Classical Mechanics

Damped Oscillators

Damped Oscillators - Problem Solving

         

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

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

A body of mass m=2 kgm = 2 \text{ kg} is oscillating on a spring with spring constant k=4 N/m. k = 4 \text{ N/m}. There is a friction proportional to the body's velocity, represented by the function f=bv=4v (N) f = -bv = -4v \text{ (N)} . Initially, the position of the body was x(t=0)=5 m  x(t=0) = 5 \text{ m } and the velocity was v(t=0)=0 m/s , v(t=0) = 0 \text{ m/s }, where xx denotes the difference in the length of the spring from its original length. What is the velocity of the body when t=π2 s ? t = \frac{\pi}{2} \text{ s } ?

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

A body of mass m=6 kgm = 6 \text{ kg} is oscillating on a spring with spring constant k=12 N/m. k = 12 \text{ N/m}. There is a friction proportional to the body's velocity, represented by the function f=bv=12v (N). f = -bv = -12v \text{ (N)}. Initially, the position of the body was x(t=0)=4 m  x(t=0) = 4 \text{ m } and the velocity was v(t=0)=0 m/s , v(t=0) = 0 \text{ m/s }, where xx 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 s ? t = 5 \text{ s } ?

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

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