Chapter 5 Problem 64

Problem

An object of mass $m$ is constrained to move in a circle of radius $r$. Its tangential acceleration as a function of time is given by $a_{\text{tan}}=b+ct^{2}$, where $b$ and $c$ are constants. If $v=v_{0}$ at $t=0$, determine the tangential and radial components of the force, $F_{\text{tan}}$ and $F_{R}$ , acting on the object at any time $t>0$.

Solution

The tangential force is simply the mass times the tangential acceleration.

$a_{T}=b+ct^{2}$ $\rightarrow F_{T}=ma_{T}=m(b+ct^{2})$

To find the radial force, we need the tangential velocity, which is the anti-derivative of the tangential acceleration. We evaluate the constant of integration so that $v=v_{0}$ at $t=0$.

$a_{T}=b+ct^{2}$ $\rightarrow v_{T}=c+bt+{\frac {1}{3}}ct^{3}$ $\rightarrow v(0)=c=v_{0}$ $\rightarrow v_{T}=v_{0}+bt+{\frac {1}{3}}ct^{3}$

$F_{R}={\frac {mv_{T}^{2}}{r}}={\frac {m}{r}}\left(v_{0}+bt+{\frac {1}{3}}ct^{3}\right)^{2}$