4. We can also have damped oscillations. Consider a horizontal mass-spring system in which there is significant friction proportional to the velocity of the mass: f = -bvx where b is a constant. This force provides the damping, or a loss of mechanical energy to thermal energy. a) Using Newton's second law find a differential equation for the position of the mass. b) To solve the differential equation, try a solution with this form: x = AeBt sin (Ct + po) where A, B, C, and do are all constants. You may assume the damping is small, which means <. What must the constants B and C be such that this is a solution? c) Plot the position as a function of time. What happens to the amplitude of the oscillations as time goes on? We won't go into more detail here, but if the damping is small (as in the case above) we call the system underdamped. However, we can also have overdamped and critically damped systems, in which damping is so large (²2) the system exponentially decays instead of oscillating.
4. We can also have damped oscillations. Consider a horizontal mass-spring system in which there is significant friction proportional to the velocity of the mass: f = -bvx where b is a constant. This force provides the damping, or a loss of mechanical energy to thermal energy. a) Using Newton's second law find a differential equation for the position of the mass. b) To solve the differential equation, try a solution with this form: x = AeBt sin (Ct + po) where A, B, C, and do are all constants. You may assume the damping is small, which means <. What must the constants B and C be such that this is a solution? c) Plot the position as a function of time. What happens to the amplitude of the oscillations as time goes on? We won't go into more detail here, but if the damping is small (as in the case above) we call the system underdamped. However, we can also have overdamped and critically damped systems, in which damping is so large (²2) the system exponentially decays instead of oscillating.
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Transcribed Image Text:4. We can also have damped oscillations. Consider a horizontal mass-spring system in which there is significant friction
proportional to the velocity of the mass:
f = -bvx
where b is a constant. This force provides the damping, or a loss of mechanical energy to thermal energy.
a) Using Newton's second law find a differential equation for the position of the mass.
b) To solve the differential equation, try a solution with this form:
x = Ae Bt sin (Ct + o)
b²
k
4m² m 2°
where A, B, C, and do are all constants. You may assume the damping is small, which means < What must the
constants B and C be such that this is a solution?
c) Plot the position as a function of time. What happens to the amplitude of the oscillations as time goes on?
We won't go into more detail here, but if the damping is small (as in the case above) we call the system underdamped.
However, we can also have overdamped and critically damped systems, in which damping is so large (≥) the system
exponentially decays instead of oscillating.
6²
4m²
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