The bending moment M of a beam, supported at one end, at a...

The bending moment $M$ of a beam, supported at one end, at a distance $x$ from the support is given by \[ M=\frac{1}{2} w L x-\frac{1}{2} w x^{2} \] where $L$ is the length of the beam, and $w$ is the uniform load per unit length. Find the point on the beam where the moment is greatest.

The bending moment $M$ of a beam, supported at one end, at a distance $x$ from the support is given by
\[
M=\frac{1}{2} w L x-\frac{1}{2} w x^{2}
\]
where $L$ is the length of the beam, and $w$ is the uniform load per unit length. Find the point on the beam where the moment is greatest.

Key Concepts

Differentiation and Critical Point Analysis

Differentiation is a fundamental calculus tool used to find the rate of change of a function. Critical point analysis involves identifying points where the first derivative of the function is zero or undefined, which are potential candidates for local maxima or minima. This method is extensively applied in analyzing the behavior of the bending moment along a beam.

Calculus-Based Optimization

Calculus-based optimization involves using differentiation to find the maximum or minimum values of a function. In the context of structural analysis, this technique is employed to determine the point along the beam where the bending moment reaches its maximum value by setting the derivative of the moment function to zero and confirming the result with the second derivative test.

Uniform Load

A uniform load, often referred to as a uniformly distributed load (UDL), is a load that is spread evenly across the entire length of the beam. This concept is key in structural analysis for calculating reactions, shear forces, and bending moments because it simplifies the load distribution into a constant per unit length.

Bending Moment

A bending moment is an internal moment in a structural member (such as a beam) that induces bending due to applied loads. It is crucial in structural analysis as it helps to determine the stresses and deformations in the beam, ensuring the design can safely support the applied forces.

Transcript

00:01 Here we're given the equation for capital m, which stands for the bending moment of a beam.

00:08 And it's equal to one half of w times l times x minus one half of w times x squared.

00:21 So x is our independent variable here because that is the distance from the support.

00:28 W and l are both some constant.

00:32 So the question is we want to find a point on the beam where the moment is greatest, so meaning we're looking for maximum m.

00:41 So anytime we're looking for a maximum or minimum, we're basically looking for the derivative first.

00:47 So let's actually rewrite this.

00:49 So we can factor out a couple of things.

00:51 So let's do one half, w, because we can factor that out.

00:57 And then i know we can factor out an x as well, but let's not do that.

01:02 So we have lx minus x squared.

01:10 Okay.

01:11 So the reason i didn't want to factor it out is because once when you do the derivative later, if you factor out the x, you would have to do the product tool.

01:19 Whereas here, it's you don't need product tool.

01:22 Okay, so let's go ahead and find a derivative of them with respect to x.

01:26 So this would be d, m, dx.

01:31 So this is equal to, well, what we factor out in the front, that's just a constant.

01:36 So let's leave that as it is.

01:38 And then now we need to take the derivative of this inside part.

01:42 So this is going to be just l minus 2x...