In Fig. 12-66, a 10 kg sphere is supported on a frictionless plane inclined at angle θ=45^∘ from

step 1 For this problem on the topic of equilibrium and elasticity, we are shown a figure of a 10 -KG s

In Fig. 12-66, a 10 kg sphere is supported on a frictionless...

Key Concepts

Static Equilibrium

This concept involves a situation where an object is at rest or moving with constant velocity, meaning that the vector sum of all forces acting on it is zero. In the context of this problem, the sphere does not accelerate, so the forces—gravity, normal force, and the cable tension—must balance exactly. Understanding static equilibrium is fundamental for solving many mechanics problems where force balances determine unknown quantities such as tension.

Force Decomposition

Force decomposition refers to breaking a force vector into components along chosen directions. In problems involving inclined planes, it is common to resolve the gravitational force into components parallel and perpendicular to the surface. This technique is crucial in accurately setting up the equilibrium equations, as it allows one to consider the effects of forces in directions that align with the geometry of the problem.

Free Body Diagram

A free body diagram is a graphical illustration used to visualize the forces acting on an object. In mechanics, drawing a free body diagram is an essential first step because it helps identify all the forces, including gravity, contact forces, and tension, as well as their directions. This visual tool supports the correct application of equilibrium conditions and aids in the proper decomposition of forces.

Tension Force

The tension force is the pulling force transmitted through a string, cable, or similar object when it is subjected to a pull. In this problem, the tension in the cable is the force which supports the sphere against gravity, in combination with the normal force exerted by the plane. Understanding how to calculate the tension is critical in many statics and dynamics problems involving support or suspension systems.

Trigonometric Analysis

Trigonometric analysis involves using sine, cosine, and other trigonometric functions to relate angles to side lengths in right triangles. In problems with inclined planes, angles play a significant role in determining the magnitude of the force components. Correct application of trigonometry is essential to relate the given angles to the physical forces in the problem and thus solve for the unknown tension.

Transcript

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