$V=\frac{1}{2}k{x}^{2}$

I want to calculate the value $x(t)$ of a particle moving in this potential, with initial conditions $x(t=0)=0$ and $v(t=0)={v}_{0}$. How would I go about doing this in simple classical mechanics?

misyjny76
2022-09-23
Answered

I have been given the potential of a simple harmonic oscillator:

$V=\frac{1}{2}k{x}^{2}$

I want to calculate the value $x(t)$ of a particle moving in this potential, with initial conditions $x(t=0)=0$ and $v(t=0)={v}_{0}$. How would I go about doing this in simple classical mechanics?

$V=\frac{1}{2}k{x}^{2}$

I want to calculate the value $x(t)$ of a particle moving in this potential, with initial conditions $x(t=0)=0$ and $v(t=0)={v}_{0}$. How would I go about doing this in simple classical mechanics?

You can still ask an expert for help

LilsGroolonip86

Answered 2022-09-24
Author has **9** answers

Solving the equation:

You can guess the solution $x=A\mathrm{cos}\sqrt{k/m}t+B\mathrm{sin}\sqrt{k/m}t$ or you can use substitution $x={e}^{\lambda t}$ and find $\lambda $

Anyway:

$x(t=0)=A\mathrm{cos}0+B\mathrm{sin}0=A=0$

$v(t=0)=-A\sqrt{k/m}\mathrm{sin}0+B\sqrt{k/m}\mathrm{cos}0=B\sqrt{k/m}={v}_{0}$

therefore:

$x(t)=\sqrt{\frac{m}{k}}{v}_{0}\mathrm{sin}\left(\sqrt{\frac{k}{m}}t\right)$

You can guess the solution $x=A\mathrm{cos}\sqrt{k/m}t+B\mathrm{sin}\sqrt{k/m}t$ or you can use substitution $x={e}^{\lambda t}$ and find $\lambda $

Anyway:

$x(t=0)=A\mathrm{cos}0+B\mathrm{sin}0=A=0$

$v(t=0)=-A\sqrt{k/m}\mathrm{sin}0+B\sqrt{k/m}\mathrm{cos}0=B\sqrt{k/m}={v}_{0}$

therefore:

$x(t)=\sqrt{\frac{m}{k}}{v}_{0}\mathrm{sin}\left(\sqrt{\frac{k}{m}}t\right)$

batejavizb

Answered 2022-09-25
Author has **4** answers

With given potential you can define force acting on your oscillator as

$F=-{\displaystyle \frac{dV}{dx}}=-kx$

This gives you Hooks law (no surprise). Than just use Newton's second law:

$ma=-kx$

Than create second order differential equation replacing a with $\ddot{x}$ and solve it with your initial conditions.

$F=-{\displaystyle \frac{dV}{dx}}=-kx$

This gives you Hooks law (no surprise). Than just use Newton's second law:

$ma=-kx$

Than create second order differential equation replacing a with $\ddot{x}$ and solve it with your initial conditions.

asked 2022-11-07

Is newton's second law a consequence of the principle of conservation of energy? How can we arrive at

net force = rate of change of momentum

using only the law of conservation of energy?

net force = rate of change of momentum

using only the law of conservation of energy?

asked 2022-09-30

Mass = m , momentum is $p=mv$. In time $\mathrm{\Delta}t$, momentum changes by $\mathrm{\Delta}p$, the rate of change of momentum is:

$\frac{\mathrm{\Delta}p}{\mathrm{\Delta}t}=\frac{\mathrm{\Delta}(mv)}{t}=m\frac{\mathrm{\Delta}v}{\mathrm{\Delta}t}$

$\frac{\mathrm{\Delta}p}{\mathrm{\Delta}t}=\frac{\mathrm{\Delta}(mv)}{t}=m\frac{\mathrm{\Delta}v}{\mathrm{\Delta}t}$

asked 2022-07-26

Two packing crates of masses 10.0 kg and 5.00 kg are connected by a light string that passes over a frictionless pulley. The 5.00-kg crate lies on a smooth incline of angle 40.0?. Find the acceleration of the 5.00-kg crate and the tension in the string.

asked 2022-07-26

asked 2022-09-06

What law is being applied when you're riding a bike then suddenly you hit a large rock.The bike stops moving but you fly over the handlebars?A. Law of Conservation of Momentum

B. Newton's First Law of Motion

C. Newton's Second Law

D. Newton's Third law

B. Newton's First Law of Motion

C. Newton's Second Law

D. Newton's Third law

asked 2022-09-29

What happens when velocities comparable to the speed of light are involved in an observation?

a. Newton’s second law of motion, F = ma , governs the motion of the object.

b. Newton’s second law of motion, F = ma , no longer governs the dynamics of the object.

c. Such velocities cannot be determined mathematically.

d. None of the above

a. Newton’s second law of motion, F = ma , governs the motion of the object.

b. Newton’s second law of motion, F = ma , no longer governs the dynamics of the object.

c. Such velocities cannot be determined mathematically.

d. None of the above

asked 2022-08-24

What is the definition for newtons second law?