Three Boxes are located on three inclined surfaces with the

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.

Solution

Here,

theta1 , theta2 , theta3

for the angle theta1

a1 = g * sin(theta1) - u * g * cos(theta1) ----(1)

for the angle theta2

a2 = g * sin(theta2) - u * g * cos(theta2)

for the angle theta3

0 = g * sin(theta3) - u * g * cos(theta3)

tan(theta3) = u

from 1

u = (g * sin(theta1) - a1 )/(g * cos(theta1))

putting in the equation

tan(theta3) = (g * sin(theta1) - a1 )/(g * cos(theta1))

theta3 = arctan((g * sin(theta1) - a1 )/(g * cos(theta1)))

the expression for the third angle is arctan((g * sin(theta1) - a1 )/(g * cos(theta1)))

from 2

u = (g * sin(theta2) - a2 )/(g * cos(theta2))

putting in the equation

tan(theta3) = (g * sin(theta2) - a2 )/(g * cos(theta2))

theta3 = arctan((g * sin(theta1) - a2 )/(g * cos(theta2)))

the expression for the third angle is arctan((g * sin(theta2) - a2 )/(g * cos(theta2)))