This problem requires us to determine the static and kinetic coefficients of friction.

Whenever there’s an inclined plane problem where something “starts to slip” or “begins sliding”, this involves the **Critical Angle**, which we have an easy equation for. Let’s check it out….

We will follow these steps:

- Draw a
**free-body diagram**of the eraser. - Resolve the weight component and write the
**ΣF=ma**equations. - Solve for the target variables.

**Step 1:** We need to represent the forces acting on the eraser in a free-body diagram as shown below.

From the free-body diagram, the ** x** component of the gravitational force,

$\mathit{N}\mathbf{=}\mathit{m}\mathit{g}\mathbf{}\mathbf{cos}\mathit{\theta}$is the normal force between the eraser and the incline.

The friction force is equal to the product of the normal force and the coefficient of friction, $\overline{){\mathit{f}}{\mathbf{=}}{\mathit{\mu}}{\mathit{N}}}$

For a stationary body, we use the static coefficient, **µ**_{s}, of friction. Static friction, **f**_{s}, is equal to the maximum force applied on an object before it starts moving.

For an object in motion, we use the kinetic coefficient, **µ**_{k}, of friction. Kinetic friction, **f**_{k}, is equal to the force required to keep an object moving with constant speed.

We are going to break the problem into two parts: **static** coefficient of friction and **kinetic** coefficient of friction.

**Step 2: **Write the **ΣF=ma** equations

**(a) **Determining the static coefficient of friction

To find the coefficient of friction, we determine the magnitude of forces acting on the eraser. The x component of gravitational force ** mg sinθ**, acts on the eraser down the incline while the frictional force, ${\mathit{f}}{\mathbf{=}}{\mathit{\mu}}{\mathit{N}}$, acts on the eraser up the slope.

To determine the coefficients of friction between rubber and various surfaces, a student uses a rubber eraser and an incline. In one experiment, the eraser begins to slip down the incline when the angle of inclination is 36.0° and then moves down the incline with constant speed when the angle is reduced to 30.0°. From these data, determine the coefficients of

(a) static friction, and

(b) kinetic friction

for this experiment.