Download Helpful relationships

Helpful relationships
Equation 1:
(average speed)
Equation 2:
(acceleration)
Equation 3: If acceleration is uniform: (and only if it is uniform!)
The next two equations are derived from Equations 1 - 3.
Equation 4:
Equation 5:
The following two equations relate to circular motion.
Equation 6:
(tangential speed)
Equation 7:
Equations 8:
(centripetal acceleration)
(force)
Equation 9:
(impulse)
The following diagram and Equation 7 and 8 are useful in the derivation of Equation 10
Equation 10:
(centripetal acceleration)
Equation 11
(work)
a)
b)
Equation 12:
(potential energy)
Equation 13:
(kinetic energy)
When loss of PE = gain of KE, and if original v is small, the final
υ can be determined by
Equation 14:
The relationship between revolutions/minute and radians/s can be determined using the
following:
Equation 15
Equation 16:
(power)
and
When the distance between parallel crosspieces is known as in the following diagram
Equation 17:
The average radius in the following diagram can be determined using Equation 18.
Equation 18:
"h" in the following diagram can be determined using Equation 19.
Equation 19:
The above relationship is derived in Appendix A that is at end of this file.
The flow rate can be determined when the speed (υ), depth (d), and width (w) are
known
Equation 20:
Equation 21:
Equation 22:
and for water
The overall v at an instance can be determined when vx and vy are
known by using the Pythagorean theorem.
Equation 23:
(coefficient of friction)
The following four equations are special examples of previously stated equations.
Equation 24:
Equation 25:
Equation 26:
Equation 27:
Equation 28:
(period of a pendulum)
Equation 29:
Pressure – altitude Conversion
1 mbar = 8.32568323 m
1 mm Hg = 11.1 m
1 mm Hg = 1.3332239 mbar
Equation 30: For Steel Venom
where D = space between chair sets; L = length of chair sets; n = number of
chair sets
Equation 31: For Steel Venom
where H = max change in height of chair sets
D = space between chair sets
n = number of chair sets
v = theoretical average vertical speed of chair sets
The above relation is derived in Appendix B
Appendix A
Height calculation using the Law of Sines
Having measured side d and angles Θ1 and Θ2 , determine the height H.
The triangle having sides d, B, and C can be thought of as consisting of two right triangles which
have a common side D. One triangle has a hypotenuse of d and an adjacent angle of Θ1 while
the other triangle has a hypotenuse of B and an adjacent angle of Θ2 – Θ1 . The common side can
be expressed in terms of the hypotenuse and adjacent angle of either triangle. Thus, we have the
following:
(A-1)
or
(A-2)
Substituting A-2 into A-1, we have the following:
(A-3)
Or
(A-4)
Equation A-4 is one form of the law of sines. It states that value of one side divided by the sine of its
opposite angle is equal to the value of another side divided the sine of its opposite angle.
The height H can be written in terms of the side B and the adjacent angle Θ2 as follows:
(A-5)
If we solve equation A-4 for B and substitute the result into equation A-5, we obtain the following:
Appendix B
Law of Conservation of Energy as applied to the Steel Venom
The “Steel Venom” is an example of a system that oscillates between all-potential energy and all-kinetic
energy. The ride is analogous to throwing a ball into the air. When the ball reaches its maximum height, it
has zero vertical velocity, and hence no kinetic energy. When ball returns to the elevation at which it left
the hand of the thrower, it has its initial kinetic energy and its initial gravitational energy. A closer
analogy would be to throw several balls into the air such that they were uniformly spaced and constrained
to a fixed path. In the real world friction is a part of the process, and so conservation of energy is an ideal
that may or may not be closely approximated.
The following equation treats the “Steel Venom” as a collection of “n” particles that are uniformly spaced
by a distance D. When the linear induction motor imparts energy to the ride, the front chair set
(designated as mass number 1) will attain a maximum elevation change of “H”. All other chair sets will
attain elevations which differ from the front chair set by some integer multiple of “D” depending upon the
position of the chair set in the string of chairs. The left-hand side of equation (1) represents the maximum
kinetic energy of the system, while the right-hand side represents the maximum potential energy.
If we assume that each set of chairs has the same mass (m), the equation number (1) can be rewritten as
follows:
If we divide equation (2) by m, and reduce the equation to its simplest form, we obtain the following: