Chapter 9: Force

9.6 Inclined planes

Let’s say you have a box of snakes sliding down a frictionless inclined plane, and you want to know the acceleration. Of course, you begin by drawing your free body diagram:

A right triangle is formed by a dashed line. One leg is horizontal, and another is vertical. The hypotenuse extends from upper left to lower right. A point on the hypotenuse is marked by a dot. Two arrows extend from the dot; one points directly down, labeled F superscript G; the other extends up and to the right, perpendicular to the hypotenuse. The second arrow is labeled F superscript N. The angle between the hypotenuse and the horizontal leg is labeled with a lowercase Greek theta. To the side is another pair of arrows; one points directly to the right, labeled x; the other points directly up, labeled y.
Free body diagram for the box of snakes on an inclined plane with negligible friction.

Then you apply Newton’s second law:

\[
\begin{align*}
F_\textit{net,x} &= ma_x & F_\textit{net,y} &= ma_y \\
F^N\sin\theta &= ma_x & F^N\cos\theta – mg &= ma_y
\end{align*}
\]

Even if we know the mass of the box and the angle of the incline, we still don’t have enough information to solve this! We do not know the magnitude of the normal force of the incline pushing on the box, nor do we know either component of the acceleration. There are three unknowns, but only two equations—this is not a solvable system.

You are free to define your coordinate system however you like. For inclined plane problems, rotate your coordinate axes to simplify the problem: orient the coordinates so that the \(x\) axis lies along the incline, as shown below. That way, any acceleration is only along the \(x\) axis and the normal force will lie along the \(y\) axis. You will need to use trig to find \(x\) and \(y\) components of the force of gravity, which still points directly downward—you can choose whatever coordinate grid you like, but you cannot change the direction of gravity!

A right triangle is formed by a dashed line. One leg is horizontal, and the other is vertical. The hypotenuse extends  from upper left to lower right. The angle in the bottom right, between the hypotenuse and the horizontal leg, is labeled with a lowercase Greek theta.

Two sets of x-y coordinate axes are shown, each with the origin located halfway along the hypotenuse. One set of axes is made with dotted line arrows; an arrow pointing straight up is labeled old y axis, and an arrow pointing directly to the right is labeled old x axis. The other set of coordinate axes is shown with solid line arrows. One arrow extends away from the triangle perpendicular to the hypotenuse; this is labeled y. The other arrow lies along the hypotenuse, pointing down and to the right; this is labeled x. The angle between the dotted-line y axis and the solid-line y axis is labeled with a lowercase Greek theta.
Rotated axes for an inclined plane. Consider complementary angles to determine the relationship between the angle \(\theta\) and the new (rotated) axes.

Now let’s return to the snakes, and find out their acceleration down the plane.

Example

You set a box of snakes at the top of a plane that is inclined at 30°. What is the box’s acceleration?

I’ll lay the coordinate axes directly on the FBD so that we can really see the relationship between the forces acting on the box and the angle of the incline. Also, I’ll define the \(+x\) direction as pointing down the ramp; this is just a matter of convenience so that the acceleration comes out positive.

FBD—Box of snakes
A right triangle is made by a dashed line. One leg of the triangle is horizontal and the other is vertical; the hypotenuse extends from upper right to lower left. The angle in the lower right corner, between the hypotenuse and the horizontal leg, is labeled with a lowercase Greek theta.

A spot on the hypotenuse is indicated with a dot. There are two thick arrows extending from this dot. One points up and to the right, perpendicular to the hypotenuse, labeled F superscript N. The other thick arrow points straight down, labeled F superscript G.

There are also two thin arrows showing coordinate axes whose origin is the dot on the hypotenuse. The x axis lies along the hypotenuse, pointing down and to the right. The y axis is perpendicular to the hypotenuse, pointing up and to the right. The angle between the y axis and the thick arrow pointing straight down is labeled with a lowercase Greek theta.
Free body diagram for a box of snakes on an inclined plane with minimal friction; the coordinate axes are rotated.
Newton’s second law

\[
\begin{align*}
F_\textit{net,x} &= ma_x \\
F^G_x &= ma_x \\
F^G\sin\theta &= ma_x \\
mg\sin\theta &= ma_x \\
a_x &= g\sin\theta \\
&= \left(9.81\ \textrm{m/s}^2\right)\sin(30^\circ) \\
&= 4.9\ \textrm{m/s}^2
\end{align*}
\]

As an aside, notice that the acceleration of the box only depends on the angle of the incline—it is independent from the mass of the box! The 17th century scientist Galileo actually did experiments on objects rolling down inclined planes and extrapolated his results to make important discoveries about objects in free fall. We’ll revisit free fall in chapter 12.

Example

A force with a magnitude of 95 N applied parallel to the surface of a 20° ramp will push a 12 kg mailbag up the ramp at constant speed. What is the magnitude of the kinetic friction between the bag and the ramp? What force parallel to the ramp will push it down the ramp at a constant speed?

From the information given, we can determine the magnitude of the force of friction that acts between the mailbag and the ramp. With that knowledge, we can find the force required to push the bag down the ramp.

FBD—Mailbag
A right triangle is made by a dashed line. One leg of the triangle is horizontal and the other is vertical; the hypotenuse extends from upper right to lower left. The angle in the lower right corner, between the hypotenuse and the horizontal leg, is labeled with a lowercase Greek theta.

A spot on the hypotenuse is indicated with a dot. There are four thick arrows extending from this dot. One points up and to the right, perpendicular to the hypotenuse, labeled F superscript N. One points straight down, labeled F superscript G. One points down and to the right, along the hypotenuse, labeled F superscript f. One points up and to the left, along the hypotenuse, labeled F superscript app.

There are also two thin arrows showing coordinate axes whose origin is the dot on the hypotenuse. The x axis lies along the hypotenuse, pointing down and to the right. The y axis is perpendicular to the hypotenuse, pointing up and to the right. The angle between the y axis and the thick arrow pointing straight down is labeled with a lowercase Greek theta.
Free body diagram for a mailbag pushed up an inclined plane. Rotated coordinate axes are shown.
Newton’s second law

Because the bag is traveling at a constant speed, its acceleration is zero. The applied force is in the negative \(x\) direction and the friction—opposing the motion of the mailbag—points in the positive \(x\) direction. We also need to consider the \(x\) component of the weight of the bag.

\[
\begin{align*}
F_\textit{net,x} &= ma_x \\
F^f – F^\textit{app} + F^G_x &= m(0) \\
F^f – F^\textit{app} + F^G\sin\theta &= 0 \\
F^f &= F^\textit{app} – mg\sin\theta \\
&= (95\ \textrm{N}) – (12\ \textrm{kg})\left(9.81\ \textrm{m/s}^2\right)\sin(20^\circ) \\
&= 54.7\ \textrm{N}
\end{align*}
\]

Now that we know the friction, let’s look at the forces when the bag is being pushed down the ramp. The free body diagram will look very similar, but with the applied force and the force of friction pointing in opposite directions as before.

\[
\begin{align*}
F_\textit{net,x} &= ma_x \\
F^\textit{app} – F^f + F^G_x &= m(0) \\
F^\textit{app} – F^f + F^G\sin\theta &= 0 \\
F^\textit{app} &= F^f – mg\sin\theta \\
&= (54.7\ \textrm{N}) – (12\ \textrm{kg})\left(9.81\ \textrm{m/s}^2\right)\sin(20^\circ) \\
&= 14.4\ \textrm{N}
\end{align*}
\]

example

A 205 kg log is pulled up a ramp by means of a rope that is parallel to the surface of the ramp. The ramp is inclined at 30° with respect to the horizontal. The kinetic frictional force is 1700 N, and the log has an acceleration of 0.8 m/s2. If your rope is rated to 2000 N, will you be able to pull the log with this rope?

FBD—Log
A right triangle is made by a dashed line. One leg of the triangle is horizontal and the other is vertical; the hypotenuse extends from upper right to lower left. The angle in the lower right corner, between the hypotenuse and the horizontal leg, is labeled with a lowercase Greek theta.

A spot on the hypotenuse is indicated with a dot. There are four thick arrows extending from this dot. One points up and to the right, perpendicular to the hypotenuse, labeled F superscript N. One points straight down, labeled F superscript G. One points down and to the right, along the hypotenuse, labeled F superscript f. One points up and to the left, along the hypotenuse, labeled F superscript T.

There are also two thin arrows showing coordinate axes whose origin is the dot on the hypotenuse. The x axis lies along the hypotenuse, pointing up and to the left. The y axis is perpendicular to the hypotenuse, pointing up and to the right. The angle between the y axis and the thick arrow pointing straight down is labeled with a lowercase Greek theta.
Free body diagram for a log being pulled up an inclined plane. Rotated coordinate axes are shown.

Note that I have defined \(+x\) to point up the ramp this time; it is generally most convenient to define \(+x\) in the direction of motion.

Newton’s second law

\[
\begin{align*}
F_\textit{net,x} &= ma_x \\
F^T – F^f – F^G_x &= ma_x \\
F^T – F^f – F^G\sin\theta &= ma_x \\
F^T &= ma_x + F^f + mg\sin\theta \\
&= (205\ \textrm{kg})\left(0.8\ \textrm{m/s}^2\right) + (1700\ \textrm{N}) + (205\ \textrm{kg})\left(9.81\ \textrm{m/s}^2\right)\sin(30^\circ) \\
&= 2.87\ \textrm{kN}
\end{align*}
\]

The tension in the rope would need to be greater than 2 kN, so this rope would break. You need a stronger rope.

Practice