Static equilibrium/tension force problem

[psneath]

Homework Statement

The attached system is in static equilibrium, and the string in the middle is exactly horizontal.
Find:
a)tension T1
b)tension T2
c)tension T3
d)angle theta

Homework Equations

I know I need to create equations that equal zero to analyze the x and y components of force as well as torque

The Attempt at a Solution

So
Fx=T3x-T1x=0 (I think the tension in T2 is zero)
Fy=T1y + T3y-m3g-m2g=0
For the torque equation I'm using the axis of rotation as the point where T1 meets m3

torque = T2R + T3yR - m2gR

When I try to solve any of these I come across way too many unknowns to solve...can someone see if I've assessed for forces properly and show me where to go from here?

Thanks!

 

[K29]

you have 4 unknowns and 3 equations which means you should be able to solve for each variable fairly easily. ( I say 4 unknowns because T3x cand T3y can both be expressed in terms of T3, with some triangular thinking.. Not sure if your torque equation is right or wrong(haven't done that stuff in a while)

 

[psneath]

Can I assume that T1y=m3g and T3y=m2g?

 

[collinsmark]

Can I assume that T1y=m3g and T3y=m2g?

Yes, you can and you should. [I assume you mean T_1y = (3.0 kg) x g, and T_3y = (2.0 kg) x g]

And it's important to understand why you can. So I encourage you to think about it for a couple of minutes.

Consider a tiny point on the string, right at the intersection of the T₁ string, the horizontal T₂ string, and the vertical string holding up the 3.0 kg mass. According to this small point of string, there are only 3 forces acting on it. So you can isolate this particular point to be a function of only these 3 forces. How many forces are acting on that particular point in the y-direction? Since nothing is accelerating, what does it tell you about those forces that exist in the y-direction (as seen from that particular point)?

[Edit: Btw, The above applies because we are dealing with light strings. If we were dealing with heavy rods, or rigid rods that clamped onto each other and held firm, things might be a little different.]

 

[rdl]

Your equations for the x and y components of force are correct. However, it seems that you have overlooked the fact that the string in the middle is exactly horizontal. This means that the tension in T2 is not zero, as it is providing a horizontal force to balance the weight of m2 and m3.

To solve for the unknowns, you will need to use the fact that the system is in static equilibrium, which means that the sum of all forces and torques must equal zero. This will give you a system of equations that can be solved simultaneously to find the unknowns.

For the torque equation, you have correctly chosen the point where T1 meets m3 as the axis of rotation. However, you will also need to consider the torque due to the weight of m2, which is acting at a distance of R/2 from the axis of rotation.

Here is how you can set up the equations to solve for the unknowns:

Sum of forces in the x-direction:
T3x - T1x - T2 = 0

Sum of forces in the y-direction:
T1y + T3y - m3g - m2g = 0

Sum of torques about the axis of rotation:
T2R + T3yR - m2g(R/2) = 0

You now have three equations with three unknowns (T1, T2, and T3). Solving these equations simultaneously will give you the values for T1, T2, and T3.

To find the angle theta, you can use the fact that the string in the middle is exactly horizontal. This means that the y-component of T2 must be equal to the weight of m2. You can use this relationship to solve for theta.

I hope this helps. Good luck with your problem!