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And you posted equation 7.5.velvetmist said:In the equation 7.4, the author is taking v0=√(C/M)*x
Sorry, I made a typo, I posted the correct one.berkeman said:And you posted equation 7.5.
Yes.berkeman said:What is the total energy of a simple spring oscillator made up of?
Yes.berkeman said:This is for a spring oscillator, right?
It doesn't say much, cause it's trying to explain spring oscillators in a general way.berkeman said:Could you please define the terms in the equation you posted?
velvetmist said:v0=√(C/M)*x
If you are going to post typeset equations, it would be worthwhile to learn a little LaTeX. Info => HowTo => https://www.physicsforums.com/help/latexhelp/velvetmist said:If you do the integral you got: View attachment 231123. Then -M*v02=C*x2. So v0=√(-C/M)*x.
Edit: i just realized i forgot to put the minus.
jbriggs444 said:If you are going to post typeset equations, it would be worthwhile to learn a little LaTeX. Info => HowTo => https://www.physicsforums.com/help/latexhelp/
For example, $$\frac 1 2 Mv^2 - \frac 1 2 Mv_0^2$$
velvetmist said:If you do the integral you got:. Then -M*v02=C*x2. So v0=√(-C/M)*x.
velvetmist said:If you do the integral you got ##\frac{1}{2} Mv^2 - \frac{1}{2} Mv_0^2 =##.
velvetmist said:Then ##-M v^2_0 = C x^2##
velvetmist said:But we got that ## x = A \sin(\phi)##
(5.12) integral. And yes, i just realized that I cropted it in a wrong way, but basically: $$\Delta T = \frac 1 2 Mv^2-\frac 1 2 Mv_0^2.$$George Jones said:Do what integral? Also, this only half of an equation!
Cause we have thatGeorge Jones said:I don't see how you got this.
I'm just taking that from the textbook. I already posted that part but i will do it again if that helps u:George Jones said:Actually, ##x = A \sin \left( \omega_0 t + \phi \right)##.
velvetmist said:Cause we have that
$$\Delta T = \frac 1 2 Mv^2-\frac 1 2 Mv_0^2 = \frac 1 2 Mv^2-\frac 1 2 Cx^2,$$
velvetmist said:I'm just taking that from the textbook.
velvetmist said:But we got that ## x = A \sin(\phi)##
Sorry, sorry, I'm really sorry, i just made a lot of typos, I meantGeorge Jones said:I agree with
$$\Delta T = \frac 1 2 Mv^2-\frac 1 2 Mv_0^2 ,$$
but I do not understand the second equals sign, I do not understand why
$$\Delta T = \frac 1 2 Mv^2-\frac 1 2 Mv_0^2$$
is also equal to
$$\frac 1 2 Mv^2-\frac 1 2 Cx^2 .$$
Sorry, i just didn't read "at t=0", which is a huge mistake, but even like that i think is kind of weird cause he have thatGeorge Jones said:No, you have not taken it from the textbook. You wrote (and used)
velvetmist said:That's because
$$ \Delta (T+V) =\Delta E = \frac 1 2 Mv^2-\frac 1 2 Mv_0^2,$$
cause here ##\Delta V=0.##
Thank you so much! I finally get it, all of my questions were between pages 149 and 150. We will see conservation of energy in three classes, that's why I was so confused about that.George Jones said:No,
$$\Delta V = \frac{1}{2} C x^2 - \frac{1}{2} C x_0^2 .$$
See equation (5.20).
An oscillator is a physical system that exhibits periodic motion, meaning it repeats the same pattern in regular intervals. This can be seen in various systems such as a pendulum, a spring, or an electronic circuit.
Oscillators conserve energy through the principle of conservation of energy, which states that energy cannot be created or destroyed, only transformed from one form to another. In an oscillator, energy is constantly being transformed between kinetic energy (motion) and potential energy (stored energy) as it oscillates back and forth.
The period of an oscillator, or the time it takes to complete one full cycle, is affected by several factors including the mass, stiffness, and length of the oscillator. In general, a heavier mass or a stiffer oscillator will have a longer period, while a longer oscillator will have a shorter period.
In a perfect system, an oscillator would continue to oscillate forever without losing any energy. However, in real-world systems, there is always some friction or resistance present that will eventually cause the oscillator to lose energy and stop oscillating.
Oscillators have many practical uses in everyday life, such as in clocks, watches, and other timekeeping devices. They are also used in musical instruments, radio and television transmitters, and many other electronic devices. Additionally, the concept of oscillation is important in fields such as engineering, physics, and biology.