Saturday, July 21, 2012

Unit 3 - Lecture 16 ==> Mass & acceleration relationship

Using the data given

  • Decomposing  $$ \Delta{x} = V_0 t + \frac{1}{2} a t^2 $$ :
    • Since the initial velocity is zero, $$ V_0 t = 0 $$
    • Thus we have $$ \Delta{x} = \frac{1}{2} a t^2 $$
    • Solving for a, $$a = \frac{2\Delta{x}}{t^2} $$
  • Calculate 4 cases for acceleration using force of 10 Newtons
    MassTimeAccelerationAcceleration Rounded
    10 kg4.471.00095591291.00
    20 kg6.320.5007210382950.50
    30 kg7.750.3329864724250.33
    40 kg8.940.2502389782240.25
  • $$\mbox{Acceleration} \varpropto \mbox{Mass}$$
    • So find a relationship where the constant c is a constant
    • Try $$ c =  a * m $$
  • Calculate 4 cases for the constant c
  • MassAccelerationConstant c
    101.0010
    200.5010
    300.3310
    400.2510
    • And no other combination results in a constant constant
    • Thus as a goes up or down, m must go down or up the same amount for F to  remain a fixed value.
    • Thus F = c * m, and we have just calculated the force of gravity. 

Conclusions

  • By observation, acceleration is directly proportional to the force applied.
  • And as mass increases, acceleration proportionality decreases.
    • $$ a \varpropto \frac{1}{m} $$
  • And since $$ F = \mbox{constant} * m $$
    • $$ c = 10 $$
    • $$ F = 10 * m $$

4 comments:

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    Replies
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      I have no ulterior motive other than (1) being useful to myself and 2) sharing that information freely without being pushy or over steping my bounds.

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    2. FYI.

      I have experience teaching. That was my last assignment before retiring from IBM. I taught Project Management.

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