3. This problem is about driven oscillating systems. As we saw in lecture, when a mass-spring system is set in motion (given a tug or a kick) and subsequently left alone, it will oscillate at a frequency that depends on the spring constant and However, what if an additional sinusoidal force with arbitrary the mass, which is called the natural frequency, wo = frequency w is applied? a) Consider a horizontal mass-spring system with an additional horizontal driving force F = Focos (wt) where Fo is a constant. For now, assume wwo. Using Newton's second law, find a differential equation for the position of the mass. b) To solve the differential equation from part a), try a solution with the form x = A cos (wt) where A is a constant. What must A be to solve the differential equation? c) The solution given in part b) shows that the mass can oscillate at the frequency of the driving force instead of its natural frequency. However, notice that the solution in part b) has no arbitrary constants to take into account different possible initial conditions, so it can't be the most general solution. The most general solution is given by x = A cos (wt) + B sin (wot +90) where B and are constants and A is the same as what you determined in part b). Show that this function indeed satisfies the differential equation you found in part a). Notice we have effectively added to the function given in part b) the solution one would expect if there was no driving force at all. d) What are B and do if at t=0 the mass is located at equilibrium and at rest? e) What happens to A if w→wo? This is called resonance. At the resonance frequency, the driving force is able to add energy to the system very, very well. However, there's something not quite right about our modeling of the system, what do you think we might need to include to make the limit w→wo more realistic? (Hint: look at problem 4.)
3. This problem is about driven oscillating systems. As we saw in lecture, when a mass-spring system is set in motion (given a tug or a kick) and subsequently left alone, it will oscillate at a frequency that depends on the spring constant and However, what if an additional sinusoidal force with arbitrary the mass, which is called the natural frequency, wo = frequency w is applied? a) Consider a horizontal mass-spring system with an additional horizontal driving force F = Focos (wt) where Fo is a constant. For now, assume wwo. Using Newton's second law, find a differential equation for the position of the mass. b) To solve the differential equation from part a), try a solution with the form x = A cos (wt) where A is a constant. What must A be to solve the differential equation? c) The solution given in part b) shows that the mass can oscillate at the frequency of the driving force instead of its natural frequency. However, notice that the solution in part b) has no arbitrary constants to take into account different possible initial conditions, so it can't be the most general solution. The most general solution is given by x = A cos (wt) + B sin (wot +90) where B and are constants and A is the same as what you determined in part b). Show that this function indeed satisfies the differential equation you found in part a). Notice we have effectively added to the function given in part b) the solution one would expect if there was no driving force at all. d) What are B and do if at t=0 the mass is located at equilibrium and at rest? e) What happens to A if w→wo? This is called resonance. At the resonance frequency, the driving force is able to add energy to the system very, very well. However, there's something not quite right about our modeling of the system, what do you think we might need to include to make the limit w→wo more realistic? (Hint: look at problem 4.)
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Transcribed Image Text:3. This problem is about driven oscillating systems. As we saw in lecture, when a mass-spring system is set in motion
(given a tug or a kick) and subsequently left alone, it will oscillate at a frequency that depends on the spring constant and
the mass, which is called the natural frequency, wo = . However, what if an additional sinusoidal force with arbitrary
frequency w is applied?
a) Consider a horizontal mass-spring system with an additional horizontal driving force
F = Focos (wt)
where Fo is a constant. For now, assume wwo. Using Newton's second law, find a differential equation for the position
of the mass.
b) To solve the differential equation from part a), try a solution with the form
x = A cos (wt)
where A is a constant. What must A be to solve the differential equation?
c) The solution given in part b) shows that the mass can oscillate at the frequency of the driving force instead of its
natural frequency. However, notice that the solution in part b) has no arbitrary constants to take into account different
possible initial conditions, so it can't be the most general solution. The most general solution is given by
x = A cos (wt) + B sin (wot +90)
where B and do are constants and A is the same as what you determined in part b). Show that this function indeed
satisfies the differential equation you found in part a). Notice we have effectively added to the function given in part b)
the solution one would expect if there was no driving force at all.
d) What are B and do if at t=0 the mass is located at equilibrium and at rest?
e) What happens to A if w→wo? This is called resonance. At the resonance frequency, the driving force is able to add
energy to the system very, very well. However, there's something not quite right about our modeling of the system, what
do you think we might need to include to make the limit w→wo more realistic? (Hint: look at problem 4.)
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 = -buz
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:
z = Aet sin (Ct + o)
6²
where A, B, C, and do are all constants. You may assume the damping is small, which means
constants B and C be such that this 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.
215
<. What must the
m
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