* / More lifting You exert a 630-N force on rope 2 in the previous problem (Figure P 7.13 ) . Wr

* / More lifting You exert a 630-N force on rope 2 in the...

$* /$ More lifting You exert a $630-\mathrm{N}$ force on rope 2 in the previous problem (Figure $\mathrm{P} 7.13 )$ . Write the two equations $(x \text { and } y)$ for the first condition of equilibrium using the pulley as the object of interest for a force diagram. Calculate $\theta_{1}$ and $\theta_{2} .$ You may need to use the identity $(\sin \theta)^{2}+(\cos \theta)^{2}=1$

Key Concepts

Equilibrium Conditions

Equilibrium conditions refer to the state where the net forces and moments acting on a system are zero. When analyzing a pulley or any object, the sum of forces in each independent direction (usually x and y) must be zero, which provides a set of equations that can be solved for unknown variables such as angles or tensions.

Free-Body Diagram

A free-body diagram (FBD) is a visual representation that isolates an object to display all the external forces acting upon it. In problems involving pulleys and ropes, drawing an FBD helps in identifying and correctly resolving the force components to set up the appropriate equilibrium equations.

Force Decomposition

Force decomposition involves breaking a force into its horizontal and vertical components using trigonometric functions. This process is crucial when forces act at an angle since the individual components in the x and y directions are used to formulate the equilibrium equations.

Trigonometric Relationships

Trigonometric relationships, specifically the definitions of sine and cosine for angles, are used to express the components of angled forces. These relationships allow the conversion between the magnitude of a force and its directional components, which is essential when setting up equilibrium equations.

Pythagorean Identity

The Pythagorean identity, expressed as sin²? + cos²? = 1, is a fundamental trigonometric relation that helps in solving for unknown angles or force components when one of the trigonometric ratios is known. This identity often appears in problems requiring the determination of angles from resolved force components.

Transcript

00:01 In this case, we're told that we now exert a 630 newton force on this rope, and we have the same engine block here, and we want to figure out what these two angles are.

00:16 So we draw our free body diagram.

00:21 Again, we have the weight of the engine, and that also the weight of the engine passes through the pulley and is in this part of the rope.

00:30 2.

00:31 So that's the weight of the engine and we have our force that we're now exerting 630 newtons.

00:39 So we can do force balances on our pulley.

00:44 In the x direction we have minus f cosine theta 2, okay the x component of this force, plus mg the weight of the engine times cosine of theta 1 so the component of that force in this direction.

01:04 This has no component in the x direction.

01:08 In the y direction we have the y component of this force which is f sign of theta and then we have the y component of this force which is mg sine of theta 1 and then we have minus mg which i've just lumped into here and pull out the weight of the engine.

01:30 So now we have we have two equations and two unknowns and we need to solve those.

01:36 And one thing we can do is take, put this guy on the other side here and then square both sides.

01:43 So that gives us f squared, cosine squared or theta 2 equals m g squared cosine squared of theta 1.

01:53 We can then take this equation, we can then take this equation, and square it, we'll move this on the other side and then square both sides...

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