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Fundamentals of Differential Equations and Boundary Value Problems
- A force of 13 pounds stretches a spring 1 foot. A mass weighing 6.4 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 1.6 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. (b) Express the equation of motion in the form (c) Find the first time at which the mass passes through the equilibrium position heading upward. (Round your answer to three decimal places.)arrow_forwardA mass weighing 16 pounds stretches a spring feet. The mass is initially released from rest from a point 2 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that 31/01 the instantaneous velocity. Find the equation of motion x(t) if the mass is driven by an external force equal to f(t) = 20 cos(3t). (Use g = 32 ft/s² for the acceleration due to gravity.) numerically equal to x(t) = e e-( ³ ) ( - 4 + cos( √4² +) - 134 -sin (V+T ;)) + 20-sin (3r) + cos (3r) x ftarrow_forwardA 1/8-kg mass is attached to a spring with stiffness 16N/m. The damping constant for the system is 2N-sec/m. If the mass is moved 3/4 to the left of the equilibrium and gives an initial leftward velocity of 2m/sec, determine the equation of motion of the mass and give its damping factor, quasiperiod and quasifrequency?arrow_forward
- An object of weight 10 kg is attached to the free end of a spring and thereby comes to its equilibrium position by stretching the spring 62.5 cm. The object is pulled down 50 cm from the equilibrium position and released at time t = 0. Assuming the damping constant is 8 and no external driving forces, determine object's motion with g = 10 m/s².arrow_forwardConsider a spring mass system with a 144 lb object attached. Suppose the object stretches the spring 18 inches in equilibrium. If the object is initially displaced 8 inches above equilibrium and given an initial velocity of -2 ft/s, find its displacement y in feet as a function of time t. Assume that this motion is undamped and that the spring is not deformed in the process. y(t) = feetarrow_forwardA force of 5 pounds stretches a spring 1 foot. A mass weighing 6.4 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 1.2 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. x(t) = ft (b) Express the equation of motion in the form x(t) = Ae-At sin Vw2 – 2?t + p P), which is given in (23) of Section 3.8. (Round o to two decimal places.) x(t) = ft (c) Find the first time at which the mass passes through the equilibrium position heading upward. (Round your answer to three decimal places.)arrow_forward
- A mass of 3 kg is suspended from a spring whose constant is 2 Nt/cm. The system is then immersed in a medium that imparts a damping force equal to 1.2 times the instantaneous velocity. The mass is released from 5 cm above the equilibrium positon with downward velocity 50 cm/sec. Determine the position of the mass at any time t.arrow_forwardWhen a 5 kg mass is attached to a spring whose constant is 80 N/m, it comes to rest in the equilibrium position. Starting at t=0, a force equal to f(t) = 35e-5t cos 7t is applied to the system. In the absence of damping, (a) find the position of the mass when t = π. (b) what is the amplitude of vibrations after a very long time?arrow_forwardA force of 4 pounds stretches a spring 1 foot. A mass weighing 3.2 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 0.4 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. x(t) = ft (b) Express the equation of motion in the form x(t) = Ae-¹t sin(√w² - 2²t + p), which is given in (23) of Section 3.8. (Round up to two decimal places.) x(t) = ft (c) Find the first time at which the mass passes through the equilibrium position heading upward. (Round your answer to three decimal places.) Sarrow_forward
- A mass of 4 kg stretches a spring 16 cm. The mass is acted on by an external force of 2 sin(4) N (newtons) and moves in a medium that imparts a viscous force of 6 N when the speed of the mass is 2 cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 4 cm/s, formulate the initial value problem describing the motion of the mass. Assume that g = 9.8 m 4u" + 6 + 245u = 2sin( 4u" + 300u +245u= 2sin( O4u" + 6 +2.45u = 2sin( 4u" + 300u +245u= 2sin( (2), u(0) = 0. u'(0) = 0.04, uin meters. sin().u(0) = 0, u' (0) = 0.04, uin meters. sin().u(0) = 0. (0) = 0.04, uin meters. n(), u(0) = 0.04, u'(0) = 0, win meters. 4u" + 300u' +2.45u = 2sin().u(0) = 0, '(0) 0.04, win meters.arrow_forwardA mass of 6 kg stretches a spring 24 cm. The mass is acted on by an external force of 5 sin() N (newtons) and moves in a medium that imparts a viscous force of 12 N when the speed of the mass is 4 cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 10 cm/s, formulate the initial value problem describing the motion of the mass. Assume that g = 9.8 6" + 12 +2.45u = 5sin( 6 +300+245u=5sin( 6u" + 300u' +245u = 5sin(). 6" + 300 +2.45u = 5sin(). 6 + 12 (0) = 0, / (0) = 0.10, in meters. (0) = 0.10, (0) = 0, uin meters. (0) = 0, (0) = 0.10, uin meters. (0) = 0. (0) = 0.10, win meters. +245u = 5sin(). (0) = 0, 1/(0) = 0.10, uin meters. n(). 10arrow_forwardA piece of chicken is boiled until its internal temperature reaches 165 °F. The chicken is then transferred to a refrigerator held at 49 °F to cool down. The internal temperature of the chicken at time t is given by T(t), and the temperature obeys Newton's law dT = -k(T - Tm) where Tm is the temperature of the refrigerator and k > 0 is a constant. After 5 minutes in the refrigerator, the internal temperature of the chicken is 155 °F. How long will it take for the temperature of the chicken to reach 70 °F?arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning