A wheel rotates around a stationary axis such that its rotation angle $ heta$ varies with time as $

Question

A wheel rotates around a stationary axis such that its rotation angle $	heta$ varies with time as $	heta(t) = 	heta_0 e^{\beta t}$, where $	heta_0$ and $\beta$ are positive constants. If the linear velocity of a point on the rim is $v$ at an instant when the angular displacement is $	heta$, which of the following expressions represents the magnitude of the total acceleration of that point at this instant?

Professor PC Thomas Classes · Answer

$\omega = \frac{d	heta}{dt} = \beta 	heta_0 e^{\beta t} = \beta 	heta$. Angular acc $\alpha = \frac{d\omega}{dt} = \beta^2 	heta$. Radius $R = v/\omega = v/(\beta 	heta)$. Tangential acc $a_t = \alpha R = (\beta^2 	heta) \frac{v}{\beta 	heta} = \beta v$ (constant!). Normal acc $a_n = v\omega = v(\beta 	heta)$. Total acc $a = \sqrt{a_t^2 + a_n^2} = \sqrt{(\beta v)^2 + (v\beta	heta)^2} = \beta v \sqrt{1 + 	heta^2}$.