For Problems 4-10, determine the motion of the spring-mass system governed by the given initial-value problem. In each case, state whether the motion is underdamped, critically damped, or overdamped, and make a sketch depicting the motion.
Want to see the full answer?
Check out a sample textbook solutionChapter 8 Solutions
EBK DIFFERENTIAL EQUATIONS AND LINEAR A
- Solve for y. V – yz = xy +xzarrow_forwardA hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring feet. The ball is started in motion from the equilibrium position with a downward velocity of 3 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second). Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t. (Note that the positive direction is down.) Take as the gravitational acceleration 32 feet per second per second. y = Hint: e^(-16t)(((1/(2sqrt(2))*e^(8sqrt(2)t)-(1/(2sqrt(2))*e^(-8sqrt(2)t))arrow_forwardA linear second-order non-homogeneous equation models this scenario: People falling at a height of 100ft above the ground attached to a 100-foot rope into a pit that's 25 ft deep and 75 feet from ground level. Spring constant of the rope is 120 lbs/ft, and air resistance is 5 times the instantaneous velocity. M is mass of person and g is gravity. Note that the pit is actually 100 ft in height: 75 from ground level plus 25 ft deep. Write this scenario as an initial value problem in matrix form. The equation is my''+5y'+120y=mgarrow_forward
- The acceleration of a moving particle at any time t is given by expression 6t – 24. Find the position equation and velocity equation if when t = 4, v = –3;and, when t = 1,S = 34.arrow_forwardA linear second-order non-homogeneous equation models this scenario: People falling at a height of 100ft above the ground attached to a 100-foot rope into a pit that's cut off at 75 feet underground. Spring constant of the rope is 120 lbs/ft, and air resistance is 5 times the instantaneous velocity. M is mass of person and g is gravity. Note that the pit is actually 100 ft deep, it's just cut off 25 ft from the bottom to make the pit 75 feet deep. What are the initial conditions of the height y(t) of the person falling at time t? The equation is my''+5y'+120y=mgarrow_forwardA hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring feet. The ball is started in motion from the equilibrium position with a downward velocity of 8 feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second) Suppose that aftert seconds the ball is y feet below its rest position. Find y in terms of t. (Note that the positive direction is down.) Take as the gravitational acceleration 32 feet per second per second. y =arrow_forward
- Solve S. a (2Ja-y-) dy aarrow_forwardThe gas equation for one mole of oxygen relates its pressure, P (in atmospheres), its temperature, T (in K), and its volume, V (in cubic decimeters, dm³): dT T 16.574. = 1 V 0.52754. 1 V2 (a) Find the temperature T and differential dT if the volume is 34 dm³ and the pressure is 0.75 atmosphere. T = 0.3879 P + 12.187 V P. (b) Use your answer to part (a) to estimate how much the pressure would have to change if the volume increased by 2.5 dm³ and the temperature remained constant. change in pressure =arrow_forwardSolve (20) x2y′′+xy′+y=0.arrow_forward
- 4. The equation of motion of a particle is s = t* – 2t3 + t2 – t, where s is in meters and t is in seconds. a) Find the velocity and acceleration as functions of t. b) Find the position, velocity, and acceleration after 1 second.arrow_forward(2 – t²) e&-t° dt еarrow_forwardAn engineer wants to determine the spring constant for a particular spring. She hangs various weights on one end of the spring and measures the length of the spring each time. A scatterplot of length (y) versus load (x) is depicted in the following figure. Inad a Is the model y = P, +B, x an empirical model or a physical law? b. Should she transform the variables to try to make the relationship more linear, or would it be better to redo the experiment? Explain.arrow_forward
- Algebra and Trigonometry (6th Edition)AlgebraISBN:9780134463216Author:Robert F. BlitzerPublisher:PEARSONContemporary Abstract AlgebraAlgebraISBN:9781305657960Author:Joseph GallianPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
- Algebra And Trigonometry (11th Edition)AlgebraISBN:9780135163078Author:Michael SullivanPublisher:PEARSONIntroduction to Linear Algebra, Fifth EditionAlgebraISBN:9780980232776Author:Gilbert StrangPublisher:Wellesley-Cambridge PressCollege Algebra (Collegiate Math)AlgebraISBN:9780077836344Author:Julie Miller, Donna GerkenPublisher:McGraw-Hill Education