. 13 Damped Oscillations An oscillation that runs down and stops is called a damped oscillation. One possible reason for dissipation of energy is the drag force due to air resistance. The forces involved in dissipation are complex, but a simple linear drag model is: 4/7/2014 Physics 132 The shock absorbers in cars and trucks are heavily damped springs. The vehicle’s vertical motion, after hitting a rock or a pothole, is a damped oscillation. 14 Damped Oscillations When a mass on a spring experiences the force of the spring as given by Hooke’s Law, as well as a linear drag force of magnitude |D| = bv, the solution is: where the angular frequency is given by: Here is the angular frequency of the undamped oscillator (b = 0). 4/7/2014 Physics 132 15 Damped Oscillations Position-versus-time graph for a damped oscillator. 4/7/2014 Physics 132 16 Damped Oscillations A damped oscillator has position x = xmaxcos(ωt + φ0), where: This slowly changing function xmax provides a border to the rapid oscillations, and is called the envelope. The figure shows several oscillation envelopes, corresponding to different values of the damping constant b. 4/7/2014 Physics 132 17 Energy in Damped Systems Because of the drag force, the mechanical energy of a damped system is no longer conserved. At any particular time we can compute the mechanical energy from: Where the decay constant of this function is called the time constant τ, defined as: The oscillator’s mechanical energy decays exponentially with time constant τ. 4/7/2014 Physics 132 18 Driven Oscillations and Resonance Consider an oscillating system that, when left to itself, oscillates at a natural frequency f0. Suppose that this system is subjected to a periodic external force of driving frequency fext. The amplitude of oscillations is generally not very high if fext differs much from f0. As fext gets closer and closer to f0, the amplitude of the oscillation rises dramatically. 4/7/2014 A singer or musical instrument can shatter a crystal goblet by matching the goblet’s natural oscillation Physics 132 frequency. 19 Driven Oscillations and Resonance The response curve shows the amplitude of a driven oscillator at frequencies near its natural frequency of 2.0 Hz. 4/7/2014 Physics 132 20 Driven Oscillations and Resonance The figure shows the same oscillator with three different values of the damping constant. The resonance amplitude becomes higher and narrower as the damping constant decreases. 4/7/2014 Physics 132 21 The graph shows how three oscillators respond as the frequency of a driving force is varied. If each oscillator is started and then left alone, which will oscillate for the longest time? A. B. C. D. The red oscillator. The blue oscillator. The green oscillator. They all oscillate for the same length of time. 66% m ... sa Th ey al l os cil l at e gr ee n os fo rt he cil la to r. ill at or . os c Th e bl ue Th e Th e re d os cil l at or . 12% 13% 9% 4/7/2014 Physics 132 22 4/7/2014 https://www.youtube.com/watch ?v=xox9BVSu7Ok Physics 132 23 General Principles 4/7/2014 Physics 132 24 General Principles 4/7/2014 Physics 132 25 Important Concepts 4/7/2014 Physics 132 26 Quiz 6 Results 35 30 25 20 15 10 5 0 0 1 2 3 4 5 6 7 8 9 10 AVG: 6.42 STDEV: 1.66 4/7/2014 Physics 132 27 Quiz 6 Solutions 1. (3 pts) Consider a single charged particle, q, that is moving through the resistor as a part of a constant steady current. On the average, the charge moves through the resistor at a constant velocity. Which of the following statements are true while the charge is moving through the resistor? A. There is a net force acting on the charge. B. The net force acting on the charge is 0. C. There is a non-zero electric force on the charge. D. We can’t say anything without knowing more information. 4/7/2014 Physics 132 28 Quiz 6 Solutions 2.1 (2 pts) A mass hanging from a spring is oscillating up and down. A graph of its height above the ground is shown at the right. The origin of this graph (y = 0, t = 0) is where the axes cross. The position of the mass at a particular instant of time is marked with the letter P. At the instant marked P, the force the spring exerts on the mass A. B. C. D. is upward is downward is zero cannot be determined from the information given. 2.2. (2 pts) The period, T, of the data shown in problem 2 is the amount of time between the time between two successive peaks. If we drew a velocity curve, we would find that the period of that curve A. B. C. D. 4/7/2014 is the same as the period of the figure in problem 2.1. is longer than the period of the figure in problem 2.1. is shorter than the period of the figure in problem 2.1. cannot be determined from the information given. Physics 132 29 Quiz 6 Solutions 3. (3 pts) Consider the inside and outside of a cell, each filled with different concentration of NaCl and separated by a membrane. The membrane has only one type of Ion channel that lets through only Na+. The resting Nernst potential of the system is -100mV. This is calculated from the equation shown at the right. Which of the following statements are true when the system is at the resting potential? k BT c2 ∆V = ln q c 1 A. Some Na accumulates on the membrane on the side with higher Na + concentration B. Some Na accumulates on the membrane on the side with lower Na+ concentration C. No Na+ accumulates on either side of the membrane. D. Some Cl- accumulates on the membrane on the side with higher Na+ concentration E. Some Cl- accumulates on the membrane on the side with lower Na+ concentration F. No Cl- accumulates on either side of the membrane 4/7/2014 Physics 132 30 Displacements on an elastic string / spring Each bit of the string can move up or down (perpendicular to its length) – transverse waves Each bit of string can also move toward/away along the string length if the string is elastic (most notable on very deformable strings such as slinky, rubber band). – longitundinal waves 4/7/2014 Physics 132 31 How do the beads move? Whiteboard, TA & LA Pulse moving to the right y x • Sketch the y position of the bead indicated by the arrow as a function of time 4/7/2014 Physics 132 32 Whiteboard, TA & LA Describing the motion of the beads • Sketch the velocity of each bead in the top figure at the time shown Pulse moving to the right y x vy x 4/7/2014 Physics 132 33 A pulse is started on the string moving to the right. At a time t0 a photograph of the string would look like figure 1 below. A point on the string to the right of the pulse is marked by a spot of paint. (x is horizontal and right, y is vertical and up) 46% 31% A. B. C. D. E. F. G. 9% 1 2 3 4 5 6 7 7 None of these 4/7/2014 Physics 132 34 3% 7 5% 6 4% 5 4 3 2 3% 1 Which graph would look most like a graph of the y displacement of the spot as a function of time? A pulse is started on the string moving to the right. At a time t0 a photograph of the string would look like figure 1 below. A point on the string to the right of the pulse is marked by a spot of paint. (x is horizontal and right, y is vertical and up) 53% A. B. C. D. E. F. G. 24% 12% 6% 1 2 3 4 5 6 7 7 None of these 4/7/2014 Physics 132 35 6 7 2% 1% 5 4 3 2% 2 1 Which graph would look most like a graph of the x velocity of the spot as a function of time? A pulse is started on the string moving to the right. At a time t0 a photograph of the string would look like figure 1 below. A point on the string to the right of the pulse is marked by a spot of paint. (x is horizontal and right, y is vertical and up) 70% A. B. C. D. E. F. G. 18% 1 2 3 4 5 6 7 7 None of these 4/7/2014 Physics 132 36 3% 0% 7 2% 5 4 3 1% 6 6% 2 1 Which graph would look most like a graph of the y velocity of the spot as a function of time? A pulse is started on the string moving to the right. At a time t0 a photograph of the string would look like figure 1 below. A point on the string to the right of the pulse is marked by a spot of paint. (x is horizontal and right, y is vertical and up) 39% 23% 11% A. B. C. D. E. F. G. 10% 1 2 3 4 5 6 7 7 None of these 4/7/2014 Physics 132 37 7 7% 6 5 5% 4 3 5% 2 1 Which graph would look most like a graph of the y force of the spot as a function of time? What controls the widths of the pulses in time and space? y t ∆t y x ΔL 4/7/2014 Physics 132 38 Width of a pulse The amount of time the demonstrator’s hand was displaced up and down determines the time width of the t-pulse, ∆t. The speed of the signal propagation on the string controls the width of the x-pulse, ∆L. – The leading edge takes off with some speed, v0. – The pulse is over when the trailing edge is done. – The width is determined by “how far the leading edge got to” before the displacement was over. 4/7/2014 Physics 132 39 What Controls the Speed of the Pulse on a Spring? To make the pulse go to the wall faster 2. 3. 4. 5. 6. 7. 8. 9. Move your hand up and down more quickly (but by the same amount). Move your hand up and down more slowly (but by the same amount). Move your hand up and down a larger distance in the same time. Move your hand up and down a smaller distance in the same time. Use a heavier string of the same length under the same tension. Use a string of the same density but decrease the tension. Use a string of the same density but increase the tension. Put more force into the wave. Put less force into the wave 4/7/2014 Physics 132 31% 21% 13% 12% 8% 7% 5% 2% 1% M ov e M you ov rh M e yo an ov ur d u e p h M you an an d ov d e r ha up do y Us ou nd u and .. rh e a a p a do Us hea nd u nd d .. vie p e a a ow r Us stri str nd d n.. ng in e o a of g o wn s Pu trin t he f th .. e t m go sa . or f t h me .. Pu e fo e s de am n. tl rc .. es e i s f nt e de or o c e th n... in e w to a th ve. e w av e 1. 40 Speed of a bead The speed the bead moves depends on how fast the pulse is moving and how far it needs to travel to stay on the string. dy = how far bead moves in time dt slope of pulse speed of bead speed of pulse dx = how far pulse moves in time dt 4/7/2014 Physics 132 41 Foothold principles: Mechanical waves Key concept: We have to distinguish the motion of the bits of matter and the motion of the pattern. Mechanism: the pulse propagates by each bit of string pulling on the next. Pattern speed: a disturbance moves into a medium with a speed that depends on the properties of the medium (but not on the shape of the disturbance) Matter speed: the speed of the bits of matter depend on both the Amplitude and shape of the pulse and pattern speed. 4/7/2014 Physics 132 42 Foothold principles: Mechanical waves Key concept: We have to distinguish the motion of the bits of matter and the motion of the pattern. Mechanism: the pulse propagates by each bit of string pulling on the next. Pattern speed: a disturbance moves into a medium with a speed that depends on the properties of the medium (but not on the shape of the disturbance) v0 = speed of pulse T = tension of spring μ = mass density of spring (M/L) Matter speed: the speed of the bits of matter depend on both the size and shape of the pulse and pattern speed. 4/7/2014 Physics 132 43 Which goes with which? Graph I Graph II Graph III 4/7/2014 Physics 132 44 The math We express the position of a bit of string at a particular time by labeling which bit of string by its x position, at x at time t the position of the string is y(x,t). Since subtracting a d from the argument of a function ( ) shifts the graph of the function to the right by an amount d, if we want to set the graph of a shape f(x) into motion at a constant speed, we just need to set d = v0t and take 4/7/2014 Physics 132 45 How do waves combine? We know how one wave moves. What happens when we get two waves on top of each other? ? 4/7/2014 Physics 132 46 What happens when they overlap perfectly? Whiteboard, TA & LA ? 4/7/2014 Physics 132 47 ? 50% 1. 2. 1. 2 1 3. 2. 3. 4/7/2014 Other Physics 132 48 1 2 3 14% 3 35% What happens after the waves collide? 47% 40% (Bounce off) 1. 2. 3. 13% 3. (Cancel) 4.4/7/2014 Other Physics 132 3 (Pass through) 2 2. 1 4. 49 1 2 3 4 1% 4 1. Whiteboard, TA & LA How about on the same side? ? 4/7/2014 Physics 132 50 The math We express the position of a bit of string at a particular time by labeling which bit of string by its x position, at x at time t the position of the string is y(x,t). Since subtracting a d from the argument of a function ( ) shifts the graph of the function to the right by an amount d, if we want to set the graph of a shape f(x) into motion at a constant speed, we just need to set d = v0t and take 4/7/2014 Physics 132 51