Re: Newton's 2nd Law problem

[Leigh Palmer <palmer@SFU.CA>]

> Newton's 2nd law:
>                    F = d(mv)/dt
> and can be expressed as:
>                    F = m dv/dt + v dm/dt
> In a question like what is the force/lift on a holicopter when the mass is
> "pushed" down at speed of v m s-1 and at a rate of m kg s-1, most texts have
> used:
>                F = v dm/dt
> Isn't the above equation for constant vel but changing mass?

You must be careful about the meaning of your symbols.
What mass is changing? There is no nuclear process going
on here.

dm/dt refers to the rate of air mass flow through the
rotor. v refers to the (change in) the downward velocity
of the air produced by the rotor

> My understanding is that the vel is accelerated by the rotor from a speed of
> 0 to v and is hence not constant.
> If I use         impulse = change in momentum
> to find the answer, it is definitely more convincing,
> since
>                    F dt = final mom - initial mom
>                            =    mv -0
> So                 F     =  mv/dt
> where the mass per unit time is known.
> I am confused, as both cases will still give the same answer. Can someone
> help?

You clearly understand the physical situation. Your only
confusion lies in the interpretation of that silly
formula (F = m dv/dt + v dm/dt) which is impossible
without a careful definition of the terms in it.

This is an example of how a student who might understand
is thrown by a "formula" which looks like an easy way to
get at the answer. The formula itself is dangerous.
Standing alone it can be a barrier to understanding. My
advice is that you never use it, but rather let it arise
naturally in the context of a particular problem - in
other words derive it whenever you need it. Under no
circumstances should you write it down in a lecture and
underline it or draw a box around it!

I ask my first year students to derive everything they
need from first principles. They don't all do it, of
course. I don't let them have a formula sheet on
examinations as some of my colleagues do, and my
students constantly complain that there is too much to
memorize, to which I respond "You're right; there is
too much to memorize. Perhaps it would be better for
you to understand it."

Some of my students understand this and appreciate it,
but this is not the royal road to popularity.

Leigh