Introductory Algebra Lesson 11 – Linear Functions, Part 2 .docxmariuse18nolet
Introductory Algebra Lesson 11 – Linear Functions, Part 2
Practice Problems
Skills Practice
1. Determine the slope-intercept form of the equation of the line between each of the following
pairs of points.
a. (4, -5) and (2, 3)
b. (-3, 2) and (1, 8)
c. (5, -9) and (5, 2)
d. (2, -1) and (-2, 3)
e. (4, 3) and (12, -3)
f. (2, -4) and (7, -4)
Introductory Algebra Lesson 11 – Linear Functions, Part 2
2. Give the equation of the linear function that generates the following table of values. Write
your answer in slope-intercept form.
x f (x)
-5 91
-2 67
1 43
4 19
9 -21
3. Give the equation of the linear function that generates the following table of values. Write
your answer in slope-intercept form.
t C(t)
5 -1250
15 -900
20 -725
35 -200
45 150
4. Give the equation of the linear function shown below. Write your answer in slope-intercept
form.
Introductory Algebra Lesson 11 – Linear Functions, Part 2
5. Give the equation of the linear function shown below. Write your answer in slope-intercept
form.
6. Give the equation of the linear function shown below. Write your answer in slope-intercept
form.
7. Give the equation of the linear function shown below. Write your answer in slope-intercept
form.
Introductory Algebra Lesson 11 – Linear Functions, Part 2
8. Give the equation of the linear function shown below. Write your answer in slope-intercept
form.
9. Give the equation of the linear function shown below. Write your answer in slope-intercept
form.
10. Give the equation of the linear function shown below. Write your answer in slope-intercept
form.
Introductory Algebra Lesson 11 – Linear Functions, Part 2
11. Give the equation of the horizontal line passing through the point (-6, 11). _______________
12. Give the equation of the vertical line passing through the point (4, 7). _______________
13. Give the equation of the x-axis. _______________
14. Give the equation of the y-axis. _______________
15. Give the equation of the line passing through the point (1, -5) that is parallel to y = 12 – 8x.
16. Give the equation of the line passing through the point (6, 0) that is parallel to y = x
2
3
9 .
17. Give the equation of the line passing through the point (10, 3) that is perpendicular to
1
5
2
xy .
18. Give the equation of the line passing through the point (-12, -1) that is perpendicular to
xy 43 .
Introductory Algebra Lesson 11 – Linear Functions, Part 2
19. Draw an accurate graph of the linear equation 2x + 3y = 6.
Slope-Intercept Form:
Slope: ___________
Vertical Intercept: ____________
Horizontal Intercept: ____________
Two additional points on the line:
____________ _____________
20. Draw an accurate graph of the function 155 yx
Slope-In.
Electric Circuits Lab Series RC Circuits Phase Angle, Phase Lag.docxpauline234567
Electric Circuits Lab
Series RC Circuits: Phase Angle, Phase Lag, and Capacitors as Integrators
I.
Objectives:
After completing this lab experiment using, you should be able to:
1. Understand the effect of frequency on capacitive reactance.
2. Measure the impedance of an RC circuit.
3. Measure the phase angle and phase lag of an RC circuit using the oscilloscope.
4. Draw the impedance and voltage phasor diagrams.
5. Understand how a capacitor integrates current.
II.
Parts List:
1. Resistor 100 Ω, 1 kΩ, 6.8 kΩ
2. Capacitors 0.1 µF, 0.01 µF
III.
Procedures:
Part I:
1.
Connect the following circuit.
Figure 1: RC Circuit
2.
Connect one DMM across the resistor and one DMM across the capacitor. Set both DMMs to read AC voltage.
Measure the voltage drop across each component. Record the result in
Table 1.
3. Use Ohm’s law to
calculate the current flowing through the resistor. Since the circuit in Figure 1 is a series RC circuit, the same current will flow through the capacitor and the resistor.
Record the result in
Table 1.
Total current, I =
4.
Calculate the capacitive reactance using Ohm’s law. Record the result in
Table 2.
Capacitive Reactance, XC =
5. Now,
calculate the capacitive reactance value using the equation below.
Record the result in
Table 1 under Computed Reactance, XC.
Capacitive Reactance,
Capacitor C1
Voltage across, R
846 mV
Voltage across, C
583 mV
Total Current, I
0.846 mA
Capacitive Reactance, XC
686 Ω
Computed Reactance, XC
Table 1: Calculated and measured values
6.
Adjust the function generator frequency following the steps in
Table 2. Use the DMM to
measure the voltage across the resistor and the capacitor.
Record your measurements below.
Frequency (in Hz)
VR
(measured)
VC
(measured)
I =
(calculated)
XC =
(calculated)
XC =
(calculated)
300
983 mV
186 mV
0.983 mA
189 Ω
1k
846 mV
583 mV
0.846 mA
686 Ω
3k
884 mV
468 mV
0.884 mA
529 Ω
5k
953 mV
302 mV
0.953 mA
317 Ω
7k
975 mV
221 mV
0.975 mA
227 Ω
9k
985 mV
174 mV
0.985 mA
177 Ω
11k
990 mV
145 mV
0.990 mA
147Ω
13k
993 mV
121 mV
0.993 mA
122 Ω
15k
994 mV
105 mV
0.994 mA
106 Ω
Table 2: Calculated and measured values
7.
Plot the graph for
Frequency vs. VC.
(Use Excel or Word to Create the Plot)
Plot 1: Frequency vs. VC
Part II:
8.
Build the circuit shown in Figure 2.
Figure 2: Series RC Circuit
9.
Set the source voltage amplitude to
1.5 Vp and
frequency to
500 Hz.
10.
Connect Channel .
Physique et Chimie de la Terre Physics and Chemistry of the .docxLacieKlineeb
Physique et Chimie de la Terre / Physics and Chemistry of the Earth 2022 / 2023
Homework
Physics of the Earth
Deadline : 10th of november
The Herglotz-Wiechert method and
Earth’s mantle seismic velocities profiles
The goal of this problem is to build a model of the P and S wave velocity profiles in the Mantle,
from travel times tables build from observations. To do this, we will use the Herglotz-Wiechert method,
a method developed by Gustav Herglotz and Emil Wiechert at the beginning of the twentieth century.
We consider a seismic ray going from point S to point A, as depicted on figure 1. We denote by ∆
the angular distance of travel (i.e. the angle ŜCA), and by T (∆) the travel time of the seismic wave
as a function of angular distance. We recall that in spherical geometry the ray parameter is defined as
p = r sin i(r)
V (r) , (1)
and is constant along a given ray. Here r is the distance from the center of the Earth, i(r) is the
incidence angle (i.e. the angle between the ray and the vertical direction at a given r), and V (r) is the
wave velocity. We denote by R = 6371 km the radius of the Earth.
∆
d∆
R
p
p + dp
i
A
A’B
S
C
rb
Figure 1 – Two rays coming from the same source S with infinitesimally different ray parameters p and p + dp. Their
angular distances of travel are ∆ and ∆+ d∆, and their travel-times are T and T + dT . The line going through points A
and B is perpendicular to both rays.
1 Constant velocity model
Let us first assume that the wave velocity V does not vary with depth.
1. Draw on a figure the ray going from a source S to a point A of the surface, without any reflexion.
This ray could represent either the P or S phase.
2. Find the expression of the travel time T along this ray as a function of ∆.
3. Find the expression of the incidence angle i of the ray at point A as a function of the epicentral
distance ∆, and then show that the ray parameter is given by
p = R
V
cos
∆
2
. (2)
1/3
Physique et Chimie de la Terre / Physics and Chemistry of the Earth 2022 / 2023
2 Linking p to T and ∆
We now turn to a more realistic model and allow for radial variations of the waves velocities.
4. By considering two rays coming from the same source with infinitesimally different ray parameters
p and p + dp, and travel times T and T + dT (Figure 1a), demonstrate that
p = dT
d∆
. (3)
Hints : (1) Since the two rays are very close, the arcs connecting A to A’, A to B, and B to A’
can be approximated as straight lines. (2) Show first that the segment AB is part of a wavefront.
What does it imply for the times of arrivals at points A and B ?
5. Check that the expressions of p and T found for the constant velocity model are consistent with
eq. (3).
3 Travel time curves and estimate of the p(∆) curves
You will find on Chamillo a file containing travel time tables obtained from the global Earth’s seis-
mological model ak135 (either a text file, AK135tables.txt, or an Excel spreadsheet, AK135tables.xlsx).
The f.
2. ANSWER STRATEGY PHYSICS QUESTIONS PAPER 3
Written Practical Questions (1 Hours 30 Minutes)
SECTION A
Section A consists of two structured questions. You need to answer all
questions from this section.
This section is allocated a total of 28 MARKS.
3. Section A (Questions 1)
1. The questions in this section are based on experiments that you should
have already done in the laboratory.
2. The questions in this section normally require you to:
- State the variables base on the experiment given. (You are normally
required state the manipulated variable, the responding variable and
the constant variable based on the aim and the procedure of the
experiment)
- Record the data and do the tabulations. (Make sure you know how to
tabulate the data correctly).
- Plot a graph. (Make sure you know how to plot a graph correctly)
- State the relationship between two variables from the graph.
3. You are advised to spend 40 minutes on this section.
4. Section A (Questions 2)
1. The questions from this section also involve the interpretation of graphs.
2. You will also need to know how to determine the gradient and the unit of
the gradient of a graph.
3. The questions from this section may also involve the calculation of
certain quantities. Make sure that you write down all the steps involved
in the spaces provided in the questions paper. (Make sure you know how
to use the value of the gradient in the calculation)
4. You may need to state the precaution of the experiment.
5. You are advised to spend 30 minutes on this section
5. SECTION B
1. Section B consists of two questions. You need to answer ONE question
only from this section.
2. This section is allocated a total of 12 MARKS.
3. You are advised to spend 30 minutes on this section.
4. The questions in this section are normally base on the diagram of a
situation in our daily lives together with a brief write-up on the
situation.
5. You will have to study the situation carefully and try to find the
variables related in the situation.
6. 6. You will also be asked to state one appropriate inference, hypothesis
for an investigation and to describe an experimental framework to test
hypothesis. In your description, you will have to state clearly the
following:
(i) Aim of the experiment
(ii) Variables involved in the experiment
(iiii) List of apparatus and materials
(iv) Arrangement of the apparatus
(v) The procedure of the experiment which includes
- the methods of controlling the manipulated variable
- the method of measuring the responding variable
- the method of repeated experiment
(vi) The way you would tabulate the data
(vii) The way you would analyse the data
7. Make sure that you describe your experiment according to the format
shown above.
7. Section A
[28 marks]
Answer all questions in this section
A student carries out an experiment to investigate the relationship between
the length of wire, l, and its resistance, R.
The arrangement of the apparatus is shown in Diagram 1.1. An ammeter, dry
cells, a rheostat, a switch and piece of constantan sire are connected in
series.
A voltmeter is used to measure the potential difference, V, across the
constantan wire between P and Q.
8. A constantan wire of length, l = 20.0 cm is connected between P and Q. When
the switch is on, the rheostat is adjusted until the ammeter reading is 0.50 A.
The voltmeter reading, V, is as shown in Diagram 1.2 on page 4.
The corresponding voltmeter reading across P and Q are shown in Diagram
1.3, 1.4, 1.5 and 1.6 page 4.
9.
10. (a) For the experiment describe on pages 2 and 3, identify:
The manipulated variable
Length of (wire) / l
[1 mark]
(b)The responding variable
Resistance / R // Potential difference / V // Voltmeter reading //
Voltage
*Rej: voltmeter
[1 mark]
(c) The constant variable
Diameter of wire // Type of wire // Current // Thickness // Cross-
sections of wire // Radius // Ammeter reading // SWG // Resistivity
// Temperature
*Rej: Size, no of battery, emf, dge
[1 mark]
11. b) Based in Diagram 1.2, 1.3, 1.4, 1.5 and 1.6 on page 4:
Record the voltmeter readings, V, in the spaces provided on page 4.
Diagram 1.2 : 0.4 V
Diagram 1.3 : 0.9 V
Diagram 1.4 : 1.3 V
Diagram 1.5 : 1.7 V
Diagram 1.6 : 2.2 V
[2 marks]
(i) Calculate the values of R for each length of wire using the formula
R = V/0.5
Diagram 1.2 : 0.8 Ω [2 marks]
Diagram 1.3 : 1.8 Ω
Diagram 1.4 : 2.6 Ω
Diagram 1.5 : 3.4 Ω
Diagram 1.6 : 4.4 Ω
12. (ii) Tabulate your results for V and R for all values of l, in the space below.
[3 marks]
l / cm V/V R / ohm
20.0 0.4 0.8
40.0 0.9 1.8
60.0 1.3 2.6
80.0 1.7 3.4
100.0 2.2 4.4
Note:
Values of l, V and R shown in the table
State the units of l, V and R correctly
The values of l, V and R are consistent to one or two decimal place.
13. (c) On the graph paper on page 6, plot a graph of R against l.
Show R on the vertical-axis and l on the horizontal-axis
State the units of the variable correctly / symbol of units
Both axes are marked with uniform (even) scale
* (1:1, 1: 2, 1:4, 1:5, 1:10)
* Rej: Odd scale
All five points are plotted correctly
* 1 (2 mm x 2 mm)
* 5 points – 2m
* 3-4 points – 1m
Best fitted straight line
* point to the line 5 mm @ 2.5
Show the minimum size of graph at least 5 (y) x 4 (x) ( 10 cm x 8 cm) square
* Start from the origin until the last point
[7 marks]
14. (d) Based on your graph in 1(c), state the relationship between R and l.
Resistance of wire (R) is directly proportional to the length of wire (l)/
R α l / l α R / Increasing linearly
[1 mark]
15.
16. 2. A student carries out an experiment to investigate the relationship
between the mass, m, of a load placed on a spring and the length, l, of the
spring. The student also determines the spring constant, k.
The result of this experiment is shown in the graph of l against m in
Diagram below.
17. (a) Based on the graph in Diagram 2.1
(i) what happens to l as m increases?
Increases // longer // extended // greater length //
bigger length // higher length
[1 mark]
(ii) determine the value of l when m = 0 g.
l = 9.5 – 10 cm
- show graphical extrapolation correctly
- state the value within acceptable range
[2 marks]
18. (b) The spring constant, k, is given by the formula k = 1/h, where h is the
gradient of the graph.
(i) Calculate the gradient, h, of the graph.
Show on the graph how you calculate h.
[3 marks]
h = (22.5 – 10)/60
= 0.208 cm g-1
- Draw a sufficiently large triangle ≥ 8 cm x 8 cm
- Correct substitution
- State the value within acceptable range and correct unit
* Reject : answer in fraction
19. (ii) Determine the value of k.
[2 marks]
k = 1/h
= 4.81 g cm-1
- Correct substitution
- State the value of k within the acceptable range
20. (c) Another identical spring is connected in series to the end of the
spring. The spring constant, k’, of the two springs in series is given by
1 1 1
the formula 1/k`k= 1/k + 1/k
k'' k
Calculate k’.
k’ = ………………………….
State the value of k’ [2 marks]
2.41 g cm-1 / 2.405 g cm-1
(d) State two precautions that can be taken to improve the accuracy of
the readings in this experiment.
Repeat readings and take average
Eye perpendicular to the scale/reading to avoid parallax error.
Ensure the spring does not swing / at rest when reading is taken.
*Rej: parallax error, parallel
[2 marks]
21. Section B
[12 marks]
Answer any one questions from this section
1. Diagram 3 shows two opaque cups, A and B, containing different
amount of water. A similar coin is placed at the bottom of each cup.
When the coin is observed from the same position, the image of the
coin in cup A cannot be seen, but the image of the coin in cup B can
be seen.
22. Based on your observation on the depth of the water and the
position of the images of the coins:
(a)State one suitable inference
Depth of water affects the position of image //
Position of image depends on the depth of water
*Note:
Must have cause without effect
RV influenced by MV
RV affected MV
MV affects RV
[1 mark]
23. (b)State one hypothesis that could be investigated.
The more the depth of water, the more the depth of the image
// the higher is the image.
*Note: Must have cause and effect
[1 mark]
24. (c) With the use of apparatus such as a tall beaker, pins and other
apparatus, describe an experiment to investigate the hypothesis
stated in 3(b)
In your description, state clearly the following;
(i) The aim of the experiment.
To investigate the relationship between the depth of water
and apparent depth // real depth and apparent depth
*Note: Relate MV and RV
(ii) The variables in the experiment.
Manipulated variable: Real depth
Responding variable: Apparent depth // Depth of image
* Note: Both must correct
Constant variable: Density of water // Refractive index //
Type of liquid
25. (iii) The list of apparatus and materials.
Beaker, Water, Pins, Set of retort stand, Metre rule // Diagram
[1 mark]
26. (iv) The arrangement of apparatus.
- State a functional arrangement of the apparatus
*Note: Functional mean experimental can be done or can get a data.
[1 mark]
27. (v) The procedures of the experiment which include the method of
controlling the manipulated variable and the method of measuring the
responding variable.
State the method to control manipulated variable
Fill beaker with water to a depth of d1 = 10 cm / any number or
symbol/letter
State the method to measure the responding variable
Move the pin outside the beaker to obtain the apparent position of the
pin in the beaker.
Measure the position of the pin from the surface of the water to the
pin.
Repeat the experiment at least 4 times
Repeat the previous steps by increasing the depth of water 15 cm, 20
cm, 25 cm and 30 cm.
[3 marks]
28. (vi) The way to tabulate the data.
Show how the data is tabulated.
*Note: Must have 2 columns
Depth of water, d / cm Apparent depth, h / cm
10
15
20
25
30
(vii) The way to analyse the data.
Apparent depth/cm
- If use symbol, must mention earlier.
- Accept conclusion, statement of variable
related.
- hαd
Depth of water/cm