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Q: Mechanics; relationship between speed on an inclined plane and gravity ( Answered,   4 Comments )
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Subject: Mechanics; relationship between speed on an inclined plane and gravity
Category: Science > Physics
Asked by: duepflischiesser-ga
List Price: $5.00
Posted: 13 Mar 2003 04:42 PST
Expires: 12 Apr 2003 05:42 PDT
Question ID: 175574
Wasn't it Galileo that dropped a feather and a cannonball on a vacuumy
day from the tower at Pisa and found that they fall at the same speed?
 So gravity acts on all objects the same, regardless of their mass. 
When flying a glider one presumes that the only force moving the
airplane forward is gravity;  the airplane "falls" towards earth, but
on an inclined plane so to speak due to its geometry.  All glider
pilots know that a heavier glider falls faster, i.e. travels forward
faster at the same sink ratio, and that's why they have water tanks in
their wings that can be filled when more speed is desired.  But if
mass and descent rates are independent, why is this?  It doesn't have
anything to do with gliders, because when skiing or bicycling with my
much lighter wife, the same principle clearly applies.
Answer  
Subject: Re: Mechanics; relationship between speed on an inclined plane and gravity
Answered By: hedgie-ga on 13 Mar 2003 19:26 PST
 
duepflischiesse

        The comments below provided useful information
        but your  question
  " If mass and descent rates are independent, why is this? "

    can be addresses in a more specific manner.
    You are right that this behavior is universal  and I will
    try to explain why in some cases the descent rate does
    and in same cases in does not depend on mass. 

     I will  describe an equation which covers both cases.

    I could paraphrase that equation in words if you would
    prefer and request that in 'clarification of the answer.'

    The equation describes the 'balance of forces' and is
    usually  presented in a quite a confusing manner:
http://www.glenbrook.k12.il.us/gbssci/phys/Class/newtlaws/u2l1d.html

  By writing the equation down will help us to avoid  semantic ambiguities
  which contribute to the confusion. The exact form of equation depends
  on specific situation.

  For example, here is the form which applies to a glider
  which is not accelerating (meaning, it's speed vector is not changing):
  http://www.grc.nasa.gov/WWW/K-12/airplane/glidvec.html

   In situation where there is acceleration, gravity and some friction
   the balance of forces can be written thus:

      Force_of_Friction +  Force_of_gravity =  Force_of_Inertia   or
       Ff               +           Fg      =           Fi


       Lets examine these forces, the terms in this equation:

       Force of Inertia is Proportional to (inertial) mass:
        Fi = Mass* acceleration

        Force of Gravity (Weight) is proportional to (gravitational) mass
        Fg = Mass * g      (g is  intensity of gravitational field)

        Force of friction depends on many things
        (size, shape, speed, friction or drag coefficient d) but not on Mass:

        Ff = speed* d   (speed of the object with respect to the medium).




        Thanks for bearing with us. Here is Your answer:

        When Ff   is small  (or absent) e.g. in the outer space,
        the equation is reduced to

           Fg = Fi


           Mass can be canceled and motion is then  independent of the mass.

           When Ff is not negligeable, then acceleration depends on mass.
           
 DEMONSTRATION
          The 'evacuated cylinders' with feather and heavier object (penny)
          http://www.pha.jhu.edu/facilities/pir/lecdemo/M-c2a.html
          are built to demonstrate this:
          As you remove the medium (air) frictional force Ff tends to zero 
          and both object  hit at the same time - showing motion is independent
          of mass.
          
          Science museums, Boston, Exploratorium have a larger version of this
          experiment.
          http://www.exploratorium.edu/snacks/falling_feather.html
     

          The story of Galielo and Tower of Pisa is believed to be apocryphal;
           He did experiments which clarified interplay of inertia and gravity. 
           but since he could not have  'removed the air'  from Pisa.,
           the cannon ball, penny and feather would not hit at the same time.


    SEARCH TERMS
    balance of forces
     (This should  be used with some 'physics term' such as 'mass' or 'inertia'
     to weed out all metaphorical uses in other fields, such as  politics)

     evacuated cylinder, free-fall

 IN CONCLUSION

     Thanks for asking this questions. This topic is confusing to many,
     partly because the 'balance of forces' is often explained in a
     confusing way in schools.

        Please feel free for ask for clarification of any term or step in
        this reasoning.

        hedgie

Clarification of Answer by hedgie-ga on 14 Mar 2003 18:04 PST
In his comment thenextguy-ga makes few good points.

    The object in free fall in the atmosphere is described by our
    general equation:
  
   Ff               +           Fg      =           Fi

  As the speed is increasing, the Fi tends to zero and we get:

   Ff + Fg =0

  In that regime, object reached the 'terminal velocity'.

  In the case of two cars on the hill I would agree with him that
  the inital acceleration will be about the same. Ff is realtively small.

  However, what he says about the flat section:

 " The truck has a larger mass for
   friction to decelerate, but the frictional force between the truck's
tires & the road is also larger because of that larger mass."
  
   should not be taken to  mean that the Ff is always  proportional to mass.
   There is a sliding friction and rolling friction and I would rather 
    measure it in a comple case, such as car then speculate.
  http://www.school-for-champions.com/science/frictionrolling2.htm

   In case of the two cars, I would expect the heavy car to roll
   farther on the flat part of the trip. 


:  duepflischiesser,

If you are happy with the answer and no further clarification is needed,
we do appreciate feedback through  rating.
Comments  
Subject: Re: Mechanics; relationship between speed on an inclined plane and gravity
From: dmooredotcom-ga on 13 Mar 2003 09:59 PST
 
I think it was two cannon balls of different weight.. and that story is disputed.
Subject: Re: Mechanics; relationship between speed on an inclined plane and gravity
From: flajason-ga on 13 Mar 2003 11:15 PST
 
Galileo's experiment was to actually roll two balls of different
weights but same porportions down an inclined plane. Unfortunately,
Galieleo lacked the precision instruments we have today to measure two
balls dropped simultaneously with any accuracy.
What you are asking is true to an extent, but the situations also deal
with fluid mechanics, friction, and momentum.
Consider a large fully weighted semi truck and a small compact car.
Place them on top of a hill with a steep incline and let them roll (in
nuetral) to the bottom where it levels out flat.
The semi will take longer to get up to speed than the compact, but it
will have more momentum when it reaches the bottom and will travel
much further when it reaches the flat stretch.
When skiing with your wife, you'll probably find that she gets off to
a bit of a head start, but with your larger mass, once you get up to
speed, it takes less effort for you to maintain your speed than it
does for her.
The same principle applies to the heavier glider. 

Hope this helps!
fla-jason
Subject: Re: Mechanics; relationship between speed on an inclined plane and gravity
From: xarqi-ga on 13 Mar 2003 11:28 PST
 
Dave Scott, during the Apollo 15 mission, dropped a hammer and a
falcon feather while on the moon - they hit the surface at the same
time.
As for your glider, see:
http://www.patprojects.org/glider/aovt/weight.htm
Subject: Re: Mechanics; relationship between speed on an inclined plane and gravity
From: thenextguy-ga on 14 Mar 2003 13:19 PST
 
I don't believe the comment about the truck vs. car on the hill is
correct.  First, the speed (in the airless, frictionless case) of each
will be the same.  It's found by conservation of energy to be the
square root of 2*g*(height difference).  The same mass appears in both
kinetic & potential energies.  True, this does ignore some small
effects - neither vehicle will reach the speed above because some of
that potential energy will go into rotational kinetic energy of the
wheels, and the wheels on cars & trucks are generally not the same
diameter or mass.

When they get to the bottom, friction will eventually stop them.  The
force of friction (rolling friction in this case) depends on the
weight of the object (more specifically, the normal force, but they're
the same magnitude on flat ground).  The truck has a larger mass for
friction to decelerate, but the frictional force between the truck's
tires & the road is also larger because of that larger mass.

When air is involved, the real point is that, at the moment something
is dropped, it will accelerate towards the Earth at "g" or 9.8 m/s^2. 
Air resistance is proportional to the object's velocity (or v^2,
sometimes) and therefore provides an opposing force which rises as the
object gains speed.  The gravitational force stays constant, but the
force of air resistance grows until the two perfectly balance
(terminal velocity).  With no further acceleration, the object falls
at constant velocity until it hits.

So, without air, mass doesn't really affect the speed of a falling or
sliding object.  With it, mass & surface area & shape all become
important.

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