Physics Equations: Friction & Inclined Planes

By John Brennan

The force resisting movement for an object at rest is called static friction; for an object in motion, it's called kinetic or sliding friction. The equations you use to calculate both kinetic and static friction are very similar and are very simple for a horizontal surface. For a slope like the side of a hill, however, there's one other consideration you must take into account.

Basic Equation

You can estimate kinetic and static friction using the following simple equation: friction = coefficient x normal force. The coefficient depends on the two surfaces -- a marble surface in contact with a steel box, for example, would have a different coefficient than a zinc box in contact with a wooden surface. It also depends on whether the object is already in motion. If it is already in motion, the coefficient is usually smaller, and thus kinetic friction is generally less than the maximum force that static friction can exert.

Normal Force

The other component of this equation is the normal force. On a flat, horizontal surface, this is just equal to the object's weight. On an inclined plane, however, the force of gravity on the object is at an angle with respect to the surface, so the force pulling the object onto the surface is only equal to part of its weight. For an inclined plane, you can calculate this normal force using the weight and some right-triangle trigonometry.

Angles

The normal is an imaginary line running perpendicular to the surface on which the object rests -- hence the name normal force. The force of gravity, however, pulls straight down on the object, so it's at an angle to the inclined plane. If you imagine the force of gravity as the hypotenuse of a right triangle, the normal force will be the length of the side opposite to the hypotenuse, and the angle between hypotenuse and opposite will be equal to the angle of the inclined plane. From this, you can calculate that the normal force will be weight x cosine of angle, where cosine is a trigonometric function you can calculate using the button on your calculator.

Calculating Static Coefficients

The simplicity of this equation makes it fairly straightforward to measure coefficients of static friction for various kinds of materials. If you have a flat sheet of plastic, wood or any other material you want to investigate, you can place an object atop it then increase the angle between your inclined plane and the floor until the object starts to slide. Measure the angle between plane and floor once this happens and take its tangent of this angle (the TAN button on your calculator) and you'll have the coefficient of static friction.

About the Author

Based in San Diego, John Brennan has been writing about science and the environment since 2006. His articles have appeared in "Plenty," "San Diego Reader," "Santa Barbara Independent" and "East Bay Monthly." Brennan holds a Bachelor of Science in biology from the University of California, San Diego.