Michael Ro
Tue, Aug-13-02, 00:02
Someone wrote:
<I've seen Power expressed as two different formulas.
P=FxV and, with little explanation, P=(FxD)/T
My limited understanding of velocity would argue the second
equation should read P=F(D/T). Please, give me a hand here.>
*** Strictly speaking, one can only define Power accurately by
using calculus (another invention by the prolific Isaac Newton
and his anti-buddy Leibniz). In its most basic expression,
power is defined as the time rate of change of work, or in
mathematical notation:
P = d/dt (W) = dW/dt
.... where the symbol d/dt refers to the time derivative or
operation of calculating the slope of the Work-Time curve at a
given instant T.
Now, work done over a given displacement (a vector), in turn,
is defined as the area under the Force-Displacement curve over
that period (i.e. over a distance s, between times T1 and T2).
In mathematical form, this is done by calculating the
"integral" of F with respect to displacement, thus:
W = Integral of (F.ds) ...
where the word "Integral" is replaced by a sort of skinny
elongated "S" (because an integral originally was defined as a
special sort of Summation).
If Force F is constant in magnitude and direction throughout
the given displacement, then we may write:
W = F.s
Then, under those conditions of constant Force, we may write:
Power P = Work/Time = W/t = F.s/t
This, in turn, becomes P = F.V
or as you quoted in your letter:
P = FxD/T = F.V
It is perfectly valid to write this as FD/T, F(D/T) or (F/T).D
- all forms like this are identical, since a cross product of
two vectors is not involved (see below).
There are several problems associated with using this
simplistic rendition of the power formula:
1. Since work is defined over a displacement, which is a
vector, then movement that starts and ends in the same
place produces a value of zero, so that work done is zero.
For example, if we use that basic equation to determine
work done over a single full repetition of a squat or bench
press, the work as calculated from the external physics
will be ZERO. Obviously, if we examine the problem from the
point of view of internal metabolism, the answer is very
different. This is why physics is very careful to
distinguish between "closed" and "open" systems. If we
examine a biological system solely in terms of external
mechanics, we will be ignoring some other processes within
the system as a whole.
2. Since Force does not remain constant in any conventional
strength training situation, we are not justified in
reducing the equation to the form where F is assumed to act
with the same magnitude and in the same direction.
3. In using that simple physics, we are assuming that no heat
is being added to or lost from the system during the
exercise, which accurately speaking, is not the case.
That is why physicists often choose not to use that form
of the power and work equations and instead rely on
changes in energy.
Generally, physicists avoid using the symbol "x" for denoting
a product, because it usually refers to a special type of
mathematical product known as a "cross product" of two vectors
acting at an angle to one another and giving a resultant that
does not lie in the plane containing the two original vectors.
The "scalar product" denoted by a period (.) refers to the
normal sort of product that we find in arithmetic and
Euclidean geometry.
For further reading on this and other aspects of the basic
biomechanics of strength training, see Ch 1.2 of
"Supertraining" 2000 and thanks to Mel Siff for the above.
<I've seen Power expressed as two different formulas.
P=FxV and, with little explanation, P=(FxD)/T
My limited understanding of velocity would argue the second
equation should read P=F(D/T). Please, give me a hand here.>
*** Strictly speaking, one can only define Power accurately by
using calculus (another invention by the prolific Isaac Newton
and his anti-buddy Leibniz). In its most basic expression,
power is defined as the time rate of change of work, or in
mathematical notation:
P = d/dt (W) = dW/dt
.... where the symbol d/dt refers to the time derivative or
operation of calculating the slope of the Work-Time curve at a
given instant T.
Now, work done over a given displacement (a vector), in turn,
is defined as the area under the Force-Displacement curve over
that period (i.e. over a distance s, between times T1 and T2).
In mathematical form, this is done by calculating the
"integral" of F with respect to displacement, thus:
W = Integral of (F.ds) ...
where the word "Integral" is replaced by a sort of skinny
elongated "S" (because an integral originally was defined as a
special sort of Summation).
If Force F is constant in magnitude and direction throughout
the given displacement, then we may write:
W = F.s
Then, under those conditions of constant Force, we may write:
Power P = Work/Time = W/t = F.s/t
This, in turn, becomes P = F.V
or as you quoted in your letter:
P = FxD/T = F.V
It is perfectly valid to write this as FD/T, F(D/T) or (F/T).D
- all forms like this are identical, since a cross product of
two vectors is not involved (see below).
There are several problems associated with using this
simplistic rendition of the power formula:
1. Since work is defined over a displacement, which is a
vector, then movement that starts and ends in the same
place produces a value of zero, so that work done is zero.
For example, if we use that basic equation to determine
work done over a single full repetition of a squat or bench
press, the work as calculated from the external physics
will be ZERO. Obviously, if we examine the problem from the
point of view of internal metabolism, the answer is very
different. This is why physics is very careful to
distinguish between "closed" and "open" systems. If we
examine a biological system solely in terms of external
mechanics, we will be ignoring some other processes within
the system as a whole.
2. Since Force does not remain constant in any conventional
strength training situation, we are not justified in
reducing the equation to the form where F is assumed to act
with the same magnitude and in the same direction.
3. In using that simple physics, we are assuming that no heat
is being added to or lost from the system during the
exercise, which accurately speaking, is not the case.
That is why physicists often choose not to use that form
of the power and work equations and instead rely on
changes in energy.
Generally, physicists avoid using the symbol "x" for denoting
a product, because it usually refers to a special type of
mathematical product known as a "cross product" of two vectors
acting at an angle to one another and giving a resultant that
does not lie in the plane containing the two original vectors.
The "scalar product" denoted by a period (.) refers to the
normal sort of product that we find in arithmetic and
Euclidean geometry.
For further reading on this and other aspects of the basic
biomechanics of strength training, see Ch 1.2 of
"Supertraining" 2000 and thanks to Mel Siff for the above.