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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