A mass weighing 16 pounds stretches a spring feet. The mass is initially released from rest from a point 3 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2/1/2 the instantaneous velocity. Find the equation of motion x(t) if the mass is driven by an external force equal to f(t) = 10 cos(3t). (Use g = 32 ft/s2 for the acceleration due to gravity.)

A mass weighing 16 pounds stretches a spring feet. The mass is initially released from rest from a point 3 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2/1/2 the instantaneous velocity. Find the equation of motion x(t) if the mass is driven by an external force equal to f(t) = 10 cos(3t). (Use g = 32 ft/s2 for the acceleration due to gravity.)

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Similar questions

A solid cylinder of mass M, radius R, and length L, rolls around a light, thin axle at its center. The axle is attached to a spring of stiffness K on either side. The springs are attached to same wall. The cylinder is brought back a distance A, parallel to the wall, and allowed to oscillate. What is the maximum speed of the oscillations of the cylinder? Answer: ((3*k*A^2)/(2*M))^1/2
Show that the function x(t) = A cos ω1t oscillates with a frequency ν = ω1/2π. What is the frequency of oscillation of the square of this function, y(t) = [A cos ω1t]2? Show that y(t) can also be written as y(t) = B cos ω2t + C and find the constants B, C, and ω2 in terms of A and ω1
Two blocks of masses m1=1.0 kg and m2=3 kg are connected by an ideal spring of force constant k=4 N/m and relaxed length L. If we make them oscillate horizontally on a frictionless surface, releasing them from rest after stretching the spring, what will be the angular frequency ω of the oscillation? Choose the closest option. Hint: Find the differential equation for spring deformation.
A box of mass 0.850 kg is attached to a spring with k = 145 N/m and set into simple harmonic motion on a frictionless, horizontal table. The amplitude of motion is 5.10 cm. (Use the exact values you enter in previous answer(s) to make later calculation(s).) (a) What is the total energy of the box–spring system? ______ J (b) What is the speed of the box when the spring is compressed by 2.00 cm? (Give your answer to at least two decimal places.) ______m/s (c) What is the kinetic energy of the box at this position? ______ J (d) What is the potential energy of the box–spring system at this position? ______ J
A 3 kilogram weight stretches a spring by 8 cm. The mass-spring system is immersed in a damping fluid which exerts a force of 0.5 N when the mass is traveling 0.25 m/sec. Write a differential equation for u(t), the displacement of the mass at time t. (use g = 9.8 m/sec ^2 .)
The equation of motion for a damped harmonic oscillator is s(t) = Ae^(−kt) sin(ωt + δ),where A, k, ω, δ are constants. (This represents, for example, the position of springrelative to its rest position if it is restricted from freely oscillating as it normally would).(a) Find the velocity of the oscillator at any time t.(b) At what time(s) is the oscillator stopped?
You exert a 183 N push the leftmost of two identical blocks of mass 262 g connected by a spring of stiffness 570 kg/s?. After pushing the block a distance 15 cm, you release it; by this time the rightmost block has moved a distance 5 cm. (a) What is the energy in the oscillations between the blocks? (b) What is the speed of the center of mass after the push? (c) What is the initial relative speed between the blocks? (d) What is the amplitude of the oscillation? (e) What are the fastest as slowest linear speeds of the blocks? Fastest: Slowest:
A ball whose mass is 1.4 kg is suspended from a spring whose stiffness is 40N/m. The ball oscillates up and down with an amplitude of 14 cm. Suppose this apparatus were taken to the Moon, where the strength of the gravitational field is only 1/6 of that on Earth. What would be the period on the Moon? (Consider carefully how the period depends on properties of the system; look at the equation.)
The equation of motion of an MSQ object of mass m attached with a spring of spring constant K is oscillating in medium of damping constant b is written as m(d²x)/dt2 +b dx/dt+Kx=D0 if wo is the 35 natural angular frequency of the free oscillator thus which of the following are true? (a) The critical damping occurs when b = v2mK (b) Non-oscillatory, a periodic motion occurs when b >2mwo (c) Oscillations about equilibrium with exponentially decaying amplitude occurs when b<2mwo an (d) Non-oscillatory equilibriums occurs when b = 2vmK and faster return to a b C
A mass-spring motion is governed by the ordinary differential equation d²x dx +b + k(t)x= F(t), dt² dt m where m is the mass, b is the damping constant, k is the spring constant, and F(t) is the external force. We consider the initial conditions x(0) = 1 and x/(0) = 0. Assume the following numerical values for this part of the project: m = 1, k = 1/5, b= 1/5, and F(t) = cos yt. 1 (a) Read section 4.10. Explain what is the resonance frequency, and then compute the resonance frequency for this mass-spring system. (b) The ODE45-solver can be used to obtain the solution curves in MATLAB. Use the script Project2_Q2.m to plot the solutions and estimate the amplitude A of the steady response for y = 0.2, 0.42, 0.6, and 0.8. (c) The script also provide you with the graph of A versus y. For what frequency 7 is the amplitude the greatest? Is it equal to that you obtained in (a)?
A 4 ft spring measures 8 ft long after it is hung with a mass weighing 8 lb. The medium through which the mass moves provides damping force equal to √2 times the instantaneous speed. Find the motion equation if the mass is initially released from the equilibrium position with a descending velocity of 5 f t/s. Calculate the time the mass reaches its extreme displacement from the equilibrium position. What is the position of the dough in that instant
A 4-foot spring measures 8 feet long after a mass weighing 8 pounds is attached to it. The medium through which the mass moves offers a damping force numerically equal to √2 times the instantaneous velocity. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of 3 ft/s. (Use g = 32 ft/s² for the acceleration due to gravity.) x(t) = Find the time at which the mass attains its extreme displacement from the equilibrium position. t= X x What is the position of the mass at this instant? The extreme displacement is x = 0.39 X feet.