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Inclined planes review

Review the key concepts and skills for inclined planes, including how to write Newton's second law for forces parallel and perpendicular to the incline. 

Key terms

Term (symbol)Meaning
Inclined planeA tilted surface, sometimes called a ramp or incline.

How to write Newton’s second law for forces on an incline

1) Draw a free body diagram for the object (see Figure 3). Remember to rotate the coordinate axes to align with the incline (see Figure 1 below).
If there is any acceleration, it will typically be along the parallel axis (labeled ) of the incline.
The perpendicular axis (labeled ) typically has no acceleration and a=0.
Figure 1. Rotating the axes to align with a ramp at angle θ.
2) Write the Newton’s second law statement for the direction of interest.
ma=ΣF OR ma=ΣF
The perpendicular direction’s equation simplifies because a=0:
m(0)=ΣF0=ΣF
3) Substitute the sum of all the forces acting in the direction of interest ( or ) for ΣF. Use your free body diagram to identify which forces are acting in the direction of interest.
Sometimes a force is completely aligned in the parallel or perpendicular direction like normal force and friction.
Some forces have components in both the parallel and perpendicular direction, such as the force of gravity. In that case, the force should be broken down into the parallel and perpendicular components (see Figure 2 below) for substitution in the net force equations.
Figure 2. An object on an incline of angle θ has weight Fg components perpendicular and parallel to the incline.
The parallel and perpendicular components of weight are (Figure 2):
Fg=FgcosθFg=Fgsinθ

Common mistakes and misconceptions

People forget the force directions. The diagram below shows the forces on an object resting on an incline.
Figure 3. The direction of forces for an object on an incline.
  • Weight Fg is straight down.
  • Normal force FN pushes perpendicular to the incline.
  • Friction Ff acts parallel to the incline.

Learn more

To view a worked example of an object sliding down a ramp, watch our video of ice accelerating down a ramp.
To check your understanding and work toward mastering these concepts, check out the exercises on forces and inclined planes.

Want to join the conversation?

  • piceratops seed style avatar for user saintfort.dan
    Is there a video or a question that explains how friction is applied to an object in motion on an incline? Say if you wish to find the force applied to an object being pushed up an incline.
    (12 votes)
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  • blobby green style avatar for user Ruppert Rapport
    Why is acceleration ZERO on an inclined plane with no friction? You didn't explain that.
    (4 votes)
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  • female robot ada style avatar for user SDN 123
    Can a non contact force like mg be ever considered to be a contact force?
    (3 votes)
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  • sneak peak green style avatar for user Carolina Tarango
    is this in radians or degrees
    (2 votes)
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  • starky ultimate style avatar for user MichelleW.
    Is the angle of the incline always equal to the angle between Fg and Fgcos?
    (2 votes)
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  • blobby green style avatar for user Cherrie-lee P. Phillip
    How does it change when a force is applied horizontally to prevent the object (box) from sliding down the ramp? Does the magnitude of the normal force change while the box is being held in place on a frictionless slope? Thanks.
    (2 votes)
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  • blobby green style avatar for user michezohalisi
    An inclined plan makes an angle of 30° with the horizontal. Find the constant force, applied parallel to the inclined plan, required to cause a 20 kg box to slide I. down the incline with acceleration 1,00 m/s2?
    II. up the incline with acceleration 1,00 m/s2?
    (1 vote)
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  • leaf green style avatar for user donnerwills
    I don't understand why and how sine and cosine come into play. I know sine is a ratio of the opposite side to hypotenuse and cosine is adjacent to hypotenuse. But how does this give us our force of gravity? What about this ratio gives it to us? What about them makes it helpful in finding it and how come one of them uses sine and the other is cosine?
    (1 vote)
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    • orange juice squid orange style avatar for user Ridho Rizqullah
      If you have an inclined force (call that F1, and you know how big that force is and also the angle), horizontal force (Fx), and vertical force (Fy) (both vertical and horizontal force have unknown magnitude), try to make a rectangle with the horizontal and vertical force, you will see that the inclined force split the rectangle and form 2 triangle.You will see that :
      sin(theta) = Fy/F1 => rearrange : Fy = F1.sin(theta)
      cos(theta) = Fx/F1 => Fx = F1.cos(theta)
      using sin and cos help you to decompose an inclined force into an easier one which is horizontal and vertical force
      (2 votes)
  • aqualine ultimate style avatar for user Courtney
    In the Forces and Inclined Planes exercise, there's a question where an object on a friction-less incline of angle θ has force F exerted on it parallel to mgsinθ. It asked me to find the equation for force F. The answer was m(a-gsinθ), and I understand that that's "mass times (acceleration minus acceleration)", but I don't understand why. if F=ma, why does this formula seem to be F=ma-mgsinθ? Is it because it's acceleration minus acceleration, which would just be a new total for acceleration, or am I missing something else?
    (1 vote)
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  • blobby green style avatar for user Alex Forger
    It says to rotate the coordinate system but then the videos don't rotate the coordinate system, so I'm confused.
    (1 vote)
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