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(II) Obtain the displacement $x$ as a function of time for the simple harmonic oscillator using the conservation of energy, Eqs. $10 .[$Hint. Integrate Eq. 11 a with $v=d x / d t .]$$\begin{array}{rlr}{E} & {=\frac{1}{2} m(0)^{2}+\frac{1}{2} k A^{2}=\frac{1}{2} k A^{2}} \\ {E} & {=\frac{1}{2} m v^{2}+\frac{1}{2} k(0)^{2}=\frac{1}{2} m v_{\max }^{2}} \\ {E} & {=\frac{1}{2} m v^{2}+\frac{1}{2} k x^{2}}\end{array}$

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$$x=A \cos (\pm \omega t+\phi)$$

Physics 101 Mechanics

Chapter 14

Oscillators

Motion Along a Straight Line

Motion in 2d or 3d

Periodic Motion

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

University of Washington

Lectures

04:01

2D kinematics is the study of the movement of an object in two dimensions, usually in a Cartesian coordinate system. The study of the movement of an object in only one dimension is called 1D kinematics. The study of the movement of an object in three dimensions is called 3D kinematics.

02:18

In physics, an oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The oscillation may be periodic or aperiodic.

04:06

The displacement of a simp…

03:25

04:40

Given that the potential e…

01:00

Find the equation of motio…

03:22

The displacement of a forc…

09:46

(a) Using the potential en…

01:28

Solve the equation of mot…

so we know if we want to find the position as a function of time of a simple harmonic oscillator. But we must use the law of conservation of energy. So we can say that we can start off by saying velocity is gonna be equal to plus or minus the square root of the spring constant divided by the mass times a squared minus X squared. So that would simply be the of formula for the velocity. Now, this is gonna equal d x over g t. This is simply the definition of philosophy, a change in position over with respect to time. Therefore, D x divided by a squared minus X squared to the positive 1/2 power is going to be equal to one plus or minus the square root of K over m times DT. And at this point, we can actually integrate and say that, um, the integral from acts not that's with this The interviewer from X not to x of D. X over a squared minus X square to the 1/2 power. Uh, this would this is gonna equal plus or minus, um, this square root of the spring constant divided by the mass times the integral from zero to t of DT. And so we can say that this rather let's, uh we can actually evaluate at this point and say that negative co sign negative our coastline rather of X over a, um, evaluated at X not an ex is gonna be equal to plus or minus the square root of K over m times t and we find that weaken. Ah, express this. We can expand this rather and say negative are co sign of X over a plus Our co sign of ex not over a will be equal to plus or minus t times the square of the spring constant divided by the mass We know that the angular velocity is equal to the square root of the spring constant divided by the mass. And we know that the co sign of our coastline rather of ex not divided by X divided by the amplitude, would be equal to five. The face constant. Therefore, we can say that negative co sign of a negative are co sign of X over a plus five. The face constant is gonna be equal to plus or minus the angular velocity times the time and rearranging and isolating X X is going to be equal a co sign of plus or minus omega T. So the angular velocity times time plus the face constant. And this is the general formula for a simple harmonic oscillator. That is the end of the solution. Thank you for watching.

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