Slider P can move along rod O A. An elastic cord P C is...

Slider $P$ can move along rod $O A$. An elastic cord $P C$ is attached to the slider and to the vertical member $B C$. Knowing that the distance from 0 to $P$ is 0.12 m and the tension in the cord is 30 N , determine $(a)$ the angle between the elastic cord and the $\operatorname{rod} O A,(b)$ the projection on $O A$ of the force exerted by cord $P C$ at point $P$.

Key Concepts

Force Decomposition

Force decomposition involves breaking a force vector into components along specified directions, usually using trigonometric functions such as sine and cosine. This is essential when analyzing how a force contributes to motion or equilibrium along different axes in mechanical systems.

Projection of Forces

The projection of a force on a line is the component of the force that acts along that particular direction. By using the cosine of the angle between the force vector and the line, one can determine how much of the force is effectively contributing to motion or resisting motion along that line.

Tension in Elastic Elements

Tension refers to the force transmitted through a stretched elastic element, such as a cord or spring. In mechanical systems, understanding tension is critical because it directly affects the balance of forces and the resulting motion of connected bodies.

Application of Trigonometry in Mechanics

Trigonometry provides the mathematical framework to relate angles and distances, which is essential in resolving forces, determining their components, and calculating angles between elements in mechanical systems. This application supports the analysis and solution of problems involving force direction and magnitude.

Transcript

00:01 Okay, so in this problem we have a rod.

00:03 Let's draw the rod in here.

00:05 This is our rod.

00:08 Supported by a wire and a hinge.

00:12 So let's see, this is the wall or the hinge, as you want to call.

00:21 So this one in here, let's put in here the wall.

00:29 We have the wire, so this is the wire.

00:35 This green line is the wire.

00:37 And what else do we have we have the length of the rod is the length is going to be represent this line in here which is just l so the length of the wire is just l what else we have the forces involved in this problem first of the first one is the tension in the wire so this is the tension t we have the center of mass of the rod with his weight pointing downwards, let's call mg.

01:19 And we have a supportive force in the hinge here that we're going to call the force h.

01:28 The last thing we have is the angle between the wire and the rod, which is theta.

01:35 And that's the entire system that we want to solve.

01:40 Okay, so now that we have established our system, the first thing we need to calculate is the tension in the wire.

01:49 But how are we going to do this? we know that is the second newton's second law for rotations.

01:59 And supposing that our rotation point is in here, the hinge, we have that the sum of all talks in the hinge, we have that the sum of all talks in the hinge.

02:12 Is going to be zero...

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