- #1
noodle_snacks
- 3
- 0
Homework Statement
A particle oscillates between the points x = 40mm and x = 160mm with an acceleration a = k(100-x) where k is a constant. The velocity of the particle is 18mm/s when x=100 and zero at x = 40mm and x = 160mm. Determine a) the value of hte constant k, b) the velocity when x = 120mm
Homework Equations
[tex]a = k(100-x)[/tex]
The Attempt at a Solution
This looked like a simple harmonic oscillator to me.
So I went:
[tex]a = 100k - kx [/tex]
[tex]\frac{d^2x}{dt^2} = 100k - kx[/tex]
Define:
[tex]\dot x = \frac{\mathrm{d}x}{\mathrm{d}t}[/tex]
Then Observe:
[tex]\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} = \ddot x = \frac{\mathrm{d}\dot {x}}{\mathrm{d}t}\frac{\mathrm{d}x}{\mathrm{d}x}=\frac{\mathrm{d}\dot {x}}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{\mathrm{d}\dot{x}}{\mathrm{d}x}\dot {x}[/tex]
Then substitute:
[tex]\frac{d\dot x}{dx}\dot x = 100k-kx [/tex]
[tex]d\dot x = (100k-kx)dx [/tex]
[tex]\int \dot x d\dot x = \int (100k-kx)dx [/tex]
[tex]\dot x^2 = 50kx - kx^2 + c[/tex]
I got that far in the manipulation, then I got stuck. Where do i go from here or what have I done wrong? My current approach is to solve for the differential then differentiate to get an equation for the velocity. Is there a better approach?
Last edited: