Final answer:
To demonstrate that the acceleration on a frictionless incline is a = g sinθ, we analyze the component of gravitational force parallel to the incline and apply Newton's second law. The mass of the object cancels out, showing the acceleration depends only on the angle of the incline and gravity.
Step-by-step explanation:
To show that the acceleration of any object down a frictionless incline that makes an angle θ with the horizontal is a = g sinθ, first we must consider the forces acting on the object. The only force exerting an influence down the slope is the component of the object's weight parallel to the incline, which can be calculated as Wx = mg sinθ, where m is the mass of the object, g is the acceleration due to gravity, and θ is the angle of the incline. Since there is no friction, no other forces act parallel to the incline to oppose this motion.
Newton's second law states that force equals mass times acceleration, or F = ma. Therefore, for the object on the incline, mg sinθ = ma. Simplifying this equation by cancelling out the mass (because it is on both sides of the equation), we are left with a = g sinθ, confirming that the acceleration is independent of mass and is dictated solely by the angle of incline and the acceleration due to gravity.