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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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