A slender rod with length L has a mass per unit length that...

A slender rod with length $L$ has a mass per unit length that varies with distance from the left end, where $x=0$ , according to $d m / d x=\gamma x,$ where $\gamma$ has units of $\mathrm{kg} / \mathrm{m}^{2}$ . (a) Calculate the total mass of the rod in terms of $\gamma$ and $L .$ (b) Use Eq. $(9.20)$ to calculate the moment of inertia of the rod for an axis at the left end, perpendicular to the rod. Use the expression you derived in part (a) to express $I$ in terms of $M$ and $L .$ How does your result compare to that for a uniform rod? Explain this comparison. (c) Repeat part (b) for an axis at the right end of the rod. How do the results for parts (b) and (c) compare? Explain this result.

Key Concepts

Mass Distribution

This concept involves understanding that mass can vary continuously along an object's length. In many physical problems, the density is given as a function of position rather than as a constant, which requires integrating the density function over the entire length to obtain the total mass.

Integration for Total Mass

Integrating the mass per unit length (or density function) across the object's length is a fundamental technique in determining the overall mass of a continuous system. This approach is critical when dealing with objects whose mass distribution is not uniform.

Moment of Inertia Calculation

The moment of inertia is a measure of an object's resistance to angular acceleration about a given axis. It is calculated by integrating the products of the small mass elements and the square of their distances from the axis of rotation. This principle applies to both uniform and non-uniform systems.

Non-uniform versus Uniform Mass Distribution

Comparing a non-uniform rod to a uniform one illustrates how variations in density affect physical properties such as moment of inertia. A non-uniform mass distribution alters the weighting of mass elements in the integration process, leading to differences in the rotational inertia compared to a uniform case.

Dependence on Axis Location

The moment of inertia of a body depends critically on the location of the axis of rotation. Changing the axis (such as from one end of the rod to the other) alters the distances of the mass elements from the axis, and understanding this dependency is key to analyzing rotational dynamics in various configurations.

Transcript

0:00 Hi, everybody.

00:01 So for a, we need to find to express the initial in terms of m and l.

00:07 Okay, so we have the derivative, dx, lambda, x.

00:14 Okay, and basically we're going to have dm equals d.

00:21 Oops.

00:23 I mean, it's still going to be dx, but let me put this a little bit better, lambda x, dx.

00:28 There we go.

00:28 I wrote it weird.

00:29 So we have capital m, we're going to need to l and zero.

00:35 And pretty much it's just going to be lambda dx, and we're just going to take the intuple, okay? of the x, so we get x squared over square, over 2, l, zero.

00:48 So this we get one half, and i do not need that one there.

00:55 One over, lambda over two, l squared.

00:58 Okay? and now for b, b is asking us right here, what is, on the x is the right rod, how it results in b and c.

01:18 So basically we're kind of just doing the exact same thing as we did as a.

01:22 So we have the rod, and we need to find the inertia of the rod.

01:29 So we have m equals one -half and the l -square.

01:34 This is of the rod.

01:36 And we have lambda.

01:37 It's going to be 2m, l square.

01:40 Okay, we're not going to save that for later.

01:42 So now we have inertia.

01:44 It's going to be equal to l0 x squared d .m...