- Uniform Circular Motion is best understood using the concepts of vector geometry.
So are many other concepts from physics, where quantities have a direction and a "strength" -- a rocket in flight, for example. It is moving in a certain direction, and it has a speed, whose value can be represented by the length of a vector.
- vector: a quantity with length and direction:
- In uniform circular motion, we could represent the position of the bob by
and
- The three vectors that are (most) important in the physics
of circular motion are the vectors of
- position,
- velocity, and
- acceleration,
each of which is a function of time.
- Animations, demonstrations:
- Uniform Circular Motion
- For all of the gory details, a nice layout of
all the physics, and a cool (but scary) example, let's
check out physclips
-- the Australian dialect makes it more fun, too!
- If a vector is in two-dimensions, then the length
is just given using the Pythagorean theorem formula:
in which case
Obviously this can be generalized to vectors in
four dimensions, five dimensions; even six
dimensions. Maybe seven dimensions, too; or even
n dimensions.
- The vectors above are given with components:
examples:
- #5, p. 663
- #9, p. 663
- #11, p. 663
- Figure 11 illustrates the so-called parallelogram
rule:
example:
Here it is in three-space:
with the obvious changes to formulas because you now have three
components, instead of two:
- Vectors with length 1 are special, called unit
vectors, and are important because they can be dedicated to
showing direction:
It's clear that we can turn any vector into a unit vector, by simply scaling it:
example:
- We use special components and the parallelogram rule to
write each vector in terms of a basis of vectors (equal
in number to the dimension of the space): for example, in
two-space, the usual basis is given by the two (perpendicular)
vectors denoted i and j:
and, in three-space,
- Theorems:
The Triangle inequality basically says that the diagonal of a
parallelogram is shorter than or equal to the sum of the two
sides:
"The shortest distance between two points is a straight
line."
In the case of Figure 17 (for vectors at right angles to each
other)
this rule is familiar in the form of the old algebraic
inequality
and the only time you'll have equality is when either a
or b is zero.
- Let's look at some more examples:
- #19, p. 664
- #29, p. 664
- #47, p. 664
- More physics:
- #58, p. 665
- First and foremost: Newton's laws of motion (in particular ), and statics.
- The parallelogram rule
- We can think of the force vectors on the cables as being
composed of vector components (as well as
coordinate components).
- #60, p. 665
- Let's use a little calculus and the parameterization of the position vector in uniform circular motion given in one of our videos
to derive the velocity and acceleration vectors, and check the video's claims.