A 143.8-N bird feeder is supported by three cables as shown in the figure below. Find the tension in each cable.
The rope on the right is 30 degrees and the rope on the left is 60 degrees. and the middle rope is connected to the bird feeder and is vertical.
How do I approach this problem?
Left 79.5
To approach this problem, you can start by breaking down the weight of the bird feeder into its components along the vertical and horizontal directions. Then, you'll need to use trigonometry to find the tension in each cable.
Here's how you can proceed:
Step 1: Resolve the weight of the bird feeder into vertical and horizontal components.
The total weight of the bird feeder is 143.8 N. The vertical component of the weight is equal to the total weight since the middle cable is connected vertically. The horizontal component is zero since the bird feeder is not being pulled horizontally.
Step 2: Determine the tensions in the cables using the resolved components.
Let's label the tensions in the cables as T1, T2, and T3 for the left, middle, and right cables, respectively.
For T1:
Since the angle between the left cable and the vertical is 60 degrees, the vertical component of T1 is given by T1 * cos(60°) = T1/2.
For T3:
Since the angle between the right cable and the vertical is 30 degrees, the vertical component of T3 is given by T3 * cos(30°) = T3 * √3/2.
The vertical components of T1 and T3 should balance the weight of the bird feeder, which is the same as the tension in the middle cable.
So, T1/2 + T3 * √3/2 = 143.8 N
Step 3: Solve the equation for T1 and T3.
To find the values of T1 and T3, you'll need to solve the equation using basic algebra. Rearrange the equation:
T1/2 = 143.8 N - T3 * √3/2
Multiply both sides by 2:
T1 = 287.6 N - T3 * √3
Now you have an equation relating the tensions T1 and T3.
Step 4: Find the remaining tension T2.
Since T2 is connected vertically, its tension is equal to the total weight of the bird feeder, which is 143.8 N.
So now you have the tensions in each cable: T1, T2, and T3.