Prop up one end of a piece of glass to form an inclined plane with a gradual angle. Start with an angle that isn't quite steep enough for a quarter to slide down. Place a quarter at the top of the incline and shove it just enough to set it in motion. It should come to a stop if the angle was set right. Gradually increase the angle and continue to test it with the quarter. Repeat until you find the angle where a small shove causes the quarter to slide all the way to the bottom without stopping. Its descent should be slow without any apparent acceleration.
Divide the rise of the inclined plane by the run. This is the coefficient of kinetic friction. For example, if the vertical height to the top of the incline is 17 centimeters, and the horizontal distance from the bottom of the incline to the point directly below the top is 58 centimeters, the coefficient of kinetic friction is 0.29.
Subtract the diameter of the quarter from the length of the inclined plane. This corresponds to the distance travelled by the quarter if the starting point is the top edge of the quarter flush with the top edge of the incline, and the ending point is the bottom edge flush with the bottom edge. For the example, use 58 centimeters.
Increase the angle so the inclined plane is steep enough that the quarter will slide freely without a shove. The exact angle of the incline isn't important, because that is what you will be deriving. Place the quarter at the starting point and release it without pushing it. Use a stopwatch to time how long it takes to travel to the ending point. For the example, use 1.5 seconds.
Calculate the acceleration of the quarter. Use the formula a = 2*d/(t^2), where "a" is acceleration, "d" is distance and "t" is time. For example, convert centimeters to meters and find a = (2*0.58)/(1.5^2) = 0.52 m/(sec^2).
Write an equation for the net force acting on the quarter. The force due to gravity can be divided into the component parallel to the incline, mg*sin(A), and the component perpendicular to the incline, mg*cos(A), where "m" is mass, "g" is the gravitational constant and "A" is the angle of the incline. Multiplying the perpendicular component times the coefficient of friction gives the frictional force, umg*cos(A), where "u" is the coefficient of friction. Therefore, the net force acting on the quarter is F = mg*sin(A) - umg*cos(A). Set that equal to the mass of the quarter times its acceleration: mg*sin(A) - umg*cos(A) = ma. Cancel out "m" and simplify to sin(A) - u*cos(A) = a/g.
Calculate the angle of the inclined plane by solving for A. First, define an angle B such that cos(B) = 1/R and sin(B) = u/R. Rewrite the previous expression as R(sin(A)/R - u*cos(A)/R) = a/g. Substitute to find R(sin(A)*cos(B) - cos(A)*sin(B)) = a/g, which simplifies to R(sin(A - B)) = a/g. According to the definition of B, R is the hypotenuse of the triangle whose other two sides have lengths 1 and u, so R = sqrt(u^2 + 1), and B = arccos(1/R). Therefore, A = arcsin(a/(gR)) + arccos(1/R). Returning to the given example, R = sqrt(0.29^2 + 1) = 1.04; therefore A = arcsin(0.52/(9.8*1.04)) + arccos(1/1.04) = 2.92 + 15.9 = 18.8 degrees.