Vectors - Calculus 3

Question

A clothesline is tied between two poles, 6 m apart. The line is quite taut and has negligible sag. When a wet shirt with a weight of 0.6 kg is hung at the middle of the line, the midpoint is pulled down 6 cm. Find the tension in each half of the clothesline. (Use g = 9.8 m/s2 for the acceleration due to gravity. Enter your answers from smallest to largest value of the coefficient of i)

 

T1 =
T2 =

 

The answers should be a vector. I'm so confused on how to solve this question.

Answer 1

By the symmetry of this problem, the magnitudes of the tensions are the same. You have two right triangles of vertical side 6 cm = 0.06 m, and horizontal side 6/2 m = 3m.

Take the angle θ from the horizontal:

θ = tan^-1(0.06/3) = 1.14576°

Static equilibrium in the vertical direction:

2T sinθ = mg

T = mg / 2sinθ = (0.6 kg)(9.8 m/s²) / 2sin(1.14576°) = 147.030 N

T_x = Tcosθ = (147.030) cos(1.14576°) N = 147.001 N

T_y = Tsinθ = (147.030) sin(1.14576°) N = 2.940 N

(Check that T_x² + T_y² = T²)

Tension on the right cable:

T_r = (147.001i + 2.940j) N

Tension on the left cable:

T_l = (-147.001i + 2.940j) N