Determine the reactions at A and B when β=50^∘

Name: Problem 81, Chapter 4: Vector Mechanics for Engineers: Statics and Dynamics
Uploaded: 2020-06-25T23:39:05-08:00
Duration: 11 min 43 s
Channel: Henry York
Description: Determine the reactions at A and B when β=50^∘

Key Concepts

Moment Equilibrium

This concept involves ensuring that the sum of all moments about any chosen point is zero. Applying moment equilibrium helps in eliminating one or more unknowns, typically allowing for the calculation of reaction forces that would otherwise be difficult to solve using force equations alone.

Force Decomposition

When forces are applied at an angle, they can be resolved into perpendicular components using trigonometric functions. This decomposition is vital for correctly applying equilibrium equations in both the horizontal and vertical directions, especially when dealing with angles like the given beta value.

Static Equilibrium

This is the fundamental condition in mechanics where a system or structure remains at rest or in uniform motion. It requires that the sum of all forces and the sum of all moments about any point be equal to zero. This concept is crucial when analyzing problems involving support reactions, as it allows the determination of unknown forces acting on a body.

Free Body Diagram

A graphical representation used to show all external forces acting on a body isolated from its surroundings. The free body diagram is essential in statics for identifying forces, moments, and their points of application, which subsequently aids in setting up the equilibrium equations required to solve for support reactions.

Reaction Forces

These are the forces exerted by supports or connections in response to loads or external forces. Recognizing the nature and direction of reaction forces is key in structural analysis, as they provide the necessary counterbalance to applied loads to maintain equilibrium.

Transcript

00:06 Problem 4 .81 is a three -force rigid body problem, where there is a 100 newton applied force at c, and the other two forces are the reactions at a and b.

00:18 Now, we know the direction of the reaction at b, since there's a roller there, which means that it must be normal to the surface the roller is in contact with.

00:31 And since it is a three -force rigid body problem, it means that all three forces must be concurrent, which means that where two forces lines of action meet, the third forces line of action will also intersect there.

00:53 I'll call that point, point d where they intersect.

00:57 And to find the exact angle of the reaction forces, a and b.

01:04 I have to do some geometry, make some triangles.

01:12 So i'm making a right triangle between d and a, get a point here that i'll call e.

01:22 And i'll make a point here called f to get a right triangle with b is.

01:26 Well.

01:31 And this triangle will be helpful as well with this point h between a, b, and h.

01:45 Now start off finding the angle of the reaction force at b.

01:54 So you have the triangle between d, f, and b.

02:07 And to find the sides of this triangle, we'll use another triangle between b, c, and d.

02:15 We know this angle is 75 degrees because he applied forces at 50 degrees and and the arm between b and c was at 25 degrees so you add them up to 75 we know bc's 0 .25 meters given in the problem and taking the tangent of the angle 75 degrees sending equal to its opposite side d b over its adjacent side 0 .25 meters we can find the side db which is equal to 0 .933 meters and so we know that the angle of the reaction force at b is going to be 65 degrees because of this right triangle here and since the arm at ac is at 25 degrees to the horizontal and using sign of that angle we can find the opposite its side fd over its hypotenuse .933 meters.

04:19 So fd is equal to .846 meters.

04:29 And using the pythagorean theorem, we can find the last side of this triangle.

04:51 The sides of these triangle, of this triangle fd, b, will be important to find the reaction force angle at a.

04:59 We did just find the reaction force angle at b, which is 65 degrees, going between b and the intersection point of the three forces at d.

05:13 So now drawing the triangle going from a to the intersection point of the lines of actions of all the forces, we need to find this angle here, which i'll call theta, and this triangle will help us do that.

05:55 Now that we know the angle between a and b is 25 degrees, given in the problem, as well as the distance of 0 .15 meters.

06:05 So taking the sign of that angle, we can find the distance b -h.

06:33 Using the pythagorean theorem, we can find the last side of that triangle, h...

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