A mass attached to a spring hanging straight down will experience a force mg due to gravity. In this case, the equation for the vertical motion is my′′ + γy′ + ky = mg (Note that y is pointing down.) To find the general solution, find the null + particular solution. A particular solution is yp(t) = mg/k. Why? (a) Determine the equilibrium position. How does it vary with mass and spring stiffness? (b) Solve the equation with m = 3,γ = 0,k = 2,g = 10 and starting values y(0) = 0,y′(0) = 0. Sketch the solution and relate it to the hanging mass. (c) What is the amplitude of the motion if the initial velocity is changed to y′(0) = 2?
A mass attached to a spring hanging straight down will experience a force mg due to gravity. In this case, the equation for the vertical motion is my′′ + γy′ + ky = mg (Note that y is pointing down.) To find the general solution, find the null + particular solution. A particular solution is yp(t) = mg/k. Why? (a) Determine the equilibrium position. How does it vary with mass and spring stiffness? (b) Solve the equation with m = 3,γ = 0,k = 2,g = 10 and starting values y(0) = 0,y′(0) = 0. Sketch the solution and relate it to the hanging mass. (c) What is the amplitude of the motion if the initial velocity is changed to y′(0) = 2?
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A mass attached to a spring hanging straight down will experience a force mg due to gravity. In this case, the equation for the vertical motion is
my′′ + γy′ + ky = mg
(Note that y is pointing down.) To find the general solution, find the null + particularsolution. A particular solution is yp(t) = mg/k. Why?
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(a) Determine the equilibrium position. How does it vary with mass and spring stiffness?
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(b) Solve the equation with m = 3,γ = 0,k = 2,g = 10 and starting values y(0) = 0,y′(0) = 0. Sketch the solution and relate it to the hanging mass.
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(c) What is the amplitude of the motion if the initial velocity is changed to y′(0) = 2?
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Step 1: Determine the given variable:
VIEWStep 2: Determine the general solution:
VIEWStep 3: 1. a) Calculation of equilibrium position:
VIEWStep 4: 2. (b) Solve the equation using given variable and initial conditions:
VIEWStep 5: Draw the graph of equation of motion and relating wit the hanging mass:
VIEWStep 6: 3 (c). Determine the amplitude of the motion if the initial velocity is changed to y′(0) = 2:
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