An object rests on a plane, with an angle of incline, θ, an acceleration due to gravity, g, and a coefficient of friction μ between the object and the plane. Which of the following gives the acceleration of the object?
A) a = g sin θ
B) a = g (sin θ – cos θ)
C) a = g (cos θ – μ sin θ)
D) a = g (sin θ – μ cos θ)
The force of gravity down the plane is given by Fg = mg sin θ. The frictional force is given by Ff = μ mg cos θ.
Thus, we can set up the overall equation:
Fnet = Fg – Ff
Applying Newton’s Second Law, we can re-write the equation as:
ma = Fg – Ff
Substituting the equations given for Ff and Fg we get:
ma = mg sin θ – μ mg cos θ
Canceling out “m” throughout the equation and factoring out the “g” leaves us with:
a = g (sin θ – μ cos θ)
Thus, choice (D) is the right answer.