Damped and Forced Harmonic Motion

In summary: Newton's second law states that the net force on an object is equal to the net acceleration of the object. In this case, the net force is the negative of the acceleration. This means that ##-\dfrac{m}{x}## is equal to the negative of the acceleration.
  • #1
FeDeX_LaTeX
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Homework Statement


http://desmond.imageshack.us/Himg812/scaled.php?server=812&filename=quesq.jpg&res=medium


Homework Equations


F = ma, F = -kx, SHM equations

The Attempt at a Solution



Here's the diagram they've done for part (b).

http://desmond.imageshack.us/Himg844/scaled.php?server=844&filename=ansr.jpg&res=medium

I'm not understanding why the 6mkv force is shown acting upwards; surely if the string is going to move up, and the water is a resistive force, then it would act downwards?

And why is the acceleration downwards? I thought it would accelerate towards the point where it reaches (x+e) where the particle would be at maximum velocity, or am I thinking of springs?

Thanks.
 
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  • #2
Considering there's an obvious mistake in the equation given in part (b), I wouldn't put too much faith in there not being other obvious errors. You're right. For the situation described, the acceleration should be upward and the damping force, downward. Also, note that the figure shows x being the distance from the lower dotted line to some arbitrary depth below, but it should be from the dotted line to the location of P. The figure is a mess.
 
  • #3
Okay, thanks. I thought I was going to have a mental breakdown.
 
  • #4
I'm bumping this because the exam is getting closer and I'm still not sure about this. I don't think this is a mistake, and I'm perhaps just missing something crucial. Every question in the textbook has the resistive force acting in the direction of motion and acceleration (seemingly) pointing the opposite direction. Now I am confused.

-It's SHM, right? So acceleration acts towards the centre of motion which is UPWARDS.
-Resistive force should therefore be acting downwards.

Are the above two statements correct?

EDIT: I re-did the question, with acceleration upwards and resistive force downwards. Got a negative co-efficient for the 6k dx/dt (the m in their 'show that' question is an error, it should cancel). So the resistive force is causing the problem. Ugh. I can't get their equation no matter what I do.
 
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  • #5
You originally asked about the drawing, which is indeed a mess. The equation, however, is fine, other than the extra m. What does Newton's second law tell you ##m\ddot{x}## is equal to?
 

FAQ: Damped and Forced Harmonic Motion

What is the difference between damped and forced harmonic motion?

Damped harmonic motion occurs when a system experiences a resistive force that decreases its amplitude over time. Forced harmonic motion is when an external force is applied to a system, causing it to vibrate at a specific frequency.

How do you calculate the damping ratio in damped harmonic motion?

The damping ratio, denoted by the Greek letter zeta (ζ), can be calculated by dividing the damping coefficient by the critical damping coefficient. The damping coefficient is the ratio of the resistive force to the velocity, while the critical damping coefficient is the minimum amount of damping required for a system to return to equilibrium without oscillating.

What is resonance in forced harmonic motion?

Resonance occurs when an external force is applied to a system at its natural frequency, causing the amplitude of the system to increase significantly. This can result in large oscillations and potential damage to the system if not controlled.

How does damping affect the frequency of oscillation in a damped harmonic motion?

Damping decreases the frequency of oscillation in a damped harmonic motion. This is because the resistive force slows down the system, causing it to take longer to complete each cycle.

What are some real-life examples of damped and forced harmonic motion?

Damped harmonic motion can be observed in a swinging pendulum or a car's suspension system. Forced harmonic motion can be seen in musical instruments, such as a guitar string being plucked, or in bridges that are designed to withstand wind vibrations.

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