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Physics 7B DLM 16 Overview
Model: Linear Transport Model
AC-6.3.3 Heat flow FNT from DLM 15 (No activity sheet) (~ 15 min)
Model: Exponential Change
AC-7.1.2 Introduction to Capacitors and Capacitance (~ 25 min) Learning Goals: • Begin to think about non-stead-state currents
- Get an introduction to a capacitor AC-7.1.3 Electrical Capacitor Discharging (~40 min) Learning Goals: • see AC 7.1.
- Get familiar with RC circuits. Act-7.1.4 Physics of Exponential Change DLM 15 FNTs (~ 60 min) Learning Goals:
- Understand how a physical system that behaves as dy(t)/dt = – λ y(t) leads to exponential
- behaviorPractice seeing how the properties of a physical system determine the constant λ
- know that 1/λ, the time constant, for a resistor-capacitor is the product RC
Announcements
- Reading Assignment - Finish reading Unit 7.
- Begin studying for the DLM activities, and practice problems. final now. Do a little bit each day by going over the FNTs, quizzes,
Physics 7B Activity 7.1.2 DLM 16
Introduction to Capacitors and Capacitance
Consider the two pieces of metal very close to each other, but not touching. Each plate
is suddenly connected to the two ends of a battery through a resister R (use a light bulb)
as shown.
The two metal plates are called a capacitor. The metal plates can accept electrons or
lose electrons.
What happens when the battery is first connected?
Usually we have thought about current in terms of positive charge, but here it is useful
to focus on the electrons themselves, since they are the charge carriers in metal
conductors.
a) When the switch is closed (connection is made) can there be a steady-state current?
Explain? Hint: think about the conductivity of air as opposed to the conductivity of
copper wire. Does air normally conduct electricity?
b) When the switch is closed (connection is made) can there be a current that lasts for a
short time? Explain?
c) Focusing on the two metal plates, what physically is happening to electrons
immediately after the switch is closed? Are they leaving a plate? Going to a plate?
What is happening to the net charge on each plate?
d) What would the time dependence of the current in the resistor be like immediately
after closing the switch? Make a sketch of the value of this current as a function of
time. What would its maximum value be? Explain.
e) Put your response to FNT 3 on the board after discussing each part in your SG.
Physics 7B Activity 7.1.4 DLM 16
**Physics of Exponential Change DLM 15 FNTs A) Experimentally seeing the fundamental condition that produces exponential behavior In your small group
- FNT 2 - using your Volume data** a) Use one of the graphs from your group members. Determine from your graphs of Vol(t) and dVol/dt, what number (with units) you would have to multiply the curve of Vol(t) to make it equal the curve of dVol/dt. That is, determine the constant λ in the relation: dVol/dt = - λ Vol(t). Fill in the following chart for several time values: time Volume dVol/dt λ
b) What do you notice about the λ? Describe in your own words what you did in this FNT, in the question above, and the significance of all this mean?” what you found. That is, answer the question, “What does **Whole class discussion
- Using your Voltage data -** Using the data from part e) of the last activity complete parts a) and b) that you just finished for the flow out of a standpipe. Whole class discussion
B) Interpreting the time constant
- The inverse of the constant λ in e-λt^ is called the time-constant; i.e., 1/λ = time-constant. In electrical phenomena, it is called the “RC time a) By how much has the voltage decayed from a series resistor-constant”. -capacitor circuit when the time is b) equal to one time constant?How many RC time constants must pass for the voltage to have decayed to less than 5 % of its original value? (use your calculator: see how much it decreases for each successive time constant) c) Use the RC time constant you found in Activity 7.1.3 part (h) and the value of the resistor you found in part (a) to calculate the actual value of the capacitor you used.
- a) Use a calculator to “experimentally” (don’t use a formula) determine how many years it would take to reduce the radioactivity at a waste site to less than 0.01 of its initial value, if the dominant radioactive isotope had a half-life of 50 years. Put your work on the board. b) Draw a graph representing this decay and label your axes. Plot a curve if the time constant were 5 years or 500 years. Whole class discussion
Physics 7B Exit Handout DLM 16
FNTs Wrap-up of Exponential Change (you derive the equation). 1) When a capacitor is discharging, the circuit is simply a capacitor and resistor hooked in series—end-to-end. From the definition of capacitance, the voltage across the capacitor ΔV is given by the expression: ΔV = q/C. You know by definition that the current I is simply the time derivative of the charge: I = dq/dt. Apply the energy-density relation from one side of the resistor to the other. Re-arrange this expression, using the substitutions suggested in the two previous sentences to get it in the form dq/dt = - λq. What is λ equal to in terms of the other electrical constants in this situation?
Beginning of Simple Harmonic Motion (SHM) The most general form of the equation that describes any object undergoing SHM is given by: y ( t ) = A sin^ $ % & (^2) T" t + #' ( ) + B.
A is the amplitude, T is the period of oscillation, φ is a constant phase factor, and B is the equilibrium value of y (t), if it is not zero. We will normally let B = 0. The next several FNTs have to do with making sense of this relationship. 2) Use the general expression above, with for all of them (all on the same graph). Your time axis should go from t = 0 to t = 10 s. B = 0, to sketch the following graphs, using the same time axis I) A = 2, T = 5, φ = 0 II) A = 2, T = 5, φ = π/2 III) A = 2, T = 5, φ = - π/ Explain in words how you knew how to “start” each graph where t = 0. 3) From your graph, describe in words the motion of the object described by the equation in FNT 2 for case II) at the particular times when t = 0 and when t = 1.25 s. What is its position and qualitativel describe its velocity. y
4) a. Using your knowledge of a mass-spring developed in 7A, relate energy to amplitude in variable form. What is the object’s maximum speed in variable form? b. For a mass maximum speed; compressing a mass of 2m to x-spring system of mass m and spring constant k, which alteration will achieve a greater max = A, or compressing a mass 4m to A = 2xmax?
5) An object moves with simple harmonic motion. If the amplitude and period are both doubled, the object’s maximum speed is a. quadrupled c. unchanged e. quartered b. doubled d. halved 6) (^) phase constant,Draw three independent f, that equals i) pictures of a mass π/4 rad ii) - -πspring system undergoing an oscillation at t = 0 with a/4 rad iii) 3 π/4 rad. For each case label the amplitude (each has amplitude A) and place a velocity vector on the mass to indicate its direction of motion.
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