### Measurement of the Coefficient of Static Friction with an Inclined Plane

A block is resting over an incline plane as shown in the figure. The angle of the incline can be changed by pulling on the string. If the block only starts sliding when the angle is greater than qc.
What is the value of the static frictional force when the inclination angle is q smaller than qc?
 a) 0 b) N c) d) e) None of the above.

and the coefficient of static friction between the two surfaces

 a) N b) c) d) e) None of the above.

Solution:

When the angle of the inclined is q, smaller than qc, the block is at rest, first picture of sequence on the left, . The forces acting on the block are the weight of the block, the normal force, and the static friction. These forces are shown on the next figure of the sequence . Based on the direction of the forces, again it is convenient to select a coordinate system with the x-axis parallel to the inclined plane and the y-axis perpendicular to the inclined   . Thus, only the vector representing the weight of the block need to be broken down into components. Just as in the case of the inclined planed studied before, the angle of the incline appears between the bottom part of the y-axis and the vector representing the weight of the block . Accordingly, the resulting components of the weight are  , along the x-axis pointing to the right; and, along the y-axis pointing down, . Finally, the corresponding FBD for the block is the last figure of the sequence . In the present case, the block is at rest.

y-component:

From the diagram, the net force in the y-direction is

Applying Newton's second law to this component (no motion present)

From the two relations for the net force, the following equation and solution is obtained,

This result allows you to calculate the normal force at any angle.

x-component:

Also the net force in the x-direction is

In this case, the block is not sliding down the inclined

Equating the two relations, the following equation is obtained

The previous results indicates how the static frictional force adjust to mach the force pulling the block down depending on the angle of the inclined.

At q equals to qc the static frictional force is maximum, .

In the maximum case, the maximum static friction is related to the normal force by the relation,

Substituting the value of the normal force at the critical angle,

As calculated above, at any angle the static frictional force is given by which evaluated at the critical angle is . Thus,

From the previous two relations an equation for the coefficient of static friction is obtained,

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com