American eels (Anguilla rostrata) are freshwater fish with long, slender bodies that we can treat as uniform cylinders 1.0m long and 10 cm in diameter. An eel compensates for its small jaw and teeth by holding onto prey with its mouth and then rapidly spinning its body around its long axis to tear off a piece of flesh. Eels have been recorded to spin at up to 14 revolutions per second when feeding in this way. Although this feeding method is costly in terms of energy, it allows the eel to feed on larger prey than it otherwise could.

A new speed of eel is found to have the same mass but one-quarter the length and twice the diameter of the American eel. How does its moment of inertia for spinning around its long axis compare to that of the American eel? The new species has

1. Half the moment of inertia as the American eel
2. The moment of inertia as the American eel
3. Twice the moment of inertia as the American eel
4. Four times the moment of inertia as the American eel

Step by step solution

01

Density of substance

A substance's density is defined as its mass per unit volume. Which is given by,

$\mathit{\rho }\mathbf{=}\frac{\mathbf{m}}{\mathbf{v}}$

Where$\rho$ is density of substance. $m$is mass of substance and $v$is volume of substance.

02

identification of given data

We have given that a new eel is found which has the same mass but one-quarter the length and twice the diameter of the American eel.

03

Finding moment of inertia for new eel.

Let ${\rho }_1$is density of American eel. Which is given by,

$\begin{array}{c}{\rho }_1=\frac{m}{v}\\ =\frac{m}{\pi {r_1}^2{h}_1}\end{array}$

Let ${\rho }_2$is density of second eel. Which is given by,

$\begin{array}{c}{\rho }_2=\frac{m}{v}\\ =\frac{m}{\pi {\left(2r_1\right)}^2\frac{h_1}{4}}\\ =\frac{m}{\pi {r_1}^2{h}_1}\end{array}$

Here we can say that density of both eel is same.

Now, we know that

$I=\frac{1}{2}m{r}^2$

Now, for American eel,

$$\begin{gathered} I_1=\frac{1}{2}{\rho }_1\pi {r_1}^2{h}_1{r_1}^2 \\ {I}_1=\frac{1}{2}{\rho }_1\pi {r_1}^4{h}_1 \end{gathered}$$

For second eel

$$\begin{gathered} I_2=\frac{1}{2}m{r_2}^2 \\ \begin{array}{c}{I}_2=\frac{1}{2}\pi {\left(2r_1\right)}^2\frac{h_1}{4}{\left(2r_1\right)}^2{\rho }_2\\ =\frac{1}{2}\pi {\left(2r_1\right)}^4\frac{h_1}{4}{\rho }_2\end{array} \end{gathered}$$

Here we have ${\rho }_1={\rho }_2$. So,

$I_2=\frac{1}{2}\pi {\left(2r_1\right)}^4\frac{h_1}{4}{\rho }_1$

$I_2=4\left(\frac{1}{2}\pi {\left(r_1\right)}^4{h}_1{\rho }_1\right)$

Now, from the above two equations

$I_2=4I_1$

So, moment of inertia of new eel is Four times the moment of inertia as the American eel

Hence, option (d) is correct.

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