Question

The equations listed together on page 38 give position as a function of time, velocity as a function of time, and velocity as a function of position for an object moving in a straight line with constant acceleration. The quantity appears in every equation. (a) Do any of these equations apply to an object moving in a straight line with simple harmonic motion? (b) Using a similar format, make a table of equations describing simple harmonic motion. Include equations giving acceleration as a function of time and acceleration as a function of position. State the equations in such a form that they apply equally to a block–spring system, to a pendulum, and to other vibrating systems. (c) What quantity appears in every equation?

Short answer

(a)No. Since the acceleration is not constant in simple harmonic motion

(b)The table is shown below.

(c) The angular frequency appears in every equation.

Step-by-step solution

Relationship between amplitude and frequency

If an oscillator is subject to a sinusoidal driving force that is described by

$F\left(t\right)=F_0\sin \omega t$

It exhibits resonance, in which the amplitude is largest when the driving frequency matches the natural frequency ${\omega }_0=\sqrt{\frac{k}{m}}$of the oscillator.

${\omega }_0=$Resonance frequency

Step 2(a): Find any of these equations apply to an object moving in a straight line with simple harmonic motion

No. Since the acceleration is not constant in simple harmonic motion, none of the equations in Table 2.2 are valid.

Step 3(b): Make a table of equations describing simple harmonic motion

Equation	Information given by equation
$x\left(t\right)=A\cos \left(\omega t+\varphi \right)$	Position as a function of time
$v\left(t\right)=-\omega A\cos \left(\omega t+\varphi \right)$	velocity as a function of time
$v\left(x\right)=\pm \omega {\left(A^2-{\omega }^2\right)}^{1/2}$	velocity as a function of position
$a\left(t\right)=-{\omega }^{2}A\cos \left(\omega t+\varphi \right)$	acceleration as a function of time
$a\left(t\right)=-{\omega }^{2}x\left(t\right)$	acceleration as a function of position

Step 4(c): Find what quantity appears in every equation

The angular frequency$\omega$ appears in every equation.