Three Boxes are located on three inclined surfaces with the friction. The fricti

Question

Three Boxes are located on three inclined surfaces with the friction. The friction coefficients are identical. Angles of inclined surfaces with the horizontal direction Theta_1 Theta_2, and Theta_3 are different. On the inclined surface with the angle Theta_1, the acceleration of the box is a_1. On the incline surface with the angle Theta_2, the acceleration is a_2. The motion on the third surface happens without any acceleration. Find an expression for third angle in terms of Theta_1, a_1, Theta a_2.

Explanation / Answer

equation of motion of the first plane

mg * sin(theta1) - k * mg * cos(theta1) = ma1

sin(theta1) - k * cos(theta1) = a1

k = tan(theta1) - a1 * sec(theta1)

equation of motion of second plane

mg * sin(theta2) - k * mg * cos(theta2) = ma2

sin(theta2) - k * cos(theta2) = a2

k = tan(theta2) - a2 * sec(theta2)

equation of motion of third plane

mg * sin(theta3) - k * g * cos(theta3) = 0

sin(theta3) - k * cos(theta3) = 0

k = tan(theta3)

tan(theta1) - a1 * sec(theta1) = tan(theta3)

theta3 = tan^-1(tan(theta1) - a1 * sec(theta1))

or

tan(theta2) - a2 * sec(theta2) = tan(theta3)

theta3 = tan^-1(tan(theta2) - a2 * sec(theta2))

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