Chapter 4, Problem 50P

(a) A box sits at rest on a rough 33° inclined plane. Draw the tree-body diagram, showing all the forces acting on the box. (b) How would the diagram change if the box were sliding down the plane? (c) How would it change if the box were sliding up the plane after an initial shove?

To determine

Part (a) To Determine:

Free body diagrams depicting all the forces acting on a box is placed on an inclined plane when the box is at rest

Answer to Problem 50P

Solution:

When the box is at rest

Explanation of Solution

When the box is at rest on a plane inclined at an angle $\theta =33°$ to the horizontal, the free body diagram is as shown below.

The weight mg of the box acts vertically downwards. The normal force $F_N$ acts perpendicular to the inclined plane. The weight mg is resolved into two components, $mg\sin \theta \text{ and }mg\cos \theta$ parallel and perpendicular to the inclined plane. Since the box is at rest, the force of friction $F_{fr}$ acts upwards parallel to the inclined plane to counter the downward force $mg\sin \theta$.

    $F_{fr}=mg\sin \theta$

To determine

Part (b) To Determine:

Free body diagrams depicting all the forces acting on a box is placed on an inclined plane when the box slides down the plane.

Answer to Problem 50P

Solution:

When the box slides down

Explanation of Solution

when the box moves down, the free body diagram is as shown below.

The force of kinetic friction $F_{fr}$ acts opposite to the direction of motion of the box. There is a net force acting on the box in the downward direction parallel to the inclined plane.

    $F_{fr}<mg\sin \theta$

To determine

Part (c) To Determine:

Free body diagrams depicting all the forces acting on a box is placed on an inclined plane when the box moves up the plane following an initial shove.

Answer to Problem 50P

Solution:

When the box moves up the plane

Explanation of Solution

(c)If the box is given an initial shove up the plane, then the free body diagram is as shown below.

Since the force of friction always acts opposite to the direction of motion of the box, in this case, the force of friction acts in the downward direction, parallel to the inclined plane.