A system of equations that I want to do with a matrix

[youngstudent16]

Homework Statement

Below are four equations, with the known quantities listed. Solve these equations to obtain an expression for $T$ in terms of known quantities only. Do the same to obtain an expression for $a$
$T-f=m_1a\hspace{5mm}N-m_1g\cos\theta=0$
$m_2g-T=m_2a \hspace{5mm} f=\mu N$

$m_1,m_2,\mu,g,\theta$ Are known quantities

Unknown quantities are $T,a,N,f$

Homework Equations

Knowing how to solve system of equations

The Attempt at a Solution

My attempt and I'm wondering if I can solve this easier with a matrix.(or if this was correct I have not done these that often)

$N=m_1g\cos\theta$
$f=\mu m_1g\cos\theta$
Now I have two equations and two unknowns
$T-\mu m_1g\cos\theta=m_1a$
$m_2g-T=m_2a$
$T=m_1a+\mu m_1g\cos\theta$
$m_2g-m_1a+\mu m_1g\cos\theta=m_2a$
$m_1\left(-a+\mu g\cos\theta\right)=m_2(a-g)$
$\frac{m_1}{m_2}+g=\frac{a}{\left(-a+\mu g\cos\theta\right)}$
$\frac{m_1}{m_2}+g+1=\frac{a}{\left(\mu g\cos\theta\right)}$
$\left(\mu g\cos\theta\right)\left(\frac{m_1}{m_2}+g+1\right)=a$Now just plug $a$ back in for the first. Is that correct and is there an easier way to do this with a matrix?

 

[techmologist]

Homework Statement

Below are four equations, with the known quantities listed. Solve these equations to obtain an expression for $T$ in terms of known quantities only. Do the same to obtain an expression for $a$
$T-f=m_1a\hspace{5mm}N-m_1g\cos\theta=0$
$m_2g-T=m_2a \hspace{5mm} f=\mu N$

$M_1,M_2,\mu,g,\theta$ Are known quantities

Unknown quantities are $T,a,N,f$

Homework Equations

Knowing how to solve system of equations

The Attempt at a Solution

My attempt and I'm wondering if I can solve this easier with a matrix.(or if this was correct I have not done these that often)

$N=m_1g\cos\theta$
$f=\mu m_1g\cos\theta$
Now I have two equations and two unknowns
$T-\mu m_1g\cos\theta=m_1a$
$m_2g-T=m_2a$
$T=m_1a+\mu m_1g\cos\theta$
$m_2g-m_1a+\mu m_1g\cos\theta=m_2a$
...

You are doing fine right up to that point. The something goes wrong. Why not collect terms with a right there and then solve for a? Everything else in that last equation is a known quantity.

I do not believe using matrices would make this calculation any shorter. Notice that most of the equations have only two of the four unknowns appearing in them. That is a good sign that simple substitution and elimination (what you are doing) is the easiest way. If most of the equations had at least three unknowns, then yeah, I would use the matrix method. That makes it much easier to keep track of what you are doing.

However, if you want to do this problem with a matrix just to get the practice, that seems like a good idea. You can check the answer you get that way with the one you get using substitution.

 

[SammyS]

Homework Statement

Below are four equations, with the known quantities listed. Solve these equations to obtain an expression for $T$ in terms of known quantities only. Do the same to obtain an expression for $a$
$T-f=m_1a\hspace{5mm}N-m_1g\cos\theta=0$
$m_2g-T=m_2a \hspace{5mm} f=\mu N$

$M_1,M_2,\mu,g,\theta$ Are known quantities

Unknown quantities are $T,a,N,f$

How are $\ m_1 \$ and $\ m_2 \$ related to $\ M_1 \$ and $\ M_2 \$ ?

 

[youngstudent16]

How are $\ m_1 \$ and $\ m_2 \$ related to $\ M_1 \$ and $\ M_2 \$ ?

I'm sorry that was a typo I will fix it

 

[youngstudent16]

You are doing fine right up to that point. The something goes wrong. Why not collect terms with a right there and then solve for a? Everything else in that last equation is a known quantity.

I do not believe using matrices would make this calculation any shorter. Notice that most of the equations have only two of the four unknowns appearing in them. That is a good sign that simple substitution and elimination (what you are doing) is the easiest way. If most of the equations had at least three unknowns, then yeah, I would use the matrix method. That makes it much easier to keep track of what you are doing.

However, if you want to do this problem with a matrix just to get the practice, that seems like a good idea. You can check the answer you get that way with the one you get using substitution.

Hmm at that point though I have the issue with
$m_2g+ \mu m_1g \cos\theta=m_2a+m_1a$
When I tried to get rid of either $m_2$ or $m_1$ I had problems
Or would this be ok to do
$m_2g+ \mu m_1g \cos\theta=\left(m_2+m_1\right)(a)$

$\frac{m_2g+ \mu m_1g \cos\theta}{m_2+m_1}=a$?

Thanks for the insight I"m taking my first physics class and this was a warm up problem we had to do and I figure we will see more of these applied to real physics so I wasn't sure if I should be using matrices more since it might be more efficient. I guess I will learn which to use with practice and your hint.

 

[Chestermiller]

Hmm at that point though I have the issue with
$m_2g+ \mu m_1g \cos\theta=m_2a+m_1a$
When I tried to get rid of either $m_2$ or $m_1$ I had problems
Or would this be ok to do
$m_2g+ \mu m_1g \cos\theta=\left(m_2+m_1\right)(a)$

$\frac{m_2g+ \mu m_1g \cos\theta}{m_2+m_1}=a$?

Thanks for the insight I"m taking my first physics class and this was a warm up problem we had to do and I figure we will see more of these applied to real physics so I wasn't sure if I should be using matrices more since it might be more efficient. I guess I will learn which to use with practice and your hint.

Yes, this is correct. But, it seems you weren't sure. If so, you need to review algebra skills.

Chet

 

[youngstudent16]

Yes, this is correct. But, it seems you weren't sure. If so, you need to review algebra skills.

Chet

I feel sure now looking over it in the moment I felt unsure. I have practiced a few other system of equations problems now and feel more adjusted to it. Thanks for all the feedback everyone. I'm sure this year I'll be asking more questions as the physics class picks up.