Two springs of constant density A spring of constant density $\delta$ lies along the helix

$$\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 2 \pi$$

a. Find $I_{z}$

b. Suppose that you have another spring of constant density $\delta$ that is twice as long as the spring in part (a) and lies along the helix for $0 \leq t \leq 4 \pi .$ Do you expect $I_{z}$ for the longer spring to be the same as that for the shorter one, or should it be different? Check your prediction by calculating $I_{z}$ for the longer spring.

## Discussion

## Video Transcript

Okay, folks. So now we're gonna talk about problem number 37 and for this problem, we're gonna be looking for the moment of inertia of a circular wire hoop about the Z axis and let me just go ahead and first of a graft a circle out for you. We have a circle, basically Ah, that that looks like this. He's got a radius of a Raisa parts of number, and it's got a constant density delta and we're gonna be looking for the moment of inertia. Now, the moment of inertia is a It is a three by three matrix. But because we're because our object is just a two dimensional object and we're only looking for the component of this matrix, along is the axis. So we can way have a very simple expression for for the moment of inertia. And that expression is just an integral of delta multiplied by r squared minus are three squared Where r three is This notation are three just means Z is the third component of the of the position vector The first component being X and the second quantities wife anyway multiplied by Yes, And because Delta is a constant I'm gonna pull the constant out. So no harsh squared minus are three squared Let me remind you something are square to just x squared plus y squared Plus these word. And now if you subtract off the Z squared you just uses half x squared plus y squared. But there's something very special about this expression because we had When you have a circle, every point on that circle X squared plus y squared gives you a constant value. And as you can probably guess, that constant values just eight. So we have a squared, Yes, but because of the fact that a is a constant, we can pull that constant out again. So we end up with DS Inter Girl Integral DS. But what is integral DS? Well, interval ds is just the circumference of the circle and the circumference is just you have you know, I don't know what the circumference of a circle is asked us to pay a So when you Ah, when you multiply the 22 days together you get two pi delta A to the power three and that's it for this video. Thank you for watching. Um but I

## Recommended Questions

A circular wire hoop of constant density $\delta$ lies along the circle $x^{2}+y^{2}=a^{2}$ in the $x y$ -plane. Find the hoop's moment of inertia about the z-axis.

Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass $M$ and radius $R$ about an axis perpendicular to the hoop’s plane at an edge.

Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass $M$ and radius $R$ about an axis perpendicular to the hoop's plane at an edge.

If a wire with linear density $ \rho(x, y, z) $ lies along a space curve $ C $, its $ \textbf{moments of inertia} $ about the $ x $-, $ y $-, and $ z $-axes are defined as

$$ I_x = \int_C (y^2 + z^2) \rho(x, y, z) ds $$

$$ I_y = \int_C (x^2 + z^2) \rho(x, y, z) ds $$

$$ I_z = \int_C (x^2 + y^2) \rho(x, y, z) ds $$

Find the moments of inertia for the wire in Exercise 35.

If a wire with linear density $ \rho(x, y) $ lies along a plane curve $ C $, its $ \textbf{moments of inertia} $ about the $ x $- and $ y $-axes are defined as

$ I_x = \int_C y^2 \rho(x, y) ds $ $ I_y = \int_C x^2 \rho(x, y) ds $

Find the moments of inertia for the wire in Example 3.

Finding a moment of inertia Find the moment of inertia

with respect to the $y$ -axis of a thin sheet of constant density

$\delta=1 \mathrm{gm} / \mathrm{cm}^{2}$ bounded by the curve $y=\left(\sin ^{2} x\right) / x^{2}$ and the interval $\pi \leq x \leq 2 \pi$ of the $x$ -axis.

Center of mass and moments of inertia for wire with variable density Find the center of mass and the moments of inertia about the coordinate axes of a thin wire lying along the curve

$$

\mathbf { r } ( t ) = t \mathbf { i } + \frac { 2 \sqrt { 2 } } { 3 } t ^ { 3 / 2 } \mathbf { j } + \frac { t ^ { 2 } } { 2 } \mathbf { k } , \quad 0 \leq t \leq 2

$$

if the density is $\delta = 1 / ( t + 1 )$

Find the moment of inertia about the $z$ -axis of a thin shell of constant density $\delta$ cut from the cone $4 x^{2}+4 y^{2}-z^{2}=0, z \geq 0,$ by the circular cylinder $x^{2}+y^{2}=2 x$ (see the accompanying figure).

(FIGURE CAN'T COPY)

Finding a moment of inertia Find the moment of inertia about

the $x$ -axis of a thin plate bounded by the parabola $x=y-y^{2}$ and

the line $x+y=0$ if $\delta(x, y)=x+y$

Finding moments of inertia Find the moment of inertia about the

$x$ -axis of a thin plate of density $\delta=1 \mathrm{gm} / \mathrm{cm}^{2}$ bounded by the circle

$x^{2}+y^{2}=4 .$ Then use your result to find $I_{y}$ and $I_{0}$ for the plate.