A box is sitting on a 2 m long board at one end. A worker picks up the board at the end with the box so it makes an angle with the ground of 35o. The coefficient of static friction between the box and the board is 0.5 and the coefficient of kinetic friction is 0.3 . Will the box slide down the ramp when it is at 35o?

Yes it will slide down the ramp

Explanation:

Let m be the mass of the box and gravitational acceleration g = 9.81m/s2, we can calculate the gravity that affects the box

W = mg

When the box is at 35 degree incline, this gravity is split into 2 components, 1 parallel to the incline (Wsin35) and the other one perpendicular with the incline (Wcos35).

The one perpendicular with the incline has an equal and opposite normal force of Wcos35

This normal force would dictate the static friction force where coefficient = 0.5. So the static friction is 0.5mgcos35

The box would slide when the parallel component of gravity wins over static friction force

mgsin35 > 0.5mgcos35

Since mg is positive we can cancel them out on both sides

sin35 > 0.5cos35

0.57 > 0.5*0.82

0.57 > 0.41

This is true so we can conclude that the box slides down the ramp

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