Solving System of Equations Homework: Trig & mv^2/L

[REVIANNA]

Homework Statement

this is actually one of the physics problems and I have boiled down the numerical to two equations.
But I have trouble manipulating equations

Homework Equations

Tsin(theta)=Fcos(theta)-mg
and
Tcos(theta)=(mv^2/Lcos(theta))-Fsin(theta)

F and T are the two unknowns

The Attempt at a Solution

I brought the terms involving m to one side and the trig functions to the other
and tried to add the the equations . But it only gets more complicated as in one eq F has a coefficient of cos(theta)+sin(theta) and in the other cos(theta)-sin(theta).same goes for coefficients of T in both equations.

 

[tommyxu3]

The form is like:
$$T\sin\theta-F\cos\theta=...$$
$$T\cos\theta+F\sin\theta=...$$
Multiplying the both ##\sin\theta## and ##\cos\theta##, and the other way around, which may help.

 

[REVIANNA]

the other way around,

I did not understand this

 

[REVIANNA]

F(cos(θ))^2 sin(θ) +Tsin^2(θ) cos(θ)
F sin^2(θ) cos(θ) -T cos^2(θ) sin(θ)

what should I do?

 

[tommyxu3]

$$T\sin\theta-F\cos\theta=...(1)$$
$$T\cos\theta+F\sin\theta=...(2)$$
What I meant are ##(1)\cdot\sin\theta+(2)\cdot\cos\theta## and ##(1)\cdot\cos\theta-(2)\cdot\sin\theta.## Can you get anything from them?

 

[REVIANNA]

It worked !
How did you think about it?
And THanks

 

[SammyS]

I did not understand this

The form is like:
$$T\sin\theta-F\cos\theta=...$$
$$T\cos\theta+F\sin\theta=...$$
Multiplying the both ##\sin\theta## and ##\cos\theta##, and the other way around, which may help.

I hope that tommyxu3 did not literally mean what he wrote.

The form he gave was good.
Here's what to do from that point.

Multiply the first equation by ##\ \sin(\theta)\ ## and the second equation by ##\ \cos(\theta) \ ##, then add the equations to eliminate F . It's essentially the method of elimination. Then solve for T .
...

 

[REVIANNA]

@SammyS I've got it, Thanks.

 

[Ray Vickson]

Homework Statement

this is actually one of the physics problems and I have boiled down the numerical to two equations.
But I have trouble manipulating equations

Homework Equations

Tsin(theta)=Fcos(theta)-mg
and
Tcos(theta)=(mv^2/Lcos(theta))-Fsin(theta)

F and T are the two unknowns

The Attempt at a Solution

I brought the terms involving m to one side and the trig functions to the other
and tried to add the the equations . But it only gets more complicated as in one eq F has a coefficient of cos(theta)+sin(theta) and in the other cos(theta)-sin(theta).same goes for coefficients of T in both equations.

It is easier if you simplify the symbolics: let ##s = \sin(\theta), c = \cos(\theta), A = mg, B = \frac{mv^2}{L} \cos(\theta)##. Then your equations read as
[tex] sT = cF - A\\
cT = -sF + B [/tex]
or
[tex] \begin{array}{rcl}
cF - sT &=& A\\
sF + cT &=& B
\end{array} [/tex]
If you know about matrices and matrix inverion you can write down the solution immediately, because in matrix form the system reads as
[tex] \pmatrix{c & s \\-s & c} \pmatrix{F\\T} = \pmatrix{A\\B} [/tex]
A crucial simplification is that ##c^2 + s^2 = 1##, because these constants are the cosine and sine of the same angle.