A coin is placed on a stationary disc at a distance of $1 \mathrm{~m}$ from the disc's centre. At time $t=0$ $\mathrm{s}$, the disc begins to rotate with a constant angular acceleration of $2 \mathrm{rad} / \mathrm{s}^2$ around a fixed vertical axis through its centre and perpendicular to its plane.
Find the magnitude of the linear acceleration of the coin at $t=1.5 \mathrm{~s}$. Assume the coin does not slip.
Find the magnitude of the linear acceleration of the coin at $t=1.5 \mathrm{~s}$. Assume the coin does not slip.
Q 32.8
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Data : $r=1 \mathrm{~m}, \alpha=2 \mathrm{rad} / \mathrm{s}^2, \omega_0=0, t=1.5 \mathrm{~s}$
$
a_{\mathrm{t}}=\alpha r=(2)(1)=2 \mathrm{~m} / \mathrm{s}^2
$
Angular speed at $t=1.5 \mathrm{~s}$,
$
\begin{aligned}
\omega & =\omega_{\mathrm{o}}+\alpha t=0+(2)(1.5)=3 \mathrm{rad} / \mathrm{s} \\
\therefore a_{\mathrm{r}} & =\omega^2 r=(3)^2(1)=9 \mathrm{~m} / \mathrm{s}^2
\end{aligned}
$
The required linear acceleration is,
$
\begin{aligned}
a & =\sqrt{a_{\mathrm{r}}^2+a_{\mathrm{t}}^2}=\sqrt{9^2+2^2}=\sqrt{85} \\
& =9.22 \mathrm{~m} / \mathrm{s}^2
\end{aligned}
$
[OR $v=u+a_1 t=0+(2)(1.5)=3 \mathrm{~m} / \mathrm{s}$
$
\left.\therefore a_{\mathrm{r}}=\frac{v^2}{r}=\frac{3^2}{1}=9 \mathrm{~m} / \mathrm{s}^2\right]
$
$
a_{\mathrm{t}}=\alpha r=(2)(1)=2 \mathrm{~m} / \mathrm{s}^2
$
Angular speed at $t=1.5 \mathrm{~s}$,
$
\begin{aligned}
\omega & =\omega_{\mathrm{o}}+\alpha t=0+(2)(1.5)=3 \mathrm{rad} / \mathrm{s} \\
\therefore a_{\mathrm{r}} & =\omega^2 r=(3)^2(1)=9 \mathrm{~m} / \mathrm{s}^2
\end{aligned}
$
The required linear acceleration is,
$
\begin{aligned}
a & =\sqrt{a_{\mathrm{r}}^2+a_{\mathrm{t}}^2}=\sqrt{9^2+2^2}=\sqrt{85} \\
& =9.22 \mathrm{~m} / \mathrm{s}^2
\end{aligned}
$
[OR $v=u+a_1 t=0+(2)(1.5)=3 \mathrm{~m} / \mathrm{s}$
$
\left.\therefore a_{\mathrm{r}}=\frac{v^2}{r}=\frac{3^2}{1}=9 \mathrm{~m} / \mathrm{s}^2\right]
$
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