A particle moves in a circle of radius $1.0 \mathrm{~cm}$ at a speed given by $\mathrm{v}=2.0 \mathrm{t}$ where $\mathrm{v}$ is in $\mathrm{cm} / \mathrm{s}$ and $\mathrm{t}$ in seconds. (a) Find the radial acceleration of the particle at $t=1 \mathrm{~s}$. (b) Find the tangential acceleration at $t=1 \mathrm{~s}$. (c) Find the magnitude of the acceleration at $t=1 \mathrm{~s}$.
(a)
Velocity of particle at $t=1 \mathrm{sec}$
$V=2 t$
$\mathrm{V}=2(1)$
$\mathrm{V}=2 \mathrm{~cm} / \mathrm{s}$
Radial acceleration
$\mathrm{A}_{\mathrm{r}}=\frac{V^{2}}{R}=\frac{(2)^{2}}{1}$
$A_{r}=4 c m / s^{2}$
(b)
Tangential acceleration
$\mathrm{A}_{t}=\frac{d v}{d t}=\frac{d(2 t)}{d t}$
$\mathrm{A}_{\mathrm{t}}=2 \mathrm{~cm} / \mathrm{s}^{2}$
(c)
$A_{n}=\sqrt{A_{r}^{2}+A_{t}^{2}}$
$A_{n}=\sqrt{(4)^{2}+(2)^{2}}$
$A_{n}=\sqrt{20} \mathrm{~cm} / \mathrm{s}$