Final answer:
The damping constant is -1.6 N·s/m and it takes approximately 0.337 s for the displacement to be 10 cm.
Step-by-step explanation:
To determine the damping constant, we need to know the equation governing the motion of the body. Assuming the body is undergoing simple harmonic motion, the equation can be written as:
m \\frac{d^2x}{dt^2} + b \\frac{dx}{dt} + kx = 0
Where m is the mass of the body, b is the damping constant, k is the spring constant, and x is the displacement of the body from equilibrium.
Since the body is in equilibrium when it reaches the speed limit, the net force acting on it must be zero. Therefore, we can write:
F_net = -kx - bv = 0
Where v is the velocity of the body at the speed limit. Rearranging the equation, we get:
b = -\\frac{kx}{v}
Substituting the given values, we have:
b = -\\frac{400\\,N/m\\times0.2\\,m}{25\\,m/s} = -1.6\\,N\\cdot s/m
Therefore, the damping constant is -1.6 N·s/m.
To find the time it takes for the displacement to be 10 cm, we can use the equation for the displacement of a body undergoing simple harmonic motion:
x(t) = A\\cos(\\omega t)
Where x(t) is the displacement at time t, A is the amplitude of the oscillation, and \\omega is the angular frequency given by:
\\omega = \\sqrt{\\frac{k}{m}}
To find the time when the displacement is 10 cm, we can set x(t) equal to 10 cm and solve for t:
0.1 = 0.2\\cos(\\sqrt{\\frac{400}{3\\times3}}\\times t)
Simplifying the equation, we get:
\\cos(\\sqrt{\\frac{400}{9}}\\times t) = \\frac{1}{3}
Using inverse cosine function, we find that:
\\sqrt{\\frac{400}{9}}\\times t = \\cos^{-1}(\\frac{1}{3})
Solving for t, we have:
t = \\frac{\\cos^{-1}(\\frac{1}{3})}{\\sqrt{\\frac{400}{9}}}
Approximating the value, we find that t ≈ 0.337 s.