Solving a Difficult System of Equations in Dynamics

[74baja]

Hi all,

I'm stuck on a system of equations I am left with at the end of a dynamics problem.

$a=-b+v$
$x=-b+vcos(30)$
$y=vsin(30)$
$171.5=20b^2+5a^2$
$a^2=x^2+y^2$
$10x-40b=0$
All in degrees.
I know I have an extra equation, but I thought I'd include it in case it is easier to solve with a certain five of them. I know it's not as simple as solving for one variable in terms of one other, plugging in and solving. I tried looking at it for ways to recombine the equations- no luck.

Thank you,
Jack

 

[berkeman]

Hi all,

I'm stuck on a system of equations I am left with at the end of a dynamics problem.

$a=-b+v$
$x=-b+vcos(30)$
$y=vsin(30)$
$171.5=20b^2+5a^2$
$a^2=x^2+y^2$
$10x-40b=0$
All in degrees.
I know I have an extra equation, but I thought I'd include it in case it is easier to solve with a certain five of them. I know it's not as simple as solving for one variable in terms of one other, plugging in and solving. I tried looking at it for ways to recombine the equations- no luck.

Thank you,
Jack

Could you clarify which quantities are variables, and which are constants? Is v a variable, for instance?

 

[Ray Vickson]

Hi all,

I'm stuck on a system of equations I am left with at the end of a dynamics problem.

$a=-b+v$
$x=-b+vcos(30)$
$y=vsin(30)$
$171.5=20b^2+5a^2$
$a^2=x^2+y^2$
$10x-40b=0$
All in degrees.
I know I have an extra equation, but I thought I'd include it in case it is easier to solve with a certain five of them. I know it's not as simple as solving for one variable in terms of one other, plugging in and solving. I tried looking at it for ways to recombine the equations- no luck.

Thank you,
Jack

You have 6 equations in 5 unknowns a, b, v, x, y, In this case, there is no solution: the equations are inconsistent.

If you leave out the fifth equation ($a^2 = x^2 + y^2$) you can fairly easily solve the remaining equations, just by expressing a,v,x,y in terms of b (using equations 1,2,3,6) and then finding b from equation 4. To see that the original system is inconsistent, just substitute the resulting solution into the missing equation 5.

 

[csleong]

I didn't really go into details to calculate the answer, and with 6 equations to find 5 variables it is more than enough and I assume that all equations are correct (i.e. either equation left out can provides same answer), let's use a coordinate system to solve this problem, plot x-y axis and for equation 5, a^2=x^2+y^2, it is a circle, from there you can easily get your answer.

Let's explain more, equation 2, x-b=vcos30, equation 3 y=vsin30, combine can get (x-b)^2+y^2=v^2, another circle, I think this can gives you a clear picture?

 

[Ray Vickson]

I didn't really go into details to calculate the answer, and with 6 equations to find 5 variables it is more than enough and I assume that all equations are correct (i.e. either equation left out can provides same answer), let's use a coordinate system to solve this problem, plot x-y axis and for equation 5, a^2=x^2+y^2, it is a circle, from there you can easily get your answer.

Let's explain more, equation 2, x-b=vcos30, equation 3 y=vsin30, combine can get (x-b)^2+y^2=v^2, another circle, I think this can gives you a clear picture?

Perhaps you did not read my reply: the equations are inconsistent, so they cannot all be correct.

 

[csleong]

Perhaps you did not read my reply: the equations are inconsistent, so they cannot all be correct.

I think you should really read what I wrote, equation 2 and 3 actually can combine to gives (x+b)^2+y^2=v^2. So there are 5 equations, 5 variables. I don't know where your inconsistent come from, and why they cannot be all correct.

Okay.. yeah previous one I wrote wrongly, it should be x+b instead of x-b.

 

[csleong]

a=−b+v (1)
x=−b+vcos30 (2)
y=vsin30 (3)
171.5=20b^2+5a^2 (4)
a^2=x^2+y^2 (5)
10x−40b=0 (6)

From equation 6 => x=4b (7)
From equation 2 => x+b=vcos30 (8)
square(equation 8) + square(equation 3) => (x+b)^2+y^2=v^2 (9)

Sub equation 7 to equation 9 => 25b^2+y^2=v^2 (10)
Sub equation 7 to equation 5 => 16b^2+y^2=a^2 (11)
equation 10 minus equation 11 => 9b^2=v^2-a^2 (12)

From equation 1 => v=a+b (13)
Sub equation 13 to equation 12 => 9b^2=(a+b)^2-a^2
8b^2-2ab=0
b=0 or a=4b

Okay I think Ray is right, my view of the "combining 2 equations" become a circle is a big mistake because of the angle of 30 degree, they are 2 linear equations and a combination of them can't make a circle.

Sorry for the stupid concept.