Equation for displacement in damped harmonic motion.

• Craptola
In summary, the equation for displacement in damped harmonic oscillations is x=C\mathrm{exp}(-\frac{b}{2m}t)\cdot\mathrm{exp}(\pm \sqrt{\frac{b^{2}}{2m}-\frac{k}{m}}t), where C is a constant depending on the initial conditions of the system. To determine the value of C, you can use a known value of x and t and solve for C. This constant is known as the constant of integration.
Craptola
This is not really a homework problem but rather a question about an equation for displacement in damped harmonic oscillations that I've come across during revision for midterms. In my notes and in various textbooks the equation is given as $$x=C\mathrm{exp}(-\frac{b}{2m}t)\cdot\mathrm{exp}(\pm \sqrt{\frac{b^{2}}{2m}-\frac{k}{m}}t)$$

I've been told that C is a constant depending on the initial conditions of the system, but I'm not sure how to go about determining the value of this constant.
Any help on this matter would be greatly appreciated.

You use a value for x that you know for some value of t. Usually ##t=0## and ##x=x_0## - you know, how far you pulled the pendulum back before you let it go?
You put the numbers into the equation and solve for C. It's a constant of integration.

1. What is the equation for displacement in damped harmonic motion?

The equation for displacement in damped harmonic motion is given by x(t) = Ae^(-ζωt)cos(ω't + φ), where x(t) is the displacement at time t, A is the initial amplitude, ζ is the damping ratio, ω is the natural frequency, ω' is the damped frequency, and φ is the phase angle.

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

The damping ratio, ζ, can be calculated by dividing the damping coefficient by the critical damping coefficient (the value at which the system is critically damped). It can also be found by dividing the actual damping coefficient by the theoretical damping coefficient (the value at which the system would be undamped).

3. What factors affect the displacement in damped harmonic motion?

The displacement in damped harmonic motion is affected by the initial amplitude, the damping ratio, the natural frequency, and the phase angle. Changes in these factors can result in different displacement values over time.

4. How does damping affect the motion in damped harmonic motion?

Damping affects the motion in damped harmonic motion by reducing the amplitude over time. This is due to the dissipation of energy caused by the damping force. As the damping ratio increases, the motion becomes more overdamped and the amplitude decreases faster.

5. Can the displacement in damped harmonic motion ever reach zero?

Yes, the displacement in damped harmonic motion can reach zero, but it may take an infinite amount of time. In an ideal damped harmonic oscillator, the displacement will decrease exponentially and eventually become zero. However, in real systems, there will always be some residual damping that prevents the displacement from reaching exactly zero.

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