Applications of Friction

Angle of Friction

The angle of friction is the angle that the resultant of the limiting friction and the normal force makes with the normal force. It is denoted by $\lambda$.

$$\tan(\lambda) = \frac{f_{s,max}}{N} = \frac{\mu_s N}{N} = \mu_s$$

So, the coefficient of static friction is equal to the tangent of the angle of friction.

Cone of Friction

The cone of friction is the cone generated by revolving the resultant of the normal force and the limiting friction force about the normal force. If the resultant of the applied forces is within the cone of friction, the object will remain in equilibrium.

Angle of Repose

The angle of repose is the maximum angle of an inclined plane at which a body can remain at rest, held by friction. Beyond this angle, the body will start to slide down. It is denoted by $\alpha$.

At the angle of repose, the component of gravity pulling the object down the incline is equal to the maximum static friction.

$$mg \sin(\alpha) = f_{s,max} = \mu_s N = \mu_s mg \cos(\alpha)$$

$$\tan(\alpha) = \mu_s$$

Thus, the angle of repose is equal to the angle of friction.

Solving Friction Problems

Here is a general strategy for solving problems that involve friction:

1. Draw a Free-Body Diagram (FBD): Draw a clear FBD for each object in the system. Show all forces acting on the object, including gravity, normal forces, applied forces, and friction.
2. Choose a Coordinate System: Choose a convenient coordinate system. It is often helpful to align one axis with the direction of motion or the inclined surface.
3. Apply Newton's Second Law: Apply Newton's Second Law ($\Sigma F = ma$) to each object, resolving the forces into their components along the chosen axes.
4. Analyze the Friction Force:
   • If the object is at rest, use the condition of static equilibrium ($\Sigma F = 0$). The static friction force will be equal and opposite to the net applied force, up to its maximum value, $f_{s,max} = \mu_s N$.
   • If the object is on the verge of moving, the static friction is at its maximum: $f_s = f_{s,max} = \mu_s N$.
   • If the object is moving, use the kinetic friction force, $f_k = \mu_k N$. The direction of kinetic friction is always opposite to the direction of velocity.
5. Solve the Equations: Solve the resulting system of equations for the unknown quantities.

Example Problem

A 10 kg block is pulled across a horizontal surface by a force of 50 N at an angle of 30° above the horizontal. If the coefficient of kinetic friction is 0.2, what is the acceleration of the block?

Solution:

1. Draw the FBD: The forces acting on the block are gravity (mg), the normal force (N), the applied force (P). Kinetic friction (f_k).

Advantages and Disadvantages of Friction

Friction is often called a 'necessary evil' because it has both useful and harmful effects.

Advantages of Friction

• Walking: Friction between our shoes and the ground allows us to walk without slipping.