Which statement best explains why the factor in the arc-length integrand carries no…

PHYSICS · Single correct · A Level · Physics · Measurements and their errors · Measurements and their errors

Problem

Which statement best explains why the factor 1+(dydx)2\sqrt{1+\left(\dfrac{dy}{dx}\right)^2} in the arc-length integrand carries no physical units, ensuring that 1+(dydx)2 dx\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}\,dx has units of length? (a) Because the differential dxdx later contributes a length unit, the factor must therefore carry units of 1/length1/\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 dydy and dxdx are both measured in length units, their ratio dydx\dfrac{dy}{dx} is dimensionless, so the entire square-root factor is unitless. (d) Because in many applications xx represents time, so dydx\dfrac{dy}{dx} is a velocity with units length/time\text{length}/\text{time}; after squaring, adding 1, and taking the square root one still has length/time\text{length}/\text{time}, which dxdx (time) converts to length.

Related theory

Browse the problem library