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# Damped Oscillators - Problem Solving

A body of mass $$m = 1 \text{ kg}$$ is oscillating on a spring with spring constant $$k = 2 \text{ N/m}.$$ There is a friction proportional to the body's velocity, represented by the function $$f = -bv = -2v \text{ (N)}$$. Initially, the position of the body was $$x(t=0) = 2 \text{ m }$$ and the velocity was $$v(t=0) = 0 \text{ m/s },$$ where $$x$$ denotes the difference in the length of the spring from its original length. What is the amplitude of the body when $$t = 2 \text{ s } ?$$

Assume that $$\sqrt{2} = 1.5$$

A body of mass $$m = 2 \text{ kg}$$ is oscillating on a spring with spring constant $$k = 4 \text{ N/m}.$$ There is a friction proportional to the body's velocity, represented by the function $$f = -bv = -4v \text{ (N)}$$. Initially, the position of the body was $$x(t=0) = 5 \text{ m }$$ and the velocity was $$v(t=0) = 0 \text{ m/s },$$ where $$x$$ denotes the difference in the length of the spring from its original length. What is the velocity of the body when $$t = \frac{\pi}{2} \text{ s } ?$$

A body of mass $$m = 3 \text{ kg}$$ is oscillating on a spring with spring constant $$k = 3 \text{ N/m}.$$ There is a friction proportional to the body's velocity, represented by the function $$f = -bv = -6v \text{ (N)}.$$ Initially, the position of the body was $$x(t=0) = 1 \text{ m }$$ and the velocity was $$v(t=0) = 0 \text{ m/s },$$ where $$x$$ denotes the difference in the length of the spring from its original length. What is the velocity of the body when $$t = 1 \text{ s } ?$$

A body of mass $$m = 6 \text{ kg}$$ is oscillating on a spring with spring constant $$k = 12 \text{ N/m}.$$ There is a friction proportional to the body's velocity, represented by the function $$f = -bv = -12v \text{ (N)}.$$ Initially, the position of the body was $$x(t=0) = 4 \text{ m }$$ and the velocity was $$v(t=0) = 0 \text{ m/s },$$ where $$x$$ denotes the difference in the length of the spring from its original length. What is the ratio of the amplitude to the initial amplitude when $$t = 5 \text{ s } ?$$

A body of mass $$m = 3 \text{ kg}$$ is oscillating on a spring with spring constant $$k = 3 \text{ N/m}.$$ There is a friction proportional to the body's velocity, represented by the function $$f = -bv = -6v \text{ (N)}$$. Initially, the position of the body was $$x(t=0) = 3 \text{ m }$$ and the velocity was $$v(t=0) = 0 \text{ m/s },$$ where $$x$$ denotes the difference in the length of the spring from its original length. What is the position of the body when $$t = 3 \text{ s } ?$$

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