Inclined Plane, mg.sin(θ) = ma?

Question

Prove that equation using F = ma

Help!

Answer 1

When a particle of mass m is placed on a smooth plane inclined to the horizontal at angle theta, its equation of motion is:

mg sin(theta) = ma

where a is the acceleration of the particle and g the acceleration due to gravity.

The forces acting on the particle are:

(a) its weight mg vertically downwards, and

(b) the normal reaction R of the plane perpendicular to the plane.

The weight mg can be divided into components:

mg cos(theta) perpendicular to the plane,

mg sin(theta) parallel to the plane.

Applying F = ma parallel to the plane gives:

mg sin(theta) = ma.

As there is no motion perpendicular to the plane, the equation for that direction is:

mg cos(theta) = R

giving the magnitude of the normal reaction R.