How To Calculate Acceleration With Friction

As a force that opposes motion, friction always reduces acceleration. Friction occurs between the interaction of an object against a surface. Its magnitude depends on the characteristics of both the surface and the object, and whether the object is moving or not. Friction may be the result of an interaction between two solid objects, but it doesn't have to be. Air drag is a type of frictional force, and you could even treat the interaction of a solid body moving on or through water as a frictional interaction.

Force is a vector quantity, which means you must consider the direction in which it acts. Two main types of frictional forces exist: the static force (F_st) and the sliding force (F_sl). Even though they act in the direction opposite to that in which an object moves, the normal force (F_N) produces these forces, which acts perpendicular to the direction of motion. F_N is equal to the weight of the object plus any additional weights. For example, if you press down on a block of wood on a table, you increase the normal force, and thus it increases the frictional force.

Both static and sliding friction depend on the characteristics of the moving body and the surface along which it moves. These characteristics are quantified in the coefficients of static (µ_st) and sliding (µ_sl) friction. These coefficients are dimensionless and have been tabulated for many common items and surfaces. Once you find the one that applies in your situation, you calculate the frictional forces using these equations:

\(F_{st}\leq\mu_{st}F_N\text{ }\F_{sl}=\mu_{sl}F_N\)

Newton's second law says that the acceleration of an object (a) is proportional to the force (F) applied on it, and the proportionality factor is the object's mass (m). If you're interested in acceleration, rearrange the equation to read:

\(a=\frac{F}{m}\)

Force is a vector quantity, which means you must consider the direction in which it acts. Two main types of frictional forces exist: the static force (F_st) and the sliding force (F_sl). Even though they act in the direction opposite to that in which an object moves, the normal force (F_N) produces these forces, which acts perpendicular to the direction of motion. F_N is equal to the weight of the object plus any additional weights. For example, if you press down on a block of wood on a table, you increase the normal force, and thus it increases the frictional force.

The total force (F) on an object subject to friction is equal to the sum of the applied force (F_app) and the frictional force (F_fr). But since the frictional force opposes motion, it's negative relative to the forward force, so:

\(F=F_{app}-F_{fr}\)

The friction force is the product of the friction coefficient and the normal force, which ​in the absence of extra downward forces​, is the weight of the object. Weight (w) is defined as the mass (m) of an object times the force of gravity (g):

\(F_N=w=mg\)

You're now ready to calculate the acceleration of an object of mass (m) subject to an applied force F_app and a frictional force. Since the object is moving, you use the coefficient of sliding friction to get this result:

\(a=\frac{F_{app}-\mu_{sl}mg}{m}\)