164

8.3The Force Vector and Simple Machines

Force is relatively easy to intuit as a vector. The force vector points in

the direction in which it is trying to accelerate the object it is acting on.

Since force vectors are so much easier to visualize than acceleration

vectors, it is often helpful to first find the direction of the (total) force

vector acting on an object, and then use that information to determine the

direction of the acceleration vector. Newton’s second law, F

total

=ma, tells us

that the two must be in the same direction.

An important application of force vectors is to analyze the forces acting

in two-dimensional mechanical systems, as in the following example.

Example: pushing a block up a ramp

Question: Figure (a) shows a block being pushed up a friction-

less ramp at constant speed by an applied force F

A

. How much

force is required, in terms of the block’s mass, m, and the angle

of the ramp,

.

.

Solution: Figure (b) shows the other two forces acting on the

block: a normal force, F

N

, created by the ramp, and the weight

force, F

W

, created by the earth’s gravity. Because the block is

being pushed up at constant speed, it has zero acceleration, and

the total force on it must be zero. From figure (c), we find

|F

A

|= |F

W

| sin

.

= mg sin

.

.

Since the sine is always less than one, the applied force is always less than

mg, i.e. pushing the block up the ramp is easier than lifting it straight up.

This is presumably the principle on which the pyramids were constructed:

the ancient Egyptians would have had a hard time applying the forces of

enough slaves to equal the full weight of the huge blocks of stone.

Essentially the same analysis applies to several other simple machines,

such as the wedge and the screw.

Discussion Questions

A. The figure shows a block being pressed diagonally upward against a wall,

causing it to slide up the wall. Analyze the forces involved, including their

directions.

Discussion question A.

(a) The applied force F

A

pushes the

block up the frictionless ramp.

(b) Three forces act on the block. Their

vector sum is zero.

(c) If the block is to move at constant

velocity, Newton’s first law says that

the three force vectors acting on it

must add up to zero. To perform vec-

tor addition, we put the vectors tip to

tail, and in this case we are adding

three vectors, so each one’s tail goes

against the tip of the previous one.

Since they are supposed to add up to

zero, the third vector’s tip must come

back to touch the tail of the first vec-

tor. They form a triangle, and since the

applied force is perpendicular to the

normal force, it is a right triangle.

F

A

.

F

W

F

N

F

A

.

F

A

F

W

F

N

.

Chapter 8Vectors and Motion