# Problem: For a simple harmonic oscillator, answer yes or no to the following questions.(a) Can the quantities position and velocity have the same sign? Yes No (b) Can velocity and acceleration have the same sign? Yes No (c) Can position and acceleration have the same sign? Yes No

###### FREE Expert Solution

The position of a simple harmonic oscillator is described as:

$\overline{){\mathbf{x}}{\mathbf{\left(}}{\mathbf{t}}{\mathbf{\right)}}{\mathbf{=}}{\mathbf{A}}{\mathbf{c}}{\mathbf{o}}{\mathbf{s}}{\mathbf{\left(}}{\mathbf{\omega }}{\mathbf{t}}{\mathbf{+}}{\mathbf{\varphi }}{\mathbf{\right)}}}$

Velocity:

$\overline{)\begin{array}{rcl}\mathbf{v}\mathbf{\left(}\mathbf{t}\mathbf{\right)}& {\mathbf{=}}& \frac{\mathbf{d}}{\mathbf{d}\mathbf{t}}\mathbf{\left[}\mathbf{A}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{\omega }\mathbf{t}\mathbf{+}\mathbf{\varphi }\mathbf{\right)}\mathbf{\right]}\\ & {\mathbf{=}}& \mathbf{-}\mathbf{A}\mathbf{\omega }\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{\omega }\mathbf{t}\mathbf{+}\mathbf{\varphi }\mathbf{\right)}\end{array}}$

Acceleration:

$\overline{)\begin{array}{rcl}\mathbf{a}\mathbf{\left(}\mathbf{t}\mathbf{\right)}& {\mathbf{=}}& \frac{\mathbf{d}}{\mathbf{dt}}\mathbf{\left[}\mathbf{-}\mathbf{A\omega sin}\mathbf{\left(}\mathbf{\omega t}\mathbf{+}\mathbf{\varphi }\mathbf{\right)}\mathbf{\right]}\\ & {\mathbf{=}}& \mathbf{-}\mathbf{A}{\mathbf{\omega }}^{\mathbf{2}}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{\omega }\mathbf{t}\mathbf{+}\mathbf{\varphi }\mathbf{\right)}\\ & {\mathbf{=}}& \mathbf{-}{\mathbf{\omega }}^{\mathbf{2}}\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\end{array}}$

###### Problem Details

For a simple harmonic oscillator, answer yes or no to the following questions.

(a) Can the quantities position and velocity have the same sign?

Yes

No

(b) Can velocity and acceleration have the same sign?

Yes

No

(c) Can position and acceleration have the same sign?

Yes

No