# Solving the Unknown Tension using Equilibrium Conditions

[sponge2006]

http://www.lasalle.edu/~longo/PHY105/labs/lab4Statics/pulleys3.gif

# The tension along the string between knots A and B is unknown. There are four distinct equilibrium conditions, each of which should allow you to calculate the magnitude of the unknown tension

1. the x components for the knot A tensions
2. the y components for the knot A tensions
3. the x components for the knot B tensions
4. the y components for the knot B tensions# Find the magnitude of the unknown tension using each condition. Calculate the percent difference (relative difference) between each value and the average. Did you obtain the same results within an acceptable amount of experimental error?

Equilibrium Data
Number Mass(g) Tension Angle X component Y Component
1 350 3.43 48 -2.295 2.549
2 310 1.568 97 -0.191 1.556
3 160 2.646 82 0.368 -2.620
4 270 3.038 42 2.258 -2.033

please check my math

tension = mass * 9.8
x component = tension * cosin angle
y = tension * sin angle

 

[jbsteele]

1. x component for knot A tensionsTension (A) = (350 * 9.8) + (310 * 9.8) + (160 * 9.8) + (270 * 9.8) = 9179.6 Newtonsx components = (-2.295*9179.6) + (-0.191*9179.6) + (0.368*9179.6) + (2.258*9179.6) = -2433.4 Newtons2. y component for knot A tensionsy components = (2.549*9179.6) + (1.556*9179.6) + (-2.620*9179.6) + (-2.033*9179.6) = -4353.4 Newtons3. x component for knot B tensionsTension (B) = Unknownx components = (-2.295*Tension (B)) + (-0.191*Tension (B)) + (0.368*Tension (B)) + (2.258*Tension (B)) = 0 Newtons4. y component for knot B tensionsy components = (2.549*Tension (B)) + (1.556*Tension (B)) + (-2.620*Tension (B)) + (-2.033*Tension (B)) = 0 NewtonsTension (B) = (2433.4 + 4353.4)/9179.6 = 3.43 NewtonsAverage = (3.43 + 3.43 + 2.646 + 3.038)/4 = 3.077 NewtonsPercent Difference = (3.43-3.077)/3.077 * 100 = 11.5%

 

[baba89]

# Based on the given data, we can use the four equilibrium conditions to calculate the magnitude of the unknown tension. Using the given equations, we can calculate the x and y components of tension for each knot and then use the Pythagorean theorem to find the magnitude of tension.

For knot A, using the x component equilibrium condition, we get:
T_A_x = -2.295 + (-0.191) = -2.486
Using the y component equilibrium condition, we get:
T_A_y = 2.549 + 1.556 = 4.105
Therefore, the magnitude of tension at knot A is:
T_A = √((-2.486)^2 + (4.105)^2) = 4.818 N

For knot B, using the x component equilibrium condition, we get:
T_B_x = 0.368 + 2.258 = 2.626
Using the y component equilibrium condition, we get:
T_B_y = -2.620 + (-2.033) = -4.653
Therefore, the magnitude of tension at knot B is:
T_B = √((2.626)^2 + (-4.653)^2) = 5.292 N

The average of these two values is:
(T_A + T_B)/2 = (4.818 + 5.292)/2 = 5.055 N

The percent difference between each value and the average is:
|4.818 - 5.055|/5.055 * 100% = 4.69%
|5.292 - 5.055|/5.055 * 100% = 4.69%

These values are within an acceptable amount of experimental error, as they are both less than 5%. This shows that the four equilibrium conditions can be used to accurately calculate the magnitude of the unknown tension.