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A rubber toy duck is at rest on an inclined plane. When the angle of inclination of the plane is increased to 36.0°, the toy duck begins to slide down the incline. Then, when the angle is decreased to 29.0°, the speed of the toy duck is constant. What are the coefficients of static and kinetic friction between the toy duck and the incline?

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Answer

toy begins to slide when the plane is at an angle of 36.0°

at this point the toy has just overcome the static friction

[tex]f_s = \mu_s N[/tex]

[tex]f_s = \mu_s mgcos \theta[/tex]

since toy begins to slide

[tex]f_s = mgsin \theta[/tex]

now on comparing

[tex] mgsin \theta = \mu_s mgcos \theta[/tex]

[tex]\mu_s = tan \theta[/tex]

[tex]\mu_s = tan (36^0)[/tex]

[tex]\mu_s = 0.726[/tex]

as the body is moving at constant velocity acceleration will be zero

so now,

[tex]m g sin \theta - f_k = ma[/tex]

[tex]m g sin \theta - \mu_k mg cos\theta = 0[/tex]

[tex]\mu_k = tan \theta[/tex]

[tex]\mu_k = tan (29^0)[/tex]

[tex]\mu_k = 0.554[/tex]

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