∙A uniform beam 4.0 m long and weighing 2500 N carries a 3500 N weight 1.50 m from the far end,

∙A uniform beam 4.0 m long and weighing 2500 N carries a...

\bullet A uniform beam 4.0 $\mathrm{m}$ long and weighing 2500 $\mathrm{N}$ carries a 3500 $\mathrm{N}$ weight 1.50 $\mathrm{m}$ from the far end, as shown in Figure $10.63 .$ It is supported horizontally by a hinge at the wall and a metal wire at the far end (a) Make a free-body diagram of the beam. (b) How strong does the wire have to be? That is, what is the minimum tension it must be able to support without breaking? (c) What are the horizontal and vertical components of the force that the hinge exerts on the beam?

Key Concepts

Center of Mass

The center of mass is the point at which the weight of an object is considered to be concentrated for the purpose of analyzing forces and torques. For uniform objects, this point is at the geometric center, simplifying the calculations of how weight forces contribute to rotations and linear effects.

Torque (Moment) Equilibrium

Torque equilibrium involves ensuring that the total clockwise moments equal the total counterclockwise moments about any pivot point. This condition is critical in analyzing systems where forces cause rotational effects, and it is used to determine unknown forces such as tensions or reaction forces.

Reaction Forces at Supports

Reaction forces are the forces exerted by supports, hinges, or connectors on a beam or structure. Analyzing these forces involves breaking them into their horizontal and vertical components and is crucial for ensuring that the structure remains in static equilibrium when subjected to external loads.

Free-Body Diagram

A free-body diagram is a simplified representation of an object showing all external forces acting on it. It helps in visualizing the forces, torques, and points of application which is essential for setting up the equations of equilibrium in statics problems.

Static Equilibrium

Static equilibrium refers to the condition in which an object remains at rest because the net forces and net torques acting on it are zero. This concept requires that the sum of all forces in every direction and the sum of all moments about any point must be balanced, ensuring the object does not accelerate or rotate.

Transcript

00:01 Hello, problem 47 has a uniform beam that's four meters long and weighs 2 ,500 newtons, and i have a 3 ,500 newton weight sitting on the end of it, and it's supported by a guide wire as well.

00:20 So let's take a look at what we got here.

00:23 We know this is a uniform board, so its weights located in the center, and it is 24.

00:35 500 newtons and that would be two meters from the end because it's a four meter board.

00:42 Now this weight here that it's supporting is 2 .5 meters from the end and it's 3 ,500 newtons.

00:51 And this is our tension wire here at a 30 degree angle and we also know it's attached here.

01:02 So we've got a horizontal and a vertical force that the hinge must exerts.

01:09 I'm going to call horizontal and vertical force.

01:15 So we're asked to find all three of these forces, tension, the horizontal and the vertical force.

01:22 We know it's an equilibrium.

01:24 So we have three conditions that must be meant.

01:28 Our net torques on this are zero.

01:31 Our forces in the x direction are zero.

01:36 And our net forces in the the y direction are zero.

01:42 So let's look first at our forces in the x direction.

01:47 We have one that's totally horizontal, the horizontal force...

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