Re: Newton's 2nd Law problem

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Fri, 11 Feb 2000, romanza wrote:

> To me, if an object has a mass which is increasing with time, the force to
> maintain its constant vel can be found from F=vdm/dt.

Not in general; it depends on how the mass is added.  For instance, If the
object picks up the mass by colliding with particles that are "at rest",
and if by v you mean *its* speed (measured, of course, in the frame in
which the particles are "at rest") then you are correct.  But perhaps the
mass is decreasing because the object of interest is simply releasing bits
and pieces (like "bombs") or increasing because someone is running
alongside and handing it glasses of water (or candy bars); then no force
is required. What if a hose is being played from behind on a skateboard
riding sponge?  Then the force required to maintain the sponge's speed is
F = (u - v)dm/dt and it would have to be applied opposite the sponge's
velocity to keep it from speeding up.  As others have said, "it depends."

> So for the rotor blade, I don't quite see why dm/dt is the rate of mass flow
> through the fluid.

As you go on to say, thinking about the rate of momentum transfer is the
key to problems like this.  A hovering helicopter receives an upward force
in the form of a reaction to the downward force it exerts on the air
which, in turn, shows up as a time rate of momentum transfer to the air.
The easiest way to think about that is as a mass flow rate times its
downward velocity increment.

Perhaps you'd be happier thinking about a specific chunk of air of mass
delta_m that changes its velocity by an amount delta_v in a time delta_t
due to the downward force from the rotors.  Newton's second law shows
that the force is equal to ma = delta_m (delta_v/delta_t).  Clearly we
can write this as delta_v (dm/dt) and we do so primarily because it is
easier to think about the "fixed change in v" and the "rate of mass flow".

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm