Simultanious equations

[James Smithson]

The bane of my life at the moment .

Currently studying statics and I love it I feel i understand it and I can get very fan on questions but when it comes to geting it to the point of two simultanious equations im lost . I find it too confusing.

Please could I post a few and have them worked out and explained so I can try copy techniques used on here ?


where μ = 0.74
where μ=0.43
I have the answers now as it says the answer when i get them wrng but it does not show the working out for the simultanious equation part so I am stuck



thank you!
James


i should have sain g is 9.8

 

[Dr.Peterson]

I think it will be most helpful if you show your attempt at one, so we can see where you are confused, and also, perhaps, what method(s) you have learned.

One thing that may help would be to evaluate all the coefficients, so that, for example, the first equation, $-\mu N+T\cos(5^\circ)-20g\sin(35^\circ)=0$ becomes $-0.74N+0.99619T-160.55=0$.

And my first thought for solving that first system of equations is to multiply the second equation by $\mu$ and add them together. Then you can easily solve for T.

 

[fresh_42]

The bane of my life at the moment .

Currently studying statics and I love it I feel i understand it and I can get very fan on questions but when it comes to geting it to the point of two simultanious equations im lost . I find it too confusing.

Please could I post a few and have them worked out and explained so I can try copy techniques used on here ?


View attachment 38707 where μ = 0.74
View attachment 38708where μ=0.43
I have the answers now as it says the answer when i get them wrng but it does not show the working out for the simultanious equation part so I am stuck



thank you!
James


i should have sain g is 9.8

What do you know about matrices, especially 2x2 matrices and their inverse?

 

[James Smithson]

I think it will be most helpful if you show your attempt at one, so we can see where you are confused, and also, perhaps, what method(s) you have learned.

One thing that may help would be to evaluate all the coefficients, so that, for example, the first equation, $-\mu N+T\cos(5^\circ)-20g\sin(35^\circ)=0$ becomes $-0.74N+0.99619T-160.55=0$.

And my first thought for solving that first system of equations is to multiply the second equation by $\mu$ and add them together. Then you can easily solve for T.

what i really need is an example of probles like this bieng solved to be honest because it confuses me soo much . ive trield taking equation one from equation 2 ive tried adding them together ive also tried to multiply together and even divide and even though i can see answers i cant work out the middle .

for example

goes to


but im there absolutly clueless to how . I was not sure what you meant by multiplying the second equation by mu dont i have to do something to both if i do it to one anyway i feel this is far beyond my levels at the moment so any advice would be appreciated. thank you !

 

[James Smithson]

What do you know about matrices, especially 2x2 matrices and their inverse?

for a 2x2 matrix


The inverse can be found using the formula

It can only be inversed if the determinant is not zero

not sure what it has to do with this but im all ears

 

[fresh_42]

Let's consider the first example
$$\begin{array}{lll} -\mu N +T \cos(5°)&=20g\sin(35°)\\ N+T\sin(5°)&=20g\cos(35°) \end{array}$$This can be written as
$$\begin{pmatrix}-\mu&\cos(5°)\\1& \sin(5°)\end{pmatrix}\cdot\begin{pmatrix}N\\T\end{pmatrix}=20g\begin{pmatrix}\sin(35°)\\ \cos(35°) \end{pmatrix}$$Now we invert the matrix on the left which is
$$\begin{pmatrix}-\mu&\cos(5°)\\1& \sin(5°)\end{pmatrix}^{-1}=\dfrac{1}{-\mu\sin(5°)-\cos(5°)}\begin{pmatrix}\sin(5°)&-\cos(5°)\\-1& -\mu\end{pmatrix}$$
Now we multiply both sides with that inverse matrix from the left and obtain

$$\begin{array}{lll} \begin{pmatrix}N\\T\end{pmatrix}&=\dfrac{20g}{-\mu\sin(5°)-\cos(5°)}\begin{pmatrix}\sin(5°)&-\cos(5°)\\-1& -\mu\end{pmatrix}\begin{pmatrix}\sin(35°)\\ \cos(35°)\end{pmatrix}\\[30pt] &=\dfrac{20g}{-\mu\sin(5°)-\cos(5°)}\begin{pmatrix}\sin(5°)\sin(35°)-\cos(5°)\cos(35°)\\ -\sin(35°)-\mu \cos(35°)\end{pmatrix}\\[30pt] &=\dfrac{20g}{-\mu\sin(5°)-\cos(5°)}\begin{pmatrix}-\cos(40°)\\ -\sin(35°)-\mu \cos(35°)\end{pmatrix}\\[30pt] &=\dfrac{20g}{\mu\sin(5°)+\cos(5°)}\begin{pmatrix}\cos(40°)\\ \sin(35°)+\mu \cos(35°)\end{pmatrix} \end{array}$$
This can be done more or less automatically and gives us the solutions for $N$ and $T.$ The other example works accordingly.

Here is a nice link that helped me a lot with matrix operations, especially as it accepts variable names such as $a,b,c$ as matrix entries: https://www.symbolab.com/solver/matrix-calculator

 

[Harry_the_cat]

James, can you solve simpler simultaneous equations? For example:

2a + 3b = 9
5a - 7b = 8