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6.1E: Spring Problems I (Exercises) - Mathematics


In the following exercises assume that there’s no damping.

Q6.1.1

1. An object stretches a spring (4) inches in equilibrium. Find and graph its displacement for (t>0) if it is initially displaced (36) inches above equilibrium and given a downward velocity of (2) ft/s.

2. An object stretches a string (1.2) inches in equilibrium. Find its displacement for (t>0) if it is initially displaced (3) inches below equilibrium and given a downward velocity of (2) ft/s.

3. A spring with natural length (.5) m has length (50.5) cm with a mass of (2) gm suspended from it. The mass is initially displaced (1.5) cm below equilibrium and released with zero velocity. Find its displacement for (t>0).

4. An object stretches a spring (6) inches in equilibrium. Find its displacement for (t>0) if it is initially displaced (3) inches above equilibrium and given a downward velocity of (6) inches/s. Find the frequency, period, amplitude and phase angle of the motion.

5. An object stretches a spring (5) cm in equilibrium. It is initially displaced (10) cm above equilibrium and given an upward velocity of (.25) m/s. Find and graph its displacement for (t>0). Find the frequency, period, amplitude, and phase angle of the motion.

6. A (10) kg mass stretches a spring (70) cm in equilibrium. Suppose a (2) kg mass is attached to the spring, initially displaced (25) cm below equilibrium, and given an upward velocity of (2) m/s. Find its displacement for (t>0). Find the frequency, period, amplitude, and phase angle of the motion.

7. A weight stretches a spring (1.5) inches in equilibrium. The weight is initially displaced (8) inches above equilibrium and given a downward velocity of (4) ft/s. Find its displacement for (t > 0).

8. A weight stretches a spring (6) inches in equilibrium. The weight is initially displaced (6) inches above equilibrium and given a downward velocity of (3) ft/s. Find its displacement for (t>0).

9. A spring–mass system has natural frequency (7sqrt{10}) rad/s. The natural length of the spring is (.7) m. What is the length of the spring when the mass is in equilibrium?

10. A (64) lb weight is attached to a spring with constant (k=8) lb/ft and subjected to an external force (F(t)=2sin t). The weight is initially displaced (6) inches above equilibrium and given an upward velocity of (2) ft/s. Find its displacement for (t>0).

11. A unit mass hangs in equilibrium from a spring with constant (k=1/16). Starting at (t=0), a force (F(t)=3sin t) is applied to the mass. Find its displacement for (t>0).

12. A (4) lb weight stretches a spring (1) ft in equilibrium. An external force (F(t)=.25sin8 t) lb is applied to the weight, which is initially displaced (4) inches above equilibrium and given a downward velocity of (1) ft/s. Find and graph its displacement for (t>0).

13. A (2) lb weight stretches a spring (6) inches in equilibrium. An external force (F(t)=sin8t) lb is applied to the weight, which is released from rest (2) inches below equilibrium. Find its displacement for (t>0).

14. A (10) gm mass suspended on a spring moves in simple harmonic motion with period (4) s. Find the period of the simple harmonic motion of a (20) gm mass suspended from the same spring.

15. A (6) lb weight stretches a spring (6) inches in equilibrium. Suppose an external force (F(t)={3over16}sinomega t+{3over8}cosomega t) lb is applied to the weight. For what value of (omega) will the displacement be unbounded? Find the displacement if (omega) has this value. Assume that the motion starts from equilibrium with zero initial velocity.

16. A (6) lb weight stretches a spring (4) inches in equilibrium. Suppose an external force (F(t)=4sinomega t-6cosomega t) lb is applied to the weight. For what value of (omega) will the displacement be unbounded? Find and graph the displacement if (omega) has this value. Assume that the motion starts from equilibrium with zero initial velocity.

17. A mass of one kg is attached to a spring with constant (k=4) N/m. An external force (F(t)=-cosomega t-2sinomega t) n is applied to the mass. Find the displacement (y) for (t>0) if (omega) equals the natural frequency of the spring–mass system. Assume that the mass is initially displaced (3) m above equilibrium and given an upward velocity of (450) cm/s.

18. An object is in simple harmonic motion with frequency (omega_0), with (y(0)=y_0) and (y'(0)=v_0). Also, find the amplitude of the oscillation and give formulas for the sine and cosine of the initial phase angle.

19. Two objects suspended from identical springs are set into motion. The period of one object is twice the period of the other. How are the weights of the two objects related?

20. The weight of one object is twice the weight of the other. How are the periods of the resulting motions related?

21. Two identical objects suspended from different springs are set into motion. The period of one motion is (3) times the period of the other. How are the two spring constants related?


WORD PROBLEMS ON SETS AND VENN DIAGRAMS

Let us come to know about the following terms in details.

n(AuB)  =  Total number of elements related to any of the two events A & B.

n(AuBuC)  =  Total number of elements related to any of the three events A, B & C. 

n(A)  =  Total number of elements related to  A.

n(B)  =  Total number of elements related to  B.

n(C)  =  Total number of elements related to  C.

Total number of elements related to A only.

Total number of elements related to B only.

Total number of elements related to C only.

Total number of elements related to both A & B

Total number of elements related to both (A & B) only.

Total number of elements related to both B & C

Total number of elements related to both (B & C) only.

Total number of elements related to both A & C

Total number of elements related to both (A & C) only.

For  two events A & B, we have

Total number of elements related to A only.

Total number of elements related to B only.


See Also:

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On these printables, students will add the lengths of the sides of the polygons to find the perimeters.


Math 101

Math 101 is a first-level course in the fulfillment of the mathematics requirement for graduation at the University of Kansas. Success in College Algebra fulfills one unit of the Critical Thinking & Quantitative Literacy General Education Goal for the KU Core and prepares students for subsequent work in a second-level mathematics course (i.e. calculus sequence or statistics). The course is designed to reinforce basic skills and deepen conceptual understanding of the algebraic principles fundamental to mathematical reasoning.

The course will focus on the study of functions through multiple representations - verbal, graphic, symbolic, and numeric. Using the basic function families: linear, absolute value, polynomial (square, square root, cube, cube root, higher degree), rational, exponential, and logarithmic, we will analyze relationships among the representations. Additional topics studied include linear systems of equations and matrices. Students will make connections between the graphs of functions, their associated equations and inequalities, and related applications.

Current students should log in to My KAP Info for the following course information:


Project work

Kick-off session on March 24 (usual lecture time and place): be there to get specific project work information.

Phase 1 review interviews will take place in the time period April 5-7. Be sure to agree on a time slot with the lecturer!

Phase 2 final goal is the Poster session on Friday, May 5, at 10-12 in the Exactum 1st floor hallway.

Project work assistants: Alexander Meaney, Zenith Purisha and Markus Juvonen.

The idea is to study an inverse problem both theoretically and computationally in teams of two students. The end product is a scientific poster that the team will present in a poster session on May 5 (details above).

The poster can be printed using the laboratory's large scale printer. Please send your poster via email as a pdf attachment to Markus Juvonen by Wednesday, 3rd May, 12 pm. Then your poster will be printed in time for the poster session.

The idea of the project work is to study an inverse problem both theoretically and computationally. The classical table of contents is recommended for structuring the first phase report and the poster:

1 Introduction
2 Materials and methods
3 Results
4 Discussion

Section 1 should briefly explain the topic in a way accessible to a non-expert.
Section 2 is for describing the data and the inversion methods used.
In section 3 the methods of Section 2 are applied to the data described in Section 2, and the results are reported with no interpretation just facts and outcomes of computations are described.
Section 4 is the place for discussing the results and drawing conclusions.

The project is either about X-ray tomography or digital image processing. You can measure a dataset yourself in the Industrial Mathematics Laboratory.

X-ray topic: you can choose your own object to image in the X-ray lab. We can offer a range of tried-and-tested objects and tools to tailor them to your liking. Also, you can come up with your own idea of the measured object. The size of the object should not much exceed the size of an egg, and the chemical composition is important as X-ray attenuation contrast arises from electron densities in the materials. Please contact Alexander Meaney to find out if your object is good for imaging.

Photographic topic: take blurred or noisy photos of suitable targets.

In the project you are supposed to apply some of the regularization methods discussed in the course. Optimally, you should have an automatic method for choosing the regularization parameter and an automatic stopping criteria for the iteration. These are both difficult requirements, so have a simple approach as plan B if a more complicated approach does not work. Also, when choosing your objects of measurement, it's good to think about mathematical models of a priori information (piecewise constant, smooth, piecewise smooth) as it affects the choice of the regularizer.

First goal: the two first sections (Introduction and Materials and Methods) should be preliminary written in LaTeX, but not necessarily in poster format. The most important things to explain are:

  • what kind of data to measure,
  • what inversion method to apply for the reconstruction, and
  • how to implement the computation.

The grade of the first goal represents 30% of the final grade of the project work. Please agree on a meeting time (in the period April 5-7) with the lecturer for reviewing and grading the first goal.

Second and final goal: poster session. The poster will be printed in size A1. You may create your own poster (from scratch), or you can use e.g. this template as a starting point and edit its layout, colors, fonts, etc. as much as you like.

You can take a look at old posters on this page.


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Practice multiplying fractions and mixed numbers with these printables.

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Calculate slopes using ordered pairs, graphs, and x/y tables.

Find the surface areas of the solid, three-dimensional shapes.

Calculate unit rates and ratios. Solve unit rate word problems involving pricing, acceleration, and linear measurement.

Calculate the volumes of cylinders, cones, spheres, and other shapes with these printables.

This page links to our full math index. You'll find thousands elementary and middle school worksheets.


Math 436: Undergraduate Algebraic Geometry

Welcome to the webpage for Math 436. This is an undergraduate algebraic geometry course.
The course will include topics on Hilbert Nullstellensatz, affine and projective varieties, smooth varieties, curves, Bezout's Theorem, and other topics as time permits.

Time/Location Tu-Th 10-11:30AM, Cupples II, L001

Prerequisites: permission of the instructor.

Textbook: There are two recommended book, both are available to download for free:

1) Ideals, Varieties, and Algorithms, an introduction to computational algebraic geometry and commutative algebra, by David Cox, John Little, Donal OShea. The book is available to download freely at WUSTL's library.
2) Undergraduate algebraic geometry by Miles Reid, available here.

Other useful books are
3) Algebraic curves by Fulton
4) Complex Algebraic Curves, by Frances Kirwan

Office hour: Tuesday 2:30-3:30, Wednesday 3-4, or By appointment. Location: Cupples I, Room 108B.

Grader: The course assistant for this course is Jeffery Norton ([email protected]).

Exams: There will be one in class midterm exam during the semester. There will also be a final exam. The midterm is tentatively scheduled for Thursday March 8. If you are unable to take the midterm exam for legitimate reasons, you will not be given a make up exam.

Grading Information: The midterm and the homework will each count for 30 percent of your grade. The final exam will count for 40 percent of your grade.

Homework: There will be weekly homework. You are encouraged to discuss the problems but the write-up must be your own.
I expect there will be 10 problem sets. The lowest homework grade will be dropped and the remaining 9 will be counted towards your final course grade.

1. Homework #1 , Due January 25 in class. Solutions
2. Homework #2 , Due February 1 in class. Solutions
3. Homework #3 , Due February 8 in class. Solutions
4. Homework #4 , Due February 15 in class. Solutions
5. Homework #5 , Due February 27 in class. Solutions
6. Homework #6 , Due March 6 in class.
7. Homework #7 , Due March 29 in class.
8. Homework #8 , Due April 12 in class.
9. Homework #9 , Due April 26 in class.
Solutions to selected problems from Homework 8 and 9


Grade Two Spring Worksheets:

  1. One number from 10 to 99 and one number from 0 to 9 (addition)(subtraction)
  2. Number sentences - fill in the + or - sign (addition/subtraction)
  3. Two digit numbers - NO CARRYING (addition) (subtraction)
  4. Two digit numbers - CARRYING
    Vertical (addition) (subtraction)
    Horizontal (addition) (subtraction)
  5. Fill in the blanks - two digit numbers with CARRYING
    Vertical (addition) (subtraction)
    Horizontal (addition) (subtraction)
  6. Two digit number MAGIC SQUARES (addition) (subtraction)
  7. Two digit number MATH TABLES (addition) (subtraction)
  8. Two digit number sentences - fill in the + or - sign (addition/subtraction)
  9. Add a column of three single digit numbers
  10. Visual multiplication exercise (multiplication)
  11. Two numbers from 0 to 10 (multiplication)
  12. Multiplication MATH TABLES (multiplication)
  13. Number sentences - fill in the + or - or x sign (addition/subtraction/multiplication)
  14. Word Problems

The Complex Monge-Ampere Operator

Instructor: Jeffrey Diller (click for contact info)
Time and place: MWF, 10:40-11:30 in 131 DeBartolo.

Abstract : The real and imaginary parts of a holomorphic function of one complex variable are harmonic. So begins a beautiful relationship between classical complex function theory and the (linear) Laplace equation in the plane. In several complex variables, however, the Laplace operator and its associated potential theory are not so relevant. Instead, one is led to consider the complex Monge-Ampere operator. This course aims to tell the much more recent story of this non-linear operator and its applications to complex analysis and geometry.

Text : I'm putting the books Pluripotential Theory by Klimek and The Complex Monge-Ampere Operator and Pluripotential Theory by Kolodziej on reserve in the math library. The former is more elementary but also a little dated by now. Mostly, however, I'll be following freely available notes and articles which I'll link to below. In particular, for the first half of the semester, I'll lean particularly heavily on some course notes by Blocki. And now I find I've been typing up some notes of my own, partly to clean up issues from lecture and partly just to make myself happy. These aren't comprehensive, but they do include some extra detail and different presentation of many basic points.


Download Free Printable Class 2 Maths Mental Mathematics Worksheets

Mathematics Worksheets for Class 2 on Mental Mathematics will help your kid remember what he/ she learned in school for a long time. Your Grade 2 kid will love to solve these engaging exercises and interesting collections of Puzzles.

CBSE Class 2 Maths Mental Mathematics Worksheets

1. Complete the following: 15 subtract 4 is ______

2. Complete the following: 14 subtract 4 is ______

3. Complete the series: 0, 1, 2, 3, _______.

4. What is 10 more than 24?

5. What is 10 less than 16?

6. What is 10 less than 22?

7. Complete the series: 0, 25, 50, 75, ______.

9. Write the numbers that comes next: 7, 14, 21, 28, _____.

11. Complete the following: Double of 9 is ________

12. What is 5 more than 15?

13. Complete the following: 20 subtract 5 is ______

15. Complete the following: 18 subtract 6 is ______

16. Is 24 x 9 = 218
A) True
B) False

17. Complete the following: 11 subtract 1 is ______

19. Solve 58*52 = ?
A) 3016
B) 3220

20. Complete the following: 7 subtract 3 is ______

CBSE Class 2 Maths Worksheet Answers

1. 11
2. 10
3. 4
4. 34
5. 6
6. 12
7. 100
8. 15
9. 35
10. 14
11. 18
12. 20
13. 15
14. 10
15. 12
16. Option B
17. 10
18. 17
19. Option A
20. 4

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6.1E: Spring Problems I (Exercises) - Mathematics

Here are the steps required for Solving Direct Variation Problems:

Step 1: Write the correct equation. Direct variation problems are solved using the equation y = kx. When dealing with word problems, you should consider using variables other than x and y, you should use variables that are relevant to the problem being solved. Also read the problem carefully to determine if there are any other changes in the direct variation equation, such as squares, cubes, or square roots.
Step 2: Use the information given in the problem to find the value of k, called the constant of variation or the constant of proportionality.
Step 3: Rewrite the equation from step 1 substituting in the value of k found in step 2.
Step 4: Use the equation found in step 3 and the remaining information given in the problem to answer the question asked. When solving word problems, remember to include units in your final answer.

Example 1 &ndash If x varies directly as y, and x = 9 when y = 6, find x when y = 15.

Example 2 &ndashIf p varies directly as the square of q, and p = 20 when q = 5, find p when q = 8.

Example 3 &ndash If c varies directly as the square root of d, and c = 6 when d = 256, find c when d = 625.

Example 4 &ndash Hooke&rsquos Law for an elastic spring states that the distance a spring stretches varies directly as the force applied. If a force of 160 newtons stretches a spring 5 cm, how much will a force of 368 newtons stretch the same spring?

Example 5 &ndash The distance a body falls from rest varies directly as the square of the time it falls (ignoring air resistance). If a ball falls 144 feet in three seconds, how far will the ball fall in seven seconds?