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To tackle the question of how the slope of an inclined plane relates to the forces acting on a body sliding down it, let's break it down step by step. You're stating that the force acting on the body parallel to the plane is three times the normal reaction force. This relationship gives us a way to determine the angle of inclination of the plane.

Understanding Forces on an Inclined Plane

When an object is on an inclined plane, it experiences two main forces: the gravitational force acting downwards and the normal force acting perpendicular to the surface of the plane. The gravitational force can be split into two components:

• Parallel to the plane (F_parallel): This is the component that causes the object to slide down.
• Perpendicular to the plane (F_normal): This is the component countered by the normal force.

Components of Gravitational Force

If we denote the weight of the object as W (which equals mg, where m is mass and g is the acceleration due to gravity), we can express these components in terms of the angle θ of the inclined plane:

• F_parallel = W * sin(θ) = mg * sin(θ)
• F_normal = W * cos(θ) = mg * cos(θ)

Setting Up the Relationship

According to your question, the force parallel to the plane is three times the normal reaction force:

F_parallel = 3 * F_normal

Substituting the expressions for these forces, we get:

mg * sin(θ) = 3 * (mg * cos(θ))

Simplifying the Equation

We can cancel out the mass m and the gravitational constant g from both sides of the equation, simplifying it to:

sin(θ) = 3 * cos(θ)

Next, we can rearrange to express this in terms of tangent:

tan(θ) = sin(θ) / cos(θ) = 3

Finding the Angle of Incline

Now that we have the tangent of the angle, we can find θ by taking the arctan:

θ = arctan(3)

Using a calculator, we can find that θ is approximately 71.57 degrees.

Conclusion

Therefore, the slope of the inclined plane, where the force acting on a body sliding down it is three times the normal reaction force, is approximately 71.57 degrees. This relationship highlights the balance of forces at play and how they relate to the angle of incline, fundamentally demonstrating the principles of dynamics in physics.