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6.1E: Spring Problems I (Exercises)


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?


Java Programming Exercises, Practice, Solution

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The best way we learn anything is by practice and exercise questions. Here you have the opportunity to practice the Java programming language concepts by solving the exercises starting from basic to more complex exercises. A sample solution is provided for each exercise. It is recommended to do these exercises by yourself first before checking the solution.

Hope, these exercises help you to improve your Java programming coding skills. Currently, following sections are available, we are working hard to add more exercises . Happy Coding!

List of Java Exercises:

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More to Come !


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Safety Considerations

Before you begin a new workout routine, it’s a good idea to talk to your doctor. Working out your core muscles shouldn’t be painful. It is likely you will feel a little sore afterward, but if you experience any sharp or long-lasting pain, talk to your doctor.

Always do a short warm-up. One idea is to march in place while swinging your arms to activate the core.

Make sure you understand the proper form for any exercise you do. Look at example photos, read written instructions, and watch videos if available.

Remember, your core is more than just your abs. Include your butt and back muscles in your workout plan for full core strength.

Sources

Ace Fitness: “Side Plank - modified.”

Ace Fitness: “Supine Dead Bug.”

Harvard Health Publishing Harvard Medical School: “Core exercise workout: 12 tips for exercising safely and effectively.”

Harvard Health Publishing Harvard Medical School: “The real-world benefits of strengthening your core.”

Self: “31 of the Best Core Exercises You Can Do at Home.”

Shape: “Why Core Strength Is So Important (It Has Nothing to Do with Sculpting a Six-Pack).”


To build up your lower back and buttocks, try lifting your back leg while standing straight. Hold a chair and raise one leg backward without bending your knee or pointing your toe. Keep your anchor leg slightly bent. Hold your position for 1 second. Do this 10-15 times with the first leg before moving on to the other one.


6.1E: Spring Problems I (Exercises)

    Course Name: Math 201: Introduction to proofs, (section A and B) Spring 2017.

Course Material

  • Textbook: We will be following two texts. Free pdf copies of both textbooks are available online. The following are the links to the versions I will be using for assigning homeworks.
    • Book of proof, by R. Hammack, second edition (Abbreviated [BP])
    • Basic Analysis: Introduction to Real Analysis, by J. Lebl. (Abbreviated [BA])
    • Course catalog description and prerequisites :(3-0) Cr. 3. F.S. Prereq: MATH 166 or MATH 166H. Logic and techniques of proof including induction. Communicating mathematics. Writing proofs about sets, functions, real numbers, limits, sequences, infinite series and continuous functions. If you are not sure if you have the expected preparation, please talk to me IN THE FIRST WEEK OF CLASS

    Course Policy

    • Homework:
      • Homeworks will be posted on the course log at the bottom of this page
      • Homeworks assigned during one week will be due on the following Friday at the beginning of the class. Please staple your homework.
      • Late homework will not be accepted except in rare and extreme circumstances. See the university webpage on course policies for more details.
      • You may work on the homeworks together, but you should write up your solutions independently. There will probably be about 14 homeworks in total, you should submit all of them. The best 12 out of the 14 you submit will count towards the final grade.


      16.1 Hooke’s Law: Stress and Strain Revisited

      Newton’s first law implies that an object oscillating back and forth is experiencing forces. Without force, the object would move in a straight line at a constant speed rather than oscillate. Consider, for example, plucking a plastic ruler to the left as shown in Figure 16.2. The deformation of the ruler creates a force in the opposite direction, known as a restoring force . Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until dissipative forces dampen the motion. These forces remove mechanical energy from the system, gradually reducing the motion until the ruler comes to rest.

      The simplest oscillations occur when the restoring force is directly proportional to displacement. When stress and strain were covered in Newton’s Third Law of Motion, the name was given to this relationship between force and displacement was Hooke’s law:

      Example 16.1

      How Stiff Are Car Springs?

      What is the force constant for the suspension system of a car that settles 1.20 cm when an 80.0-kg person gets in?

      Strategy

      Solution

      Substitute known values and solve k k size 12 <> :

      Discussion

      Energy in Hooke’s Law of Deformation

      In order to produce a deformation, work must be done. That is, a force must be exerted through a distance, whether you pluck a guitar string or compress a car spring. If the only result is deformation, and no work goes into thermal, sound, or kinetic energy, then all the work is initially stored in the deformed object as some form of potential energy. The potential energy stored in a spring is PE el = 1 2 kx 2 PE el = 1 2 kx 2 size 12 <"PE" rSub < size 8<"el">> = < <1>over <2>> ital "kx" rSup < size 8<2>> > <> . Here, we generalize the idea to elastic potential energy for a deformation of any system that can be described by Hooke’s law. Hence,

      Example 16.2

      Calculating Stored Energy: A Tranquilizer Gun Spring

      We can use a toy gun’s spring mechanism to ask and answer two simple questions: (a) How much energy is stored in the spring of a tranquilizer gun that has a force constant of 50.0 N/m and is compressed 0.150 m? (b) If you neglect friction and the mass of the spring, at what speed will a 2.00-g projectile be ejected from the gun?

      Strategy for a

      (a): The energy stored in the spring can be found directly from elastic potential energy equation, because k k size 12 <> and x x size 12 <> are given.

      Solution for a

      Strategy for b

      Because there is no friction, the potential energy is converted entirely into kinetic energy. The expression for kinetic energy can be solved for the projectile’s speed.

      Solution for b

      Discussion

      (a) and (b): This projectile speed is impressive for a tranquilizer gun (more than 80 km/h). The numbers in this problem seem reasonable. The force needed to compress the spring is small enough for an adult to manage, and the energy imparted to the dart is small enough to limit the damage it might do. Yet, the speed of the dart is great enough for it to travel an acceptable distance.

      Check Your Understanding

      Envision holding the end of a ruler with one hand and deforming it with the other. When you let go, you can see the oscillations of the ruler. In what way could you modify this simple experiment to increase the rigidity of the system?

      Solution

      You could hold the ruler at its midpoint so that the part of the ruler that oscillates is half as long as in the original experiment.

      Check Your Understanding

      If you apply a deforming force on an object and let it come to equilibrium, what happened to the work you did on the system?

      Solution

      It was stored in the object as potential energy.

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        The following exercises will help you gain greater understanding about how subordinating conjunctions work. Choose the best answer to complete each sentence.

        _________ the basement flooded, we spent all day cleaning up.

        I don’t want to go to the movies ­­­_____________ I hate the smell of popcorn.

        I paid Larry, ___________ garden design work is top-notch.

        ___________ spring arrives, we have to be prepared for more snow.

        _____________ the alarm goes off, I hit the snooze button.


        EXERCISES AND PROBLEMS FORCE AND MOTION

        Solution is in the Student Solutions Manual.
        wvw Solution is available on. the World Wide Web at:
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        Solution is available on the Interactive Learning Ware.

        Newton’s Second Law

        1E. If the I kg standard body has an acceleration of 2.00 at 20° to the positive direction of the x axis, then what are (a) the x component and (b) the y component of the net force on it, and (c) what is the net force in unit-vector notation?

        2E. Two horizontal forces act on a 2.0 kg chopping block that can slide over a friction less kitchen ‘counter, which lies in an plane. One force is . Find the acceleration of the chomping block in !Init-vector notation

        3E. Only two horizontal forces act on a 3.0 kg body. One force is. 9.0 N, acting due east, and the other is 8.0 N. acting 62° north of west. What is the magnitude of the body’s acceleration?

        41:. While two forces act on it, a particle is to move at the constant velocity . One of the forces is . What is the other force?

        5f. Three forces act on a particle that moves with unchanging velocity. What is the third force?

        6P. There are two forces on the 2.0 kg box in the overhead view of but only one shown. The figure also shows the acceleration of the box. Find the second force (a) in unit vector notation and as (b) a magnitude and (c) a direction.

        7P. Three astronauts, propelled by jet backpacks, push and guide a 120 kg asteroid toward a processing dock, exerting the forces.

        8P. An overhead view of a 12 kg tire that is to be pulled by three ropes. One force , with magnitude 50 N) is indicated. Orient the other two forces so that the magnitude of the resulting acceleration of the tire is least, and find that magnitude.

        Some Particular Forces

        9£. (a) An 11.0 kg salami is supported by a cord that runs to a spring scale, which is supported.

        10E. A block with a weight of 3.0 N is at rest on a horizontal surface. A upward force is applied to the block by means of an attached vertical string. What are the magnitude and the direction of the force of the block on the horizontal surface.

        11. A certain particle has a weight of 22 N at a point . What are its (a) weight and (b) mass at a point ? What are its (c) weight and (d) mass if it is moved to a point in space.

        Applying Newton’s laws

        12E. When a nucleus captures a stray neutron. it must bring the neutron 10 a stop within the diameter of the nucleus by means of the strong force. That force, which “glues” the nucleus together. is approximately zero outside the nucleus. Suppose that a stray neutron with an initial speed is just barely captured by a nucleus with diameter d = 1.0 X 10-14 m. Assuming that the strong force on the neutron is constant. find the magnitude of that force.

        13e.A 50 kg passenger rides in an elevator that starts from rest on the ground floor of a building at I = 0 and rises to the top floor during a 10 s interval. The acceleration of the elevator as a function of the time is shown. where positive values of the acceleration mean that it is directed upward. Give the magnitude and direction of the following forces: (a) the maximum force on the passenger from the floor. (b) the’minimum force on the passenger from the floor. and (c) the maximum force on the floor from the passenger.

        14E. Sun jamming. A sun yacht is a spacecraft with a large sail that is pushed by sunlight. Although such a push is tiny in everyday circumstances, it can be large enough to send the spacecraft outward from the Sun on a cost-free but slow trip. Suppose that the spacecraft has a mass of 900 kg and receives a push.

        15E. The tension at which a fishing line snaps is commonly called the line’s “strength.What minimum strength is needed for a line that is to stop a salmon of weight 85 N in I I em if the fish is initially drifting. Assume a constant deceleration.

        16E. A car that weighs 1.30 X 104 N is initially moving at a speed of 40 km/h when the brakes are applied and the car is brought 10 a stop in 15 m.

        17E. An electron with a speed at 1.2 X 107 m/s moves horizontally into a region where a constant vertical force of 4.5 X 10-16 N acts on it. The mass of the electron is 9.1 I X 10-31 kg.

        18E. A car traveling at 53 km/h hits a bridge abutment. A passenger in the car moves forward a distance of 65 cm (with respect to the road) while being brought to rest by an inflated air bag.

        19E. Tarzan, who weighs 820 N, swings from a cliff at the end of a 20 m vine that hangs from a high tree limb and initially makes an angle of 220 with the vertical. Immediately after Tarzan steps off the cliff, the tension in the vine is 760 N.

        20P. A 50 kg skier is pulled up a friction less ski slope that makes an angle of 8.00 with the horizontal by holding onto a that moves parallel to the slope.

        21P. Two blocks are in contact on a friction less table. A horizontal force is applied to the larger block.

        22P. A 1400 kg jet engine is fastened to the fuselage of a passenger jet by just three bolts (this is the usual practice). Assume that each bolt supports one-third of the load.

        23P. An elevator and its load have a combined mass of 1600 kg. Find the tension in the supporting cable when the elevator, originally moving downward.

        24P. Figure 5-36 shows four penguins that are being playfully pulled along very slippery (friction less) ice by a curator.

        25P. An 80 kg person is parachuting and experiencing a downward acceleration. The mass of the parachute is 5.0 kg. ,(a) What is the upward force on the open parachute from the air?

        26P Three blocks are connected and pulled to the right on a horizontal friction less table by a force will a magnitude .

        27p. one of Jupiter’s moons. If the engine provides an upward force (thrust) of 3260 N, the craft descends at constant speed if the engine provides only 2200 N, the craft accelerates downward.

        28P. A worker drags a crate across a factory floor by pulling on a rope tied to the crate .The worker exerts a force of 450 N on the rope, which is inclined at 38° to the horizontal, and the floor exerts a horizontal force of 125 N that opposes the motion.

        29P. A motorcycle and 60.0 kg rider accelerate at up a ramp inclined 10° above the horizontal. (a) What is the magnitude of the net force acting on the rider? (b) What is the magnitude of the force on the rider from the motorcycle.

        30P. An 85 kg man lowers himself to the ground from a height of 10.0 m by holding onto a rope that runs over a friction less pulley to a 65 kg sandbag. With what speed does the man hit the ground if he started from rest.


        Want a free copy of PTC Mathcad Express? Download your copy today.

        We introduce new features of PTC Mathcad within the context of math and science problems. You will follow step-by-step instructions and actually try out the new concepts within Mathcad. You may want to simply read through the Tutorial, but spending a bit more time working through some of the exercises and problems will help you fully understand how to best use PTC Mathcad to solve your own problems.

        Worksheets marked with use PTC Mathcad premium features, otherwise they are Express compatible. All worksheets created in PTC Mathcad Prime 3.0.

        In this section, we will deal with the more complicated problem of analyzing two sets of data points that are related in some way. We will graph the data sets to look at the relationship, try out a theoretical model, and then summarize our results.

        ChapterWorksheet NameDescription
        Mathcad Basics Mathcad Basics - Tutorial In this chapter, you will learn to do simple calculations, write text, and organize your work into a coherent solution.

        Mathcad Basics - Exercises In this section, we will introduce another way of selecting regions, how to edit text, and where to find trigonometric functions.

        Mathcad Basics - Problems You are now ready to try some problems on your own. In all Problem sections, you will work out problems so that you can test your new abilities with Mathcad.
        Variables and Units Variables and Units - Tutorial In this chapter, you will learn Mathcad skills that will totally change the way you approach problems.

        Variables and Units - Exercises In the exercises that follow you will practice using variables instead of the values they represent.

        Variables and Units of Measurement You are now ready to try some problems on your own. In all Problem sections, you will work out problems so that you can test your new abilities with Mathcad.
        Solving Equations Solving Equations - Tutorial In this chapter, we will explore how Mathcad can be used to help you manipulate and solve equations. You will learn that Mathcad can solve an equation for any variable, substitute an expression into an equation, and even solve multiple equations for several variables simultaneously.

        Solving Equations - ExercisesIn this exercise section you will practice solving equations with the methods you learned in the Tutorial to solve complex problems involving more than one equation.

        Solving Equations - ProblemsAs you work on the problems in this section, try to use all the Mathcad features you have learned so far and practice the problem-solving strategies we have advised along the way.

        Solving Equations - Supplemental ExercisesIn this supplemental section there is an exercise to provide more practice with the skills you learned about in the Tutorial and the first half of the exercises in this section.

        Solving Equations - Supplemental ProblemsIn this supplemental section there is another problem that uses the symbolic equal sign and commands that you learned about in the Tutorial and the Exercises.
        Graphing Graphing - TutorialIn this Tutorial, you will learn how to graph a function in the context of an example.

        Graphing - ExercisesWith these Exercises, we will practice graphing functions, but you will also learn how to format your graphs. Specially, we will talk about adding markers to a graph, changing the appearance of a curve, and adding a legend or title.

        Graphs - ProblemsMaking graphs takes practice, so there are three problems in this section.
        Advanced Problem Solving Advanced Problem Solving - Tutorial In this chapter, we are not going to teach many new features of Mathcad. You have learned quite a bit so far and now we want to show you how Mathcad can be used to understand the concepts of a problem more completely.

        Advanced Problem Solving - ExercisesThe exercise that follows involves more difficult physics than we have covered in previous sections, and you will notice that the depth of our analysis increases as well.

        Advanced Problem Solving - Problems This section provides an opportunity to apply the new problem-solving skills you have learned.

        Advanced Problem Solving - Supplemental ExerciseIn this supplemental section there is an exercise to provide more practice with the skills you learned about.
        Data Analysis Data Analysis - TutorialIn this chapter we cover some tools that are used in all areas of science: the tools of data analysis.

        Data Analysis - Exercises

        Data Analysis - ProblemsThis section contains two closely related problems that deal with data analysis. You will need to use descriptive statistics, graph data, and use linear regression to fit a straight line to the data.
        Advanced Problem Solving Final Problem - The Spring ConstantIn this final section, you work through several problems that all deal with the same physical system: a mass hanging from a spring.

        h1
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