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Definition/Summary
An object (typically a "mass on a spring") which has a position (or the appropriate generalization of position) which varies sinusoidally in time.
Equations
[tex]
x(t)=A\sin(\omega t)+B\cos(\omega t)
[/tex]
[tex]
\omega^2 =\frac{k}{m}
[/tex]
Extended explanation
According to Hooke's law and Newton's 2nd Law, a point mass of mass [itex]m[/itex] attached to a spring of spring constant [itex]k[/itex] obeys the equation
[tex]
m\frac{d^2 x}{dt^2}=-kx\;,\qquad(1)
[/tex]
where [itex]x[/itex] is the position of the point mass.
The solution of equation (1) is given by
[tex]
x(t)=A\sin(\omega t)+B\cos(\omega t)\;,\qquad(2)
[/tex]
where A and B are constants that may be chosen so that x(t) satisfies the appropriate initial conditions, and
where
[tex]
\omega=\sqrt{\frac{k}{m}}\;.
[/tex]
For example, in terms of the initial position [itex]x_0[/itex] and initial velocity [itex]v_0[/itex], equation (2) can be written as
[tex]
x(t)=\frac{v_0}{\omega}\sin(\omega t)+x_0\cos(\omega t)\;.
[/tex]
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
An object (typically a "mass on a spring") which has a position (or the appropriate generalization of position) which varies sinusoidally in time.
Equations
[tex]
x(t)=A\sin(\omega t)+B\cos(\omega t)
[/tex]
[tex]
\omega^2 =\frac{k}{m}
[/tex]
Extended explanation
According to Hooke's law and Newton's 2nd Law, a point mass of mass [itex]m[/itex] attached to a spring of spring constant [itex]k[/itex] obeys the equation
[tex]
m\frac{d^2 x}{dt^2}=-kx\;,\qquad(1)
[/tex]
where [itex]x[/itex] is the position of the point mass.
The solution of equation (1) is given by
[tex]
x(t)=A\sin(\omega t)+B\cos(\omega t)\;,\qquad(2)
[/tex]
where A and B are constants that may be chosen so that x(t) satisfies the appropriate initial conditions, and
where
[tex]
\omega=\sqrt{\frac{k}{m}}\;.
[/tex]
For example, in terms of the initial position [itex]x_0[/itex] and initial velocity [itex]v_0[/itex], equation (2) can be written as
[tex]
x(t)=\frac{v_0}{\omega}\sin(\omega t)+x_0\cos(\omega t)\;.
[/tex]
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!