A 4 ×8 -ft sheet of plywood weighing 34 lb has been temporarily placed among three pipe supports

A 4 ×8 -ft sheet of plywood weighing 34 lb has been...

Key Concepts

Static Equilibrium Conditions

This concept refers to an object being at rest or moving with constant velocity, which requires that the sum of all external forces and the sum of all moments acting on the object are zero. In statics problems, understanding these conditions is crucial for setting up the equations that allow for solving for unknown forces or reactions.

Free-Body Diagram

A free-body diagram is a schematic representation of an object separated from its environment, with all external forces and moments acting on it depicted as vectors. This visualization is essential in statics to identify all the forces, including gravitational loads, support reactions, and any other applied forces, thus facilitating the formulation of equilibrium equations.

Reaction Forces

Reaction forces are the forces exerted by supports, constraints, or surfaces that resist the external loads applied to a body. In problems involving multiple supports, such as a beam resting on collars or leaning against a surface, determining these forces involves applying equilibrium conditions to ensure that the body remains in static equilibrium.

Moment Equilibrium

Moment equilibrium, or rotational equilibrium, is achieved when the sum of the moments about any pivot point is zero. By taking moments about a strategic point, one can eliminate certain unknowns, making it simpler to solve for the remaining reaction forces. This is particularly useful in cases with multiple supports to distribute the applied load.

Center of Gravity

The center of gravity is the point at which the entire weight of an object can be considered to act. For uniformly distributed loads, the weight is often assumed to act at the geometric center of the object. Incorporating the correct location of the center of gravity is essential when calculating moments and ensuring the proper balance of forces.

Transcript

00:01 So the free body diagram for this question looks a little complicated, so i'm just going to ask you guys to bear with me.

00:07 Because we have a 4x8 sheet of plywood that is acting at an angle similar to that.

00:20 And we have three supports along this acting at these three points.

00:33 And over here we have a, we have b.

00:39 And here we have c.

00:41 We know from this length of the board is 8 feet, so from the end to point c is 3 feet, which means the rest of the way over here is 5 feet.

00:52 We know when the length of the board that is tilted up at this angle, we know this is 4 feet, and we know the shadow of casts 3 .75 feet, and we know the height is h, that is the same over here h is at a 90 degree angle right there we have the center of mass takes place right here in the middle which is g which is where the weight is acting down the weight we will call w and then we have the reaction forces at each of the supports so we at a we have a x coming in there and a y coming out a x and a y and we have bx coming in and bx coming in and b y and b y and we have bx coming in and b y by going up.

01:51 We also know that the spacing between all this, we have between a and the end is one foot.

01:56 Between a and b, there is four feet, which leads between b and the end.

02:00 There's three feet of space.

02:03 G is acting halfway long, so it acts on the ground from half of the 3 .75 shadow to 1 .875.

02:11 We're not going to draw that.

02:13 So we want to find the react.

02:14 Oh, we also have the opposite cx, 4 is happening.

02:20 At point c.

02:22 So we'll define the coordinate axis up here where we have x, y, and z.

02:35 So now what we want to do is find the height h that the board is elevated at.

02:42 So we can do this simply by using pythagorean's theorem, drawing a right triangle, and we know the shadow cast is 3 .75 feet, and we know the length of the board is 4 feet.

02:56 H so we know that 3 .75 squared plus h squared is equal to 4 squared and once you solve for all that you get that h is equal to 1 .39 feet which isn't a solution but it's needed for later questions so the first thing we want to do is solve for the reaction at point c what's going on there we're going to take that, solve for that by finding the moments about the xxs, which they have to sum t equal to zero in order for the system to be at rest.

03:38 So what that equation looks like is cx acting at a distance of height h and the weight of the system, which is acting down, acting at the halfway point, which is that 3 .75 shadow divided by two, since we want the halfway point between that, those two things.

04:09 And we said that equal to zero.

04:11 So we know that w is equal to 34 pounds...

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