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19-08-2016 05:08:00 +0000
please find the mechanics impedence
The equation of motion for a point mass is given as 3x+8x+12x=12cos(15t+phase constant)
Chima Sunday (Tutor)
This quest is a little bit confusing or may be it was not culled/copied rightly. I don't really understand why 3x+8x+12x was not just written as 23x?
That is by the way anyway, mathematically, Mechanical Impedance(MZ) is defined as the ratio of force(F) applied at a point to its resulting velocity(v) at that point.
Thus MZ = F/v
recall that from Newton F = ma, where m = mass and from question m is a point charge. This is a theoretical mass that has zero dimensions thus a zero volume but with a non zero mass
and a = acceleration
Recall also that from the theory of the application of calculus that V = dx/dt and a = dv/dt
So we are going to borrow this knowledge
Thus 23x = 12cos(15t + θ)
Where θ = phase constant
x = 12/23cos(15t + θ)
v = dx/dt = 12/23×[-sin(15t + θ)]×15
= 12×15/23×[-sin(15t + θ)]
Also a = dv/dt = 12×15/23[-cos(15t + θ)]×15
= 12×15××/23[-cos(15t + θ)]
Thus, since the particle is a "point mass" we can rightly right
MZ = ma/v = m×12×15×15/23×[-cos(15t + θ)] divided by 12×15/23×[-sin(15t + θ)]
MZ = 15m[cos(15t + θ)/sin(15t + θ)]
rcall that sinθ/cosθ = tanθ, so cosθ/sinθ = 1/tanθ
So MZ = 15m/tan(15t+θ)
I take responsibility of any mistake that may be contained in this solution. I also welcome your constructive/intelligent critique, corrections and suggestions/comments. Thank you.
19-08-2016 11:52:00 +0000
please the formula I know always start with y=A SIN(X-VT) and so on but I have never seen the formula with y=A COS(whatever) so please can you list the formula of y=Acos......... please I need it
21-08-2016 13:28:00 +0000
Lol! I will try and make this explanation simple for you. when the formular starts with y=A SIN(X-VT) you should know that the oscillation starts from resting position (i.e from origin), but you should also know that it can't always start from rest sometimes it may start from some distance on the Y-axis. View this link for more clarification. http://www.regentsprep.org/regents/math/algtrig/att7/sinuso37.gif and http://www.regentsprep.org/regents/math/algtrig/att7/sinuso39.gif The first image is for cosine formmular and the second image is for sine formular. Really have more to explain on this, but i think you should have this basic knowledge for now as you progresses in your study you will get to know more and will one day laugh just as i did at your "the formula I know always start with." and beside you should know from your trigonometry knowledge that we can convert sine to cosine. Cheers!
05-09-2016 14:03:00 +0000
recall that http://www.regentsprep.org/regents/math/algtrig/att7/graphv14.gif
05-09-2016 14:07:00 +0000
also note that inverse of tan is cot so we can equally write MZ = 15mcot(15t+phase constant)
20-10-2016 09:36:00 +0000
sure this is correct?
14-02-2017 16:25:00 +0000