Consider the 667 N weight held by two cables shown below. The left-hand cable had tension 890 N and makes an angle of è with the wall. The right-hand cable had tension 960 N and makes an angle of è1 with the ceiling.
a) What is the angle è1 which the right- hand cable makes with respect to the ceiling? Round up to four significant digits. Express unit as ”degrees”.
b) What is the angle è which the left-hand cable makes with respect to the wall? Round up to four significant digits. Express unit as ”degrees”.
The process has stumbled me for a while, but am certain that I need to use sin^2 + cos^2=1 somewhere along the way. I appreciate your help.
To find the angles è1 and è, we can use the concept of equilibrium. In this case, since the weight is being held by two cables, the sum of the forces in the horizontal and vertical directions should cancel out the weight.
Let's break down the forces acting on the weight in the horizontal and vertical directions:
For the horizontal direction:
Tension1 * cos(è) = Tension2 * cos(è1) (since the horizontal forces balance each other)
For the vertical direction:
Tension1 * sin(è) + Tension2 * sin(è1) = Weight
We know the weight is 667 N, Tension1 is 890 N, and Tension2 is 960 N.
Now let's solve the equations to find the angles:
a) To find angle è1:
From the horizontal force equation:
Tension1 * cos(è) = Tension2 * cos(è1)
890 N * cos(è) = 960 N * cos(è1)
Divide both sides by 960 N:
cos(è1) = 890 N * cos(è) / 960 N
è1 = arccos(890 N * cos(è) / 960 N)
Use this equation to calculate the value of è1, rounding to four significant digits. Don't forget to convert the result to degrees.
b) To find angle è:
From the vertical force equation:
Tension1 * sin(è) + Tension2 * sin(è1) = Weight
890 N * sin(è) + 960 N * sin(è1) = 667 N
Rearrange the equation by isolating sin(è):
sin(è) = (667 N - 960 N * sin(è1)) / 890 N
è = arcsin((667 N - 960 N * sin(è1)) / 890 N)
Use this equation to calculate the value of è, rounding to four significant digits. Convert the result to degrees.
Remember to substitute the value of è1 calculated in part a) into the equation to find the value of è.