Answer to Question #245102 in Mechanics | Relativity for Suhas Adiga

Question

Consider the motion of a particle along a spiral given by r=R-Aθ [read as R minus A(theta)].The position of particle at time t=0 is r=R and θ=0. The speed of particle decreases linearly with time as v(t) =u - kt from time t=0 to t=u/k. What is the velocity as function of time in polar coordinates?

Answer 1

We have to use the definitions for r and v to find the velocity as a function of time in polar coordinates

$r=R-A\theta; R, A=cte \\ v= u-kt; u,k=cte \\ \text{we also have } v=\cfrac{dr}{dt}=\cfrac{d}{dt}(R-A\theta)=-A\cfrac{d\theta}{dt} \\ \text{which implies that } -A\cfrac{d\theta}{dt}=u-kt \\ \implies \text{and finally } \cfrac{d\theta}{dt}=\cfrac{kt-u}{A}$

In conclusion, we find that $\cfrac{d\theta}{dt}=\cfrac{kt-u}{A}$, where k, u and A are constants.

Reference:

• Sears, F. W., & Zemansky, M. W. (1973). University physics.