$$\text { Find the required quantity given the wire that lies along the curve } r \text { and has density } \delta .$$
-Moment of inertia $\mathrm { I } _ { \mathrm { Z } }$ about the $\mathrm { z }$-axis, where $\mathbf { r } ( \mathrm { t } ) = ( 5 \cos \mathrm { t } ) \mathbf { i } + ( 5 \sin \mathrm { t } ) \mathbf { j } , 0 \leq \mathrm { t } \leq \frac { \pi } { 2 } ; \delta = 2 ( 1 + \sin 2 \mathrm { t } )$
A) $\mathrm { I } _ { \mathrm { Z } } = 10 \left( \frac { \pi + 2 } { 2 } \right)$
B) $\mathrm { I } _ { Z } = 250 \left( \frac { \pi } { 2 } \right)$
C) $\mathrm { I } _ { \mathrm { Z } } = 0$
D) $\mathrm { I } _ { \mathrm { Z } } = 250 \left( \frac { \pi + 2 } { 2 } \right)$

Question

Quizplus · Accepted Answer

We start by using the formula for the moment of inertia about the z-axis, which is given by
$$\mathrm{I_Z}=\int_C \delta r^2 \cos^2	heta ds$$
where $	heta$ is the angle between the tangent to the curve and the z-axis, $s$ is the arc length, and $r$ is the distance from the z-axis to the point on the curve.

In our case, we have $r=5$ and $ds=\sqrt{\left(\frac{dx}{dt}ight)^2+\left(\frac{dy}{dt}ight)^2}dt=5dt$, since $x=5\cos t$ and $y=5\sin t$.

To compute $	heta$, we note that the tangent to the curve is given by
$$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=	an t,$$
so
$$\cos	heta=\frac{1}{\sqrt{1+	an^2	heta}}=\frac{1}{\sqrt{1+\left(\frac{dy}{dx}ight)^2}}=\frac{1}{\sqrt{1+	an^2 t}}=\frac{1}{\sqrt{1+\frac{\sin^2 t}{\cos^2 t}}}=\cos t,$$
since $0\leq t\leq \frac{\pi}{2}$ and $\cos t\geq 0$.

Putting all these pieces together, we get
\begin{align*}
\mathrm{I_Z}&=\int_0^{\frac{\pi}{2}} \delta r^2 \cos^2 t ds\
&=\int_0^{\frac{\pi}{2}} 2(1+\sin 2t)25\cos^2 t dt\
&=50\int_0^{\frac{\pi}{2}} (\cos^2 t+\sin 2t\cos^2 t) dt\
&=25\int_0^{\frac{\pi}{2}} (1+\cos 2t+\cos 2t\cos 2t) dt\
&=25\left[t+\frac{\sin 2t}{2}+\frac{1}{4}\sin 4tight]_0^{\frac{\pi}{2}}\
&=25\left(\frac{\pi}{2}+1ight).
\end{align*}
Therefore, the answer is D.

Find the required quantity given the wire that lies along the curve r and has density δ.\text { Find the required quantity given the wire that lies along the curve } r \text { and has density } \delta . Find the required quantity given the wire that lies along the curve r and has density δ.

$\text { Find the required quantity given the wire that lies along the curve } r \text { and has density } \delta .$