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)

**Explanation**

The force of gravity down the plane is given by F_{g} = mg sin θ. The frictional force is given by F_{f} = μ mg cos θ.

Thus, we can set up the overall equation:

F_{net} = F_{g} – F_{f}

Applying Newton’s Second Law, we can re-write the equation as:

ma = F_{g} – F_{f}

Substituting the equations given for F_{f} and F_{g} 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.

Why is the force of friction using the cos(theta) doesn’t friction oppose the motion (acting in the horizontal direction using sin)? So should friction as use the horizontal component and sin(theta)?

What a great question! You certainly would think that friction would act parallel to the plane, and to a certain degree, you’re right. But the key is that the equation for frictional force is always Ffriction = (coefficient of friction) (normal force), in which the coefficient can be either static or kinetic, depending on the scenario. Note, though, that normal force always acts perpendicular to the surface. For inclined planes, mgsin(theta) generally represents the component of force pointing parallel to the plane, so mgcos(theta) gives our perpendicular, or normal, force. Frictional force, then, is simply this value multiplied by the coefficient that we need to use.