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

Question

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

LilsGroolonip86 · Accepted Answer

Solving the equation:You can guess the solution   x  =  A  cos  ⁡      k          /        m    t  +  B  sin  ⁡      k          /        m    t or you can use substitution   x  =      e          λ      t       and find   λAnyway:  x  (  t  =  0  )  =  A  cos  ⁡  0  +  B  sin  ⁡  0  =  A  =  0  v  (  t  =  0  )  =  −  A      k          /        m    sin  ⁡  0  +  B      k          /        m    cos  ⁡  0  =  B      k          /        m    =      v    0  therefore:  x  (  t  )  =            m      k            v    0    sin  ⁡      (                  k        m              t    )

batejavizb · Accepted Answer

With given potential you can define force acting on your oscillator as  F  =  −                    d        V                    d        x              =  −  k  xThis gives you Hooks law (no surprise). Than just use Newton&#039;s second law:  m  a  =  −  k  xThan create second order differential equation replacing a with             x      ¨       and solve it with your initial conditions.

I have been given the potential of a simple harmonic oscillator: V=1/2kx^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?

Answered question

Answer & Explanation

New Questions in Force, Motion and Energy