Solving Analytic Problems: 3 Difficulties & Solutions

[jenkirk]

Hey I am new to this site, i used to have george jones as a proff he was an amazing guy but now this new proff i don't really get, i am having trouble with proofs, i don't know how to do them well i have 3 problems I am having trouble with
given that the amplitude of a damped harmonic oscillator drops to 1/e of its initial value after n cycles, show the ratio of period of oscillation to the period of the same oscillator with no damping is,
T (damped)/T(undamped) = square root (1+ 1/(4pi squared n squared)

The second question is: the terminal speed of a freely falling ball is v. when the ball is supported by a light elastic spring the spring stretches by an amount x, show the natural frewuency of oscillation is:
w (omega) = (square root g/x) - g/(2v)

haha hey I am jen by the way

 

[Andrew Mason]

Hey I am new to this site, i used to have george jones as a proff he was an amazing guy but now this new proff i don't really get, i am having trouble with proofs, i don't know how to do them well i have 3 problems I am having trouble with
given that the amplitude of a damped harmonic oscillator drops to 1/e of its initial value after n cycles, show the ratio of period of oscillation to the period of the same oscillator with no damping is,
T (damped)/T(undamped) = square root (1+ 1/(4pi squared n squared)

What have you done? Can you write the expressions for $\omega$ for the damped oscillator ($m\ddot{x}+ b\dot{x} + kx = 0$) and $\omega_0$ for the undamped oscillator ($m\ddot{x} + kx = 0$).?

The second question is: the terminal speed of a freely falling ball is v. when the ball is supported by a light elastic spring the spring stretches by an amount x, show the natural frewuency of oscillation is:
w (omega) = (square root g/x) - g/(2v)

Can you determine the coefficients "b" and "k" in $m\ddot{x}+ b\dot{x} + kx = 0$?

AM

 

[jenkirk]

im trying

hello, i have attempted the problem many times hahha, i know how to prove that w = square root (g/x) that is the easy part and the original question was w = square root of (g/x - (g squared /(4vsquared))) so i figured w squared then is equal to g/x - g squared /(4vsquared) trying to work backwards and i know that they start with
-kx + mg= 0 and k= mg/x so w= square root of mg/xm or g/x
im thinking its somewhere in this equation that i should be able to configure the right answer but i don't know
and for the first problem we have never done anythin with amplitude and period before so i just don't know where to put in 1/e?

p.s. how do you put in those symbols?

 

[George Jones]

given that the amplitude of a damped harmonic oscillator drops to 1/e of its initial value after n cycles

What is the expression for the position of a particle undergoing damped harmonic motion?

What is the (time-dependent) amplitude of a damped harmonic oscllator?

how do you put in those symbols?

It's LaTeX. You can look at examples by clicking on the mathematics in any post that uses it. Also, the tread Introducing https://www.physicsforums.com/showthread.php?t=8997" is very useful.

It takes a while to get the hang of LaTeX, and it can be a pain to use, but it does produce nice results.