Consider a block of mass M a resting
on a horizontal, friction-free surface. One end of a coil spring
is attached to the block; the other end of the spring is attached
to the wall. If the block is pulled to one side and released,
the force of the spring will be the only force acting on it in
the x direction in the object will accelerate in the ex direction.
(A net force in the y direction is 0 in these forces will not
become a factor in this problem)
W hat we would like to do with his displays the block services
a from its equilibrium position and released the block. We would
like to know relationships for position-, velocity-, acceleration
of this block was a function of time as well as the velocity
of the block as a function of position. |
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The law of conservation of energy suggests that the energy
stored in the spring at maximum comprression (or extension) manifests
itself as potential or kinetic energy whenever the block is somewhere
between these two extreme points |
line 1 |
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Line one is expanded here |
2 |
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Solve line 2 for v. This equation yields velocity as
a function of position. See equations, page 1, line 4. |
3 |
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One way to define velocity |
4 |
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Substitute line 3 into line 4 |
5 |
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Rearrange and prepare to integrate. I can never remember
the trick, See http://www.exambot.com/cgi/reference/
show.cgi/math/intc/mint/inv_trig_sub.ref |
6 |
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ta-dah |
7 |
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We need to establish some initial condition by assigning
a value for C.
This says we shall start timing the block when it is farthest
to the right |
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Let x = A when T = 0 |
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the Let statement isconsistent with
our initial conditions |
9 |
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The "things" on either
side of the equation in line 10 are angles. If two angles are
equal then their sines are equal |
10 |
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line 11 will clean up with an identity.
See
http://www.exambot.com/cgi/reference/show.cgi/math/trigid.ref |
11 |
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Here is an equation for position
as a function of time. See page 1, line 3. TAke a derivitive
wrt time to get line 13 |
12 |
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A derivative of this line gets to
acceleration.
See page one line 1 |
13 |
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Line 14 has no equivalent on page 1 |
14 |
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Line 12 tells us that when t = 0, X = A. Let us calculate
the elapsed time until the next moment when x = A. Return to
line 12. Let x = A |
15 |
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In line 15, if x = A  must
be an angle whose cosine = 1. Zero radians is a trivial solution
(the block has not moved yet). The next angle for whihch this
is true is 2 pi radians |
16 |
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Let us solve line 16 for t and while we are at it, let
us rename it T.
This is called period, the time for one
cycle. |
17 |
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Line 18 shows a different arrangement of line 17. The
left side is called angular frequency (unit = 1/s). The left
side can be substituted for the right hand side in lines 12,
13, & 14 above |
18 |
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