5. (a) A vertical spring is stretched 2 meters downward after a mass of 4 kg is attached to it. Write a differential equation that describes the motion of the spring. The mass starts 1m below equilibrium and is given an initial velocity of 2m/s in the upward direction. Specifically, give an IVP for u(t), the distance of the mass from equilibrium. Assume there is no damping, nor external force, and that the acceleration due to gravity is g = 10m/s. (b) Is the spring-mass system described by the following equation underdamped, overdamped, or critically damped? u'' + 6u' + 25u = 0. (4) (c) Determine the value of m for which the following spring-mass system described by the following equation will undergo resonance. mu'' + 50u = 250 cos(5t). (5)
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We know that the force exerted by the spring is proportional to the displacement from equilibrium, and that the force exerted by gravity is proportional to the mass and the acceleration due to gravity. Therefore, we can write: F = -kx - mg where F is the net Show more…
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After a mass $m$ is attached to a spring, it stretches $s$ units and then hangs at rest in the equilibrium position as shown in FIGURE 1.3.18(b). After the spring/mass system has been set in motion, let $x(t)$ denote the directed distance of the mass beyond the equilibrium position. As indicated in Figure $1.3 .18(\mathrm{c})$, assume that the downward direction is positive, that the motion takes place in a vertical straight line through the center of gravity of the mass, and that the only forces acting on the system are the weight of the mass and the restoring force of the stretched spring. Use Hooke's law: The restoring force of a spring is proportional to its total elongation. Determine a differential equation for the displacement $x(t)$ at time $t$.
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A mass $m$ is attached to the end of a spring whose constant is $k$. After the mass reaches equilibrium, its support begins to oscillate vertically about a horizontal line $L$ according to a formula $h(t) .$ The value of $h$ represents the distance in feet measured from $L$ See FIGURE 3.8.21. (a) Determine the differential equation of motion if the entire system moves throughamedium offering a damping force numerically equal to $\beta(d x / d t)$. (b) Solve the differential equation in part (a) if the spring is stretched 4 feet by a weight of 16 pounds and $\beta=2$, $h(t)=5 \cos t, x(0)=x^{\prime}(0)=0$
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Note: Neglect the mass of the spring in all problems in this section. A $2.00$ -kg mass is attached to a spring and placed on a horizontal, smooth surface. A horizontal force of $20.0 \mathrm{~N}$ is required to hold the mass at rest when it is pulled $0.200 \mathrm{~m}$ from its equilibrium position (the origin of the $x$ axis). The mass is now released from rest with an initial displacement of $x_{i}=0.200 \mathrm{~m}$, and it subsequently undergoes simple harmonic oscillations. Find (a) the force constant of the spring, (b) the frequency of the oscillations, and (c) the maximum speed of the mass. Where does this maximum speed occur? (d) Find the maximum acceleration of the mass. Where does it oc- cur? (e) Find the total energy of the oscillating system. Find (f) the speed and (g) the acceleration when the displacement equals one third of the maximum value.
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