Answer:
Please red the answer below
Step-by-step explanation:
In order to determine the length of each cable you use the Newton second law for each component of the forces involved in the situation.
For the x component you have:
(1)
T1: tension of the first cable = 1500N
T2: tension of the second cable = 800N
θ1: angle between the horizontal and the first cable
θ2: angle between the horizontal and the first cable
For the y component you have:
(2)
W: weight of the camera = Mg = (140kg)(9.8m/s^2) = 1372N
You can squared both equations (1) and (2) and the sum the two equations:
![T_1^2cos^2\theta_1=T_2^2cos^2\theta_2\\\\T_1^2sin^2\theta_1=T_2^2sin^2\theta_2-2WT_2sin\theta_2+W^2](https://tex.z-dn.net/?f=T_1%5E2cos%5E2%5Ctheta_1%3DT_2%5E2cos%5E2%5Ctheta_2%5C%5C%5C%5CT_1%5E2sin%5E2%5Ctheta_1%3DT_2%5E2sin%5E2%5Ctheta_2-2WT_2sin%5Ctheta_2%2BW%5E2)
Then, you sum the equations:
(3)
Next, you use the following identity:
![sin^2\theta+cos^2\theta=1](https://tex.z-dn.net/?f=sin%5E2%5Ctheta%2Bcos%5E2%5Ctheta%3D1)
and you obtain in the equation (3):
![T_1^2=T_2^2-2WT_2sin\theta_2+W^2\\\\sin\theta_2=\frac{T_2^2-T_1^2+W^2}{2WT_2}=\frac{(800N)^2-(1500)^2+(1372N)^2}{2(800N)(1372N)}=0.066\\\\\theta_2=sin^{-1}0.066=27.23\°](https://tex.z-dn.net/?f=T_1%5E2%3DT_2%5E2-2WT_2sin%5Ctheta_2%2BW%5E2%5C%5C%5C%5Csin%5Ctheta_2%3D%5Cfrac%7BT_2%5E2-T_1%5E2%2BW%5E2%7D%7B2WT_2%7D%3D%5Cfrac%7B%28800N%29%5E2-%281500%29%5E2%2B%281372N%29%5E2%7D%7B2%28800N%29%281372N%29%7D%3D0.066%5C%5C%5C%5C%5Ctheta_2%3Dsin%5E%7B-1%7D0.066%3D27.23%5C%C2%B0)
With this values you can calculate the value of the another angle, by using the equation (1):
![\theta_1=cos^{-1}(\frac{T_2cos(27.23\°)}{T_1})=cos^{-1}(\frac{(800N)(cos27.23\°)}{1500N})\\\\\theta_1=61.69\°](https://tex.z-dn.net/?f=%5Ctheta_1%3Dcos%5E%7B-1%7D%28%5Cfrac%7BT_2cos%2827.23%5C%C2%B0%29%7D%7BT_1%7D%29%3Dcos%5E%7B-1%7D%28%5Cfrac%7B%28800N%29%28cos27.23%5C%C2%B0%29%7D%7B1500N%7D%29%5C%5C%5C%5C%5Ctheta_1%3D61.69%5C%C2%B0)
Now, you can calculate the length of each cable by using the information about the width of the football field. You use the following trigonometric relation:
![l_1cos\theta_1=40-d\\\\l_2cos\theta_2=d\\\\](https://tex.z-dn.net/?f=l_1cos%5Ctheta_1%3D40-d%5C%5C%5C%5Cl_2cos%5Ctheta_2%3Dd%5C%5C%5C%5C)
d: distance to the right side of the field
By using the cosine law you can fins a system of equation and then you can calculate the values of l1 and l2.