Simple Harmonic Motion

Excerpts

• Simple Harmonic Motion

  But if $d\theta$ be taken sufficiently small, $\sin d\theta$, and $d\theta$, may be made as nearly in a ratio of equality as we please, and $1 - \cos d\theta$ may be made as small a part as we please, either of $d\theta$ or $\sin d\theta$.

• Simple Harmonic Motion

  we have equations, which, though never exactly true, are such that by making $d\theta$ sufficiently small, the errors may be made as small parts of $d\theta$ as we please.

• Simple Harmonic Motion

  [The motion of the point $M$ or the point $N$ is called in physics a *simple harmonic motion*.]

• Simple Harmonic Motion

  If the angle so described be always increased by equal angles in equal portions of time, the angular velocity is said to be uniform, and is measured by the number of angular units described in a unit of time.

• Simple Harmonic Motion

  The same considerations of velocity which have been applied to the motion of a point along a line may also be applied to the motion of a line round a point.

Equations

• Simple Harmonic Motion

  x = r \cos\theta

  The abscissa of the moving point P is r times the cosine of the angle θ it has described from A.

• Simple Harmonic Motion

  y = r \sin\theta

  The ordinate of the moving point P is r times the sine of the angle θ it has described from A.

• Simple Harmonic Motion

  dx = r \sin\theta\, d\theta

  For a sufficiently small increment dθ, the infinitely small increment dx of the abscissa is approximately r sin θ times dθ; the error can be made a vanishingly small part of dθ.

• Simple Harmonic Motion

  dy = r \cos\theta\, d\theta

  For a sufficiently small increment dθ, the infinitely small increment dy of the ordinate is approximately r cos θ times dθ; the error can be made a vanishingly small part of dθ.

• Simple Harmonic Motion

  \theta = \phi t

  If the angle described grows at a non-uniform rate, the angle at time t is φ times t, and the angular velocity is then the rate φ.

• Simple Harmonic Motion

  \frac{dx}{dt} = r \sin\theta\, \frac{d\theta}{dt}

  The rate of change of the abscissa with time equals r sin θ times the angular velocity dθ/dt; it becomes exact in the limit of the ratios.

• Simple Harmonic Motion

  \frac{dy}{dt} = r \cos\theta\, \frac{d\theta}{dt}

  The rate of change of the ordinate with time equals r cos θ times the angular velocity dθ/dt; it becomes exact in the limit of the ratios.

• Simple Harmonic Motion

  a \sin\theta

  The velocity of the abscissa x is a sin θ, where a is the uniform speed of P, so the point M moves with a variable velocity that is sin θ of the velocity of P.

• Simple Harmonic Motion

  a \cos\theta

  The velocity of the ordinate y is a cos θ, so the point N moves from O with a velocity that is cos θ of the velocity of P.

Problems

No exercises in this chapter.