Question

an object with a mass of m kilograms is placed on a ramp at an angle of θ degrees from the horizontal. assuming the ramp is frictionless, what must be the acceleration of the object down the ramp?
mg sin θ
mg cos θ
tan θ
g sin θ
g cos θ
tan^{-1}(θ)

Explanation:

Step1: Analyze the forces acting on the object

The gravitational force acting on the object is \(F = mg\). When the object is on an inclined plane (ramp) at an angle \(\theta\) with the horizontal, we resolve the gravitational force into two components. The component of the gravitational force along the ramp is \(F_{\parallel}=mg\sin\theta\) and the component perpendicular to the ramp is \(F_{\perp}=mg\cos\theta\).

Step2: Apply Newton's second law

Newton's second law is \(F = ma\). Along the ramp, the net force \(F_{net}=mg\sin\theta\) (since the ramp is frictionless, there is no opposing force along the ramp). Substituting \(F_{net}\) into \(F = ma\), we have \(mg\sin\theta=ma\).

Step3: Solve for acceleration \(a\)

Divide both sides of the equation \(mg\sin\theta = ma\) by \(m\). Canceling out \(m\) (since \(m
eq0\)), we get \(a = g\sin\theta\).

Answer:

\(g\sin\theta\)