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# What are inclines?

Surfaces usually aren't perfectly horizontal. Learn how to deal with slopes!

## What are inclines?

Slides at the park, steep driveways, and shipping truck loading ramps are all examples of inclines. Inclines or inclined planes are diagonal surfaces that objects can sit on, slide up, slide down, roll up, or roll down.
Inclines are useful since they can reduce the amount of force required to move an object vertically. They're considered one of the six classical simple machines.

## How do we use Newton's second law when dealing with inclined planes?

In most cases, we solve problems involving forces by using Newton's second law for the horizontal and vertical directions. But for inclines, we're typically concerned with the motion parallel to the surface of an incline so it's often more useful to solve Newton's second law for the directions parallel to and perpendicular to the inclined surface.
This means that we will typically be using Newton's second law for the directions perpendicular $\perp$ and parallel $\parallel$ to the surface of the inclined plane.
${a}_{\perp }=\frac{\mathrm{\Sigma }{F}_{\perp }}{m}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{a}_{\parallel }=\frac{\mathrm{\Sigma }{F}_{\parallel }}{m}$
Since the mass typically slides parallel to the surface of the incline, and does not move perpendicular to the surface of the incline, we can almost always assume that ${a}_{\perp }=0$.

## How do we find the $\perp$‍  and $\parallel$‍  components of the force of gravity?

Since we're going to be using Newton's second law for the directions perpendicular to and parallel to the surface of the incline, we'll need to determine the perpendicular and parallel components of the force of gravity.
The components of the force of gravity are given in the diagram below. Be careful, people often mix up whether they should use $\text{sine}$ or $\text{cosine}$ for a given component.

## What is the normal force ${F}_{N}$‍  for an object on an incline?

The normal force ${F}_{N}$ is always perpendicular to the surface exerting the force. So an inclined plane will exert a normal force perpendicular to the surface of the incline.
If there is to be no acceleration perpendicular to the surface of the incline, the forces must be balanced in the perpendicular direction. Looking at the forces shown below, we see that the normal force must equal the perpendicular component of the force of gravity, to ensure that the net force is equal to zero in the perpendicular direction.
In other words, for an object sitting or sliding on an incline,
${F}_{N}=mg\mathrm{cos}\theta$

## What do solved examples involving inclines look like?

### Example 1: Snowy sliding sled

A child slides down a snowy hill on a sled. The angle the hill makes with respect with horizontal is $\theta ={30}^{o}$ and the coefficient of kinetic friction between the sled and the hill is ${\mu }_{k}=0.150$. The combined mass of the child and sled is .
What is the acceleration of the sled down the hill?
We'll start by drawing a force diagram.
We can use Newton's second law in the direction parallel to the incline to get,

### Example 2: Steep driveway

A person is building a house and wants to know how steep she can make her driveway and still park on it without slipping. She knows the coefficient of static friction between her tires and the concrete driveway is $0.75$.
What is the maximum angle from horizontal that the person can make her driveway and still park her car on it without slipping?
We'll start by using Newton's second law for the parallel direction.

## Want to join the conversation?

• Wouldn't the driveway be a case where we can't round up? At 37 degrees, the car would slip down the driveway, since the maximum was 36.86... degrees.
• If I were to build my drive way based on this math I would choose to round down though ;-)
• I don't understand the angle part. Is the angle of incline the same as the angle pictured between the mg and mgcos?
• Yes , the angle of incline and the angle between mg and mgcos are the same. Lets take a simple example. Say , let the angle of incline be 30 , then draw the mg force downwards so as it forms a right angled triangle including angle of incline 30 . In that triangle the other two angles will be 90 ( the right angled part ) and 60. Now draw the mgcos force from the same point as mg. We see the mgcos is perpendicular to the surface of incline , where one angle we already found out was 60 , therefore the angle between mg and mgcos has to be 90-60 = 30 , which is equal to the angle of incline.
draw the forces on the body on the incline , take this example and solve , your doubt will be dispelled. Thank you !
• I am literally so lost on the last example. Like how did sine theta get on the other side without being negative? Wouldn't you have had to subtract it from the side it was on with the cosine theta and the friction coefficient? Also why did we do tan negative one all of a sudden? I don't feel like there was a good explanation for that.
(1 vote)
• For the first question you asked, let's take a look.
0=sin(theta)-coefficient (cos theta) (hopefully this makes sense to you :/ )
-sin(theta)=-coefficient (cos theta)
sin(theta)=coefficient (cos theta) Is this makes sense to you? It should! :)
The next question was that why did (sin theta) / (cos theta) became tan(theta). right?
Sine theta is equal to opposite/hypotenuse. Cosine theta is equal to adjacent/hypotenuse. If you do (sin theta) / (cos theta), the hypotenuse thing will cancel out and it is left with opposite/adjacent, which is same as tangent(theta)! Yay!
Hopefully, this makes sense to you. It took a long time to type all these.
• In example 1, wouldn't all the snow piling up in front of the sled change things?
• It would but there is no indication in the question about snow piling up . So, you don't need to think about that when solving this problem.
• Why do you cancel both the masses in the numerator for 1 mass in the denominator?
• When you have something of the form (a*x + a*y)/(a*z) this is equivelent to ((a*x)/(a*z)) + ((a*y)/(a*z)) and you can see that the a's in the numerator and denominator to the left of the + sign cancel as do the numerator and denominator to the right of the + sign giving you (x/z) + (y/z)) which is also equivalent to (x + y)/z
• Which way does the force of friction act on an inclined plane if the force acting on the mass is in a horizontal direction and the mass is moving?
• friction always opposes motion.
• can somebody explain why there should be max. steepness for max. static friction!