5.7 Free-Body Diagrams

Having discussed at length what is and is not a force, we are ready to assemble our knowledge about force and motion into a single diagram called a free-body diagram. You will learn in the next chapter how to write the equations of motion directly from the free-body diagram. Solution of the equations is a mathematical exercise—possibly a difficult one, but nonetheless an exercise that could be done by a computer. The physics of the problem, as distinct from the purely calculational aspects, are the steps that lead to the free-body diagram.

A free-body diagram, part of the pictorial representation of a problem, represents the object as a particle and shows all of the forces acting on the object.

TACTICS BOX 5.3 Drawing a free-body diagram

1 Identify all forces acting on the object. This step was described in Tactics Box 5.2.

2 Draw a coordinate system. Use axes that match the direction of motion.

3 Represent the object as a dot at the origin of the coordinate axes. This is the particle model.

4 Draw vectors representing each of the identified forces. This was described in Tactics Box 5.1. Be sure to label each force vector.

5 Draw and label the net force vector F⃗net. Draw this vector beside the diagram, not on the particle. Or, if appropriate, write F⃗net=0⃗. Then check that F⃗net points in the same direction as the acceleration vector a⃗ on your motion diagram.

Workbook Exercises 24–29

EXAMPLE 5.4 An elevator accelerates upward

An elevator, suspended by a cable, speeds up as it moves upward from the ground floor. Identify the forces and draw a free-body diagram of the elevator.

MODEL Model the elevator as a particle.

VISUALIZE

FIGURE 5.21 Free-body diagram of an elevator accelerating upward.

A sketch of a person standing in an elevator suspended by a cable. Step 1: Identify all forces acting on the object.
The figure illustrates the steps for creating a free-body diagram of the forces acting on an elevator using the x y coordinate system.

REVIEW The coordinate axes, with a vertical y-axis, are the ones we would use in a pictorial representation of the motion. The elevator is accelerating upward, so F⃗net must point upward. For this to be true, the magnitude of T⃗ must be larger than the magnitude of F⃗G. The diagram has been drawn accordingly.

EXAMPLE 5.5 An ice block shoots across a frozen lake

Bobby straps a small model rocket to a block of ice and shoots it across the smooth surface of a frozen lake. Friction is negligible. Draw a pictorial representation of the block of ice.

MODEL Model the block of ice as a particle. The pictorial representation consists of a motion diagram to determine a⃗, a force-identification picture, and a free-body diagram. The statement of the situation implies that friction is negligible.

VISUALIZE

FIGURE 5.22 Pictorial representation for a block of ice shooting across a frictionless frozen lake.

The figure illustrates the motion diagram, the force-identification picture, and the free-body diagram of a block of ice shooting across a frictionless frozen lake.

REVIEW The motion diagram tells us that the acceleration is in the positive x-direction. According to the rules of vector addition, this can be true only if the upward-pointing n⃗ and the downward-pointing F⃗G are equal in magnitude and thus cancel each other. The vectors have been drawn accordingly, and this leaves the net force vector pointing toward the right, in agreement with a⃗ from the motion diagram.

EXAMPLE 5.6 A skier is pulled up a hill

A tow rope pulls a skier up a snow-covered hill at a constant speed. Draw a pictorial representation of the skier.

MODEL This is Example 5.2 again with the additional information that the skier is moving at constant speed. The skier will be modeled as a particle in mechanical equilibrium. If we were doing a kinematics problem, the pictorial representation would use a tilted coordinate system with the x-axis parallel to the slope, so we use these same tilted coordinate axes for the free-body diagram.

VISUALIZE

FIGURE 5.23 Pictorial representation for a skier being towed at a constant speed.

The figure illustrates the motion diagram, the force-identification picture, and the free-body diagram of a skier being towed uphill at a constant speed.

REVIEW We have shown T⃗ pulling parallel to the slope and f⃗k, which opposes the direction of motion, pointing down the slope. n⃗ is perpendicular to the surface and thus along the y-axis. Finally, and this is important, the gravitational force F⃗G is vertically downward, not along the negative y-axis. In fact, you should convince yourself from the geometry that the angle θ between the F⃗G vector and the negative y-axis is the same as the angle θ of the incline above the horizontal. The skier moves in a straight line with constant speed, so a⃗=0⃗ and, from Newton’s second law, F⃗net=0⃗. Thus we have drawn the vectors such that the y-component of F⃗G is equal in magnitude to n⃗. Similarly, T⃗ must be large enough to match the negative x-components of both f⃗k and F⃗G.

Free-body diagrams will be our major tool for the next several chapters. Careful practice with the workbook exercises and homework in this chapter will pay immediate benefits in the next chapter. Indeed, it is not too much to assert that a problem is half solved, or even more, when you complete the free-body diagram.