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Finding the Tension of Two Strings with Different Angles

(This problem is #4 from the Forces packet).

You are given a system that is at rest; you know the mass of the object, and the two angles of the strings. In this example problem, there are two strings, one with an angle of 25 degrees, and the other with an angle of 65 degrees, and a mass: 5 kilograms. Label the tension from the strings as T1 and T2, respectively. 

The first thing to notice is that since the system is at rest, the force vectors in the x and y directions balance each other out.

So, you can make an equation regarding the X and Y forces. T1cos(25) = T2cos(65), since the X-vector of each string equals force * cos(theta). T1sin(25) + T2sin(65) = Fg

You only certainly know one force: Fg.

Fg = mass * gravity, which is 5 * 9.8 = 49. So you can make these equations:

 
T1 sin(25) + T2 sin(65) = 49
T1 cos(25) = T2 cos(65)

So, by expressing T2 in terms of T1 and plugging it back into the first equation, we can find a value for T1, then find the value for T2.

T2 = T1 cos(25) / cos(65)
T2 = 2.145 T1

T1 sin(25) + (2.145 T1) sin(65) = 49
(sin(25) + 2.145 * sin(65)) T1 = 49
2.366 T1 = 49
T1 = 20.708

So, now you plug T1 into the first equation.

T2 = 2.145(20.708)
T2 = 44.4193

Now, you get your final answer: Tension in String 1 is 20.708 N and Tension in String 2 is 44.4193 N

David Witten

An ATWOOD's Machine