Physics

Problem

Which statement best explains why the factor

\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}

in the arc-length integrand carries no physical units, ensuring that

\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}\,dx

has units of length? a) Because the differential

dx

later contributes a length unit, the factor must therefore carry units of

1/\text{length}

so that the product is a length. b) Because taking the square root of a squared quantity removes both the power and the units, automatically leaving a unit-free factor. c) Because

dy

and

dx

are both measured in length units, their ratio

\dfrac{dy}{dx}

is dimensionless, so the entire square-root factor is unitless. d) Because in many applications

x

represents time, so

\dfrac{dy}{dx}

is a velocity with units

\text{length}/\text{time}

; after squaring, adding 1, and taking the square root one still has

\text{length}/\text{time}

, which

dx

(time) converts to length.

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