Lifting an Object of Changing Mass

Date: 7/28/96 at 20:28:34
From: Ramon Ponte
Subject: Lifting an Object of Changing Mass

Suppose you grab the end of a chain that weights 3lb/ft and lift it 
straight up off the floor at a constant speed of 2ft/s.

(a) Determine the upward force F you must exert as a function of S 
    (height)

(b) How much work do you do in lifting the top of the chain to 
    s = 4ft?

Strategy: Treat the part of the chain you have lifted as an object 
that is gaining mass

     0      -     hand
     0      |
     0      |3ft
   00000    |
  0000000   -    chain


-Ramon Ponte

Date: 8/1/96 at 11:23:40
From: Doctor Anthony
Subject: Re: Lifting an Object of Changing Mass

Newton's second law in its general form is 

F = d(mv)/dt  This is normally written m*dv/dt because mass is 
usually constant. When mass is varying we need to look at what is 
happening in more detail.

Consider a body moving so that at time t, its mass is m, its velocity 
is v and the resultant force acting on the body is F.

Suppose that, at this instant, an extra mass dm moving at velocity u 
is added to the body so that, at time t+dt, the body has mass m+dm and 
velocity v+dv

Just before the addition of dm the momentum of the body is  mv  and 
the momentum of dm is dm.u

Immediately afterwards, the momentum of the augmented body is 
(m+dm)(v+dv)   So the increase in momentum in time dt is

(m+dm)(v+dv) - mv - dm.u  = mv + m.dv + v.dm + dm.dv - mv - dm.u
                          = m.dv + dm(v-u) + dm.dv

The impulse of the force acting on the body at the same time interval 
is F.dt,  and so we have:

     F.dt = m.dv + (v-u)dm + dm.dv

as dt -> 0 this becomes      F = m.dv/dt + (v-u)dm/dt

If the mass increment, dm, has no velocity of its own before becoming 
part of the body, i.e. u = 0, the equation becomes

         F = m.dv/dt + v.dm/dt 

Now consider what happens as we lift the end of the chain at a 
constant velocity of 2 ft/sec.  In above Formula F is the RESULTANT 
force, so it is the combination of P the pull exerted by your hand 
upwards, and mg, the force of gravity acting downwards.  At time t, a 
length s feet is off the floor so the mass of this is 3s lbs, and the 
gravitational pull is 3s.g lbals

So F = P - 3s.g = 3s.dv/dt + 2.dm/dt     But dv/dt = 0 so this becomes

P - 3s.g = 2.dm/dt  and dm/dt = 6 (2 ft/sec = 6 lbs/sec mass increase)  

P - 3s.g = 2*6 = 12

       P = 3s.g + 12   lbals

If we want the force in lbs we divide by g = 32 to get

       P = 3s + 12/32  lbs

       P = 3s + 3/8  lbs

(b) Work done in lifting the end to  4 ft.

  W.D = INT(0 to 4)[P.ds]

      = INT(0 to 4)[(3s + 3/8)ds]
  
      =  (3/2)s^2 + (3/8)s   between 0 and 4

      =  (3/2)*16 + (3/8)*4

      = 24 + 3/2   lbs.ft

      = 25.5  lbs.ft

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/