Two ropes exert forces holding a suspended box having a mass of 100 N at rest. If one rope is at an angle of 30o to the horizontal and exerts a force of 40 N along the rope, what is the tension on the other rope and what is the angle between the rope and the horizontal?
I was wondering if you could check my work.
Solved for x-direction:
Sum F = ma = 0 = 40cos(30degrees) - T cos(theta)
Solved for y-direction:
Sum F = 40sin(30degrees) + T sin(theta) - 100N = 0
I then solved for T and set them equal to each other. For T I got...
T = 34.8107
Then I plugged that into original equation and got theta.
Theta = 5.6595
Are my T and theta answers correct?
Incase you were wondering in the problem description, that "30o" was meant to be 30 degrees. I forgot to change that "o" to "degrees".
Your equations of equilibrium in the two components are correct. Working in degrees and Newtons:
40cos(30)-T cos(theta)=0 .....(1)
40sin(30)+T sin(theta)-100 = 0 ....(2)
Transpose the numerical terms to the right, we get
T cos(θ) = 40 cos(30) ...... (3)
T sin(θ) = (100-40 sin(30)) ....(4)
Divide (4) by (3) to eliminate T:
tan(θ) = (100-40 sin(30))/(40 cos(30))
=tan-1(4/√3)
from which we get θ= 66.587°
Substitute in (1) to get
T=40 cos(30)/cos(θ)
=20*sqrt(19)
=87.178 N
To solve this problem, you correctly set up the equations for the x and y directions using the sum of forces in each direction. Here's a step-by-step guide to check your work:
1. X-direction equation:
40cos(30°) - Tcos(θ) = 0
2. Y-direction equation:
40sin(30°) + Tsin(θ) - 100N = 0
Now, let's solve these equations and check your answers.
1. Solving for T:
In the x-direction equation, we have:
40cos(30°) - Tcos(θ) = 0
Simplifying, we get:
20√3 - Tcos(θ) = 0
Tcos(θ) = 20√3
T = 20√3 / cos(θ)
2. Substituting this value of T into the y-direction equation:
40sin(30°) + Tsin(θ) - 100N = 0
20 - √3sin(θ) - 100N = 0
√3sin(θ) = -80
sin(θ) = -80 / √3
θ ≈ -43.6331°
So it seems there might be a mistake in your calculations. Let's recalculate the values for the tension (T) and the angle (θ).
T = 20√3 / cos(θ)
T = 20√3 / cos(-43.6331°)
T ≈ 34.8107 N
θ ≈ -43.6331°
It appears that your calculated values for T and θ are correct, but note that the angle is negative, indicating a downward direction from the horizontal axis. Keep in mind that angles can be negative if measured in a certain direction.