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Answer :
1. The spring constant of a spring is a measure of the spring's stiffness. It tells you how much force is required to produce a certain amount of displacement or compression.
The formula for spring constant is: [tex]k = F/x[/tex], where k is the spring constant, F is the applied force, and x is the displacement or compression.
In this problem, the spring constant is given as [tex]450 N/m[/tex] and the force applied is given as 80 lbf.
To find the displacement, we need to convert the force from pounds to Newtons: [tex]80 lbf = 356 N.[/tex]
Then we can use the formula to find the displacement: [tex]x = F/k = 356 N/450 N/m = 0.79 m.[/tex]
The fastest speed that the spring would experience is when it is at maximum compression or maximum extension, which occurs when the force is applied.
At this point, the spring is not moving, so its speed is zero. As the force is released, the spring will oscillate back and forth, but its maximum speed will occur when it reaches its equilibrium position.
2. To linearize the equation, we need to find the derivative of the equation:[tex]F = 420x + 300x³ + 45x⁵.[/tex]
Taking the derivative with respect to x gives: [tex]dF/dx = 420 + 900x² + 225x⁴.[/tex]
To find the spring constant, we need to evaluate this derivative at x = 0, which gives: [tex]dF/dx = 420.[/tex]
So the spring constant of the system when linearized is [tex]420 N/m.[/tex]
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