Module 1 lesson 8
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This document contains notes from a math lesson on representing proportional relationships with equations. It includes examples of proportional word problems involving gas mileage and drawing portraits. Homework assignments are listed and the lesson covers finding the constant of proportionality, writing equations for proportional relationships, using equations to solve problems algebraically and arithmetically, and interpreting points on a proportional graph.
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This document contains notes from a math lesson on representing proportional relationships with equations. It includes examples of writing equations to relate variables that are proportional, such as miles driven to gallons of gas. It also discusses using the constant of proportionality k to write equations in the form y=kx and solving problems algebraically and arithmetically. There are practice problems for students to complete as homework.
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This document appears to be notes from a math lesson on representing proportional relationships with equations. It includes examples of finding the constant of proportionality from data sets and using it to write equations relating variables. Students are asked to determine constants of proportionality, write proportional equations, and solve problems algebraically and arithmetically using the equations. The document emphasizes that the constant of proportionality is the unit rate and can be used to write an equation relating proportional variables.
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11 - 3
Experiment 11
Simple Harmonic Motion
Questions
How are swinging pendulums and masses on springs related? Why are these types of
problems so important in Physics? What is a spring’s force constant and how can you measure
it? What is linear regression? How do you use graphs to ascertain physical meaning from
equations? Again, how do you compare two numbers, which have errors?
Note: This week all students must write a very brief lab report during the lab period. It is
due at the end of the period. The explanation of the equations used, the introduction and the
conclusion are not necessary this week. The discussion section can be as little as three sentences
commenting on whether the two measurements of the spring constant are equivalent given the
propagated errors. This mini-lab report will be graded out of 50 points
Concept
When an object (of mass m) is suspended from the end of a spring, the spring will stretch
a distance x and the mass will come to equilibrium when the tension F in the spring balances the
weight of the body, when F = - kx = mg. This is known as Hooke's Law. k is the force constant
of the spring, and its units are Newtons / meter. This is the basis for Part 1.
In Part 2 the object hanging from the spring is allowed to oscillate after being displaced
down from its equilibrium position a distance -x. In this situation, Newton's Second Law gives
for the acceleration of the mass:
Fnet = m a or
The force of gravity can be omitted from this analysis because it only serves to move the
equilibrium position and doesn’t affect the oscillations. Acceleration is the second time-
derivative of x, so this last equation is a differential equation.
To solve: we make an educated guess:
Here A and w are constants yet to be determined. At t = 0 this solution gives x(t=0) = A,
which indicates that A is the initial distance the spring stretches before it oscillates. If friction is
negligible, the mass will continue to oscillate with amplitude A. Now, does this guess actually
solve the (differential) equation? A second time-derivative gives:
Comparing this equation to the original differential equation, the correct solution was
chosen if w2 = k / m. To understand w, consider the first derivative of the solution:
−kx = ma
a = −
k
m
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
x
d 2x
dt 2
= −
k
m
x x(t) = A cos(ωt)
d 2x(t)
dt 2
= −Aω2 cos(ωt) = −ω2x(t)
James Gering
Florida Institute of Technology
11 - 4
Integrating gives
We assume the object completes one oscillation in a certain period of time, T. This helps
set the limits of integration. Initially, we pull the object a distance A from equilibrium and
release it. So at t = 0 and x = A. (one.
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Sim variation
This document provides guidance on using a strategic intervention material (SIM) to learn about combined variation. It begins with reminders on using the SIM independently and thoroughly. The least mastered skill is solving problems involving combined variation, where a variable depends on two or more other variables in a direct and inverse relationship. Examples are provided of writing equations to represent relationships like "W varies jointly as c and the square of a and inversely as b." Several practice activities involve writing equations and solving for unknowns. The document concludes with an assessment to check understanding of combined variation concepts.
11 - 3 Experiment 11 Simple Harmonic Motio.docx
11 - 3
Experiment 11
Simple Harmonic Motion
Questions
How are swinging pendulums and masses on springs related? Why are these types of
problems so important in Physics? What is a spring’s force constant and how can you measure
it? What is linear regression? How do you use graphs to ascertain physical meaning from
equations? Again, how do you compare two numbers, which have errors?
Note: This week all students must write a very brief lab report during the lab period. It is
due at the end of the period. The explanation of the equations used, the introduction and the
conclusion are not necessary this week. The discussion section can be as little as three sentences
commenting on whether the two measurements of the spring constant are equivalent given the
propagated errors. This mini-lab report will be graded out of 50 points
Concept
When an object (of mass m) is suspended from the end of a spring, the spring will stretch
a distance x and the mass will come to equilibrium when the tension F in the spring balances the
weight of the body, when F = - kx = mg. This is known as Hooke's Law. k is the force constant
of the spring, and its units are Newtons / meter. This is the basis for Part 1.
In Part 2 the object hanging from the spring is allowed to oscillate after being displaced
down from its equilibrium position a distance -x. In this situation, Newton's Second Law gives
for the acceleration of the mass:
Fnet = m a or
The force of gravity can be omitted from this analysis because it only serves to move the
equilibrium position and doesn’t affect the oscillations. Acceleration is the second time-
derivative of x, so this last equation is a differential equation.
To solve: we make an educated guess:
Here A and w are constants yet to be determined. At t = 0 this solution gives x(t=0) = A,
which indicates that A is the initial distance the spring stretches before it oscillates. If friction is
negligible, the mass will continue to oscillate with amplitude A. Now, does this guess actually
solve the (differential) equation? A second time-derivative gives:
Comparing this equation to the original differential equation, the correct solution was
chosen if w2 = k / m. To understand w, consider the first derivative of the solution:
−kx = ma
a = −
k
m
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
x
d 2x
dt 2
= −
k
m
x x(t) = A cos(ωt)
d 2x(t)
dt 2
= −Aω2 cos(ωt) = −ω2x(t)
James Gering
Florida Institute of Technology
11 - 4
Integrating gives
We assume the object completes one oscillation in a certain period of time, T. This helps
set the limits of integration. Initially, we pull the object a distance A from equilibrium and
release it. So at t = 0 and x = A. (one.
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Module 1 lesson 8
- 1. Module 1 Lesson 8.notebook 1 September 16, 2014 09/16/13 Module 1, Lesson 8 Representing Proportional Relationships with Equations Homework: Lesson 8 Problem Set #1, 3, 4, 5 Do Now Quiz 1 Tuesday 9/23 Exit Ticket from Lesson 7
- 2. Module 1 Lesson 8.notebook 2 September 16, 2014
- 3. Module 1 Lesson 8.notebook 3 September 16, 2014 Exit Ticket
- 4. Module 1 Lesson 8.notebook 4 September 16, 2014 S.27
- 5. Module 1 Lesson 8.notebook 5 September 16, 2014 Problem Set Solutions (continued) S.27
- 6. Module 1 Lesson 8.notebook 6 September 16, 2014 Problem Set Solutions (continued) S.27
- 7. Module 1 Lesson 8.notebook 7 September 16, 2014 S.28
- 8. Module 1 Lesson 8.notebook 8 September 16, 2014 S.29
- 9. Module 1 Lesson 8.notebook 9 September 16, 2014 a. Find the constant of proportionality and explain what it represents in this situation. Gallons Miles Driven y x 8 224 10 280 4 112
- 10. Module 1 Lesson 8.notebook 10 September 16, 2014 b. Write the equation that will relate the miles driven to the number of gallons of gas. Steps to figure out the equation. 1. Which unit is the dependent variable? 2. Which unit is the independent variable? 3. Find the k. Constant of Proportionality = k = 4. Plug in the value of k to y = kx. The constant x is known as the independent variable and y as the dependent variable,.
- 11. Module 1 Lesson 8.notebook 11 September 16, 2014 c. Knowing that there is a half gallon left in the gas tank when the light comes on, will she make it to the nearest gas station? Explain why or why not. d. Using the equation found in part b, determine how far your mother can travel on 18 gallons of gas. Solve the problem in two ways. Using Arithmetic Using Algebra
- 12. Module 1 Lesson 8.notebook 12 September 16, 2014 e. Using the equation found in part b, determine how many gallons of gas would be needed to travel 750 miles.
- 13. Module 1 Lesson 8.notebook 13 September 16, 2014 S.30 a. Write several ordered pairs from the graph and explain what each coordinate pair means in the context of this graph.
- 14. Module 1 Lesson 8.notebook 14 September 16, 2014 b. Write several equations that would relate the number of portraits drawn to the time spent drawing the portraits. c. Determine the constant of proportionality and explain what it means in this situation.
- 15. Module 1 Lesson 8.notebook 15 September 16, 2014 Problem Set S.31
- 16. Module 1 Lesson 8.notebook 16 September 16, 2014 Problem Set S.32
- 17. Module 1 Lesson 8.notebook 17 September 16, 2014
- 18. Module 1 Lesson 8.notebook 18 September 16, 2014 Closing Question How can unit rate be used to write an equation relating two variables that are proportional? The unit rate is the constant of proportionality, k. After computing the value for k, it may be substituted in place of k in the equation y = kx. The constant of proportionality can be multiplied by the independent variable to find the dependent variable, and the dependent variable can be divided by the constant of proportionality to find the dependent variables.
- 19. Module 1 Lesson 8.notebook 19 September 16, 2014