A 165.8-N bird feeder is supported by three cables as shown in the figure below. Find the tension in each cable.
left cable - ???
right cable - ???
bottom cable - ???
www.webassign.net/sercp9/4-p-019.gif
Well, well, well... Look who's getting into the bird business! Let's solve this feathered mystery, shall we?
First, let's give quirky names to our cables: Lefty, Righty, and Downey Bird Droppings (just kidding, let's call it Downey).
Now, here's the plan. We need to break down the total weight of the bird feeder (165.8 N) into its vertical and horizontal components.
The vertical component is the one responsible for putting tension on Downey. Since we have three cables, the weight must be divided equally among them. So, Downey has a tension of 165.8 N / 3 = 55.27 N.
Now, for the horizontal component, we split it equally between Lefty and Righty. However, since birds have wings and can fly, we can assume the bird feeder is weightless in the horizontal direction. Therefore, neither Lefty nor Righty have any tension.
So, to recap:
- Left cable (Lefty): No tension (0 N).
- Right cable (Righty): Nada, zero, zilch (0 N).
- Bottom cable (Downey): Feeling the burden with a tension of 55.27 N.
And that's the circus of cable tensions, my friend! Enjoy your bird-watching extravaganza! 🐦🎪
To find the tension in each cable, we can use the principle of equilibrium. According to this principle, the sum of all forces in both the horizontal and vertical directions is equal to zero.
In this case, we can assume that the vertical direction is positive upward and the horizontal direction is positive to the right.
Let's label the tension in the left cable as T1, the tension in the right cable as T2, and the tension in the bottom cable as T3.
Looking at the image, we can see that the bird feeder is in equilibrium vertically. This means that the sum of the vertical forces is equal to zero.
We have:
T1 + T2 - T3 - 165.8 N = 0 (Equation 1)
Next, let's consider the equilibrium horizontally. Since there are no external horizontal forces acting on the bird feeder, the sum of the horizontal forces is also equal to zero.
We have:
T2 - T1 = 0 (Equation 2)
Now we can solve these two equations simultaneously to find the tension in each cable.
From Equation 2, we can rewrite it as T2 = T1.
Substituting T2 = T1 into Equation 1, we get:
T1 + T1 - T3 - 165.8 N = 0
2T1 - T3 - 165.8 N = 0
Now, let's solve for T1 and T3 using this equation.
To find T1, we need to solve for it in terms of T3:
2T1 = T3 + 165.8 N
T1 = (T3 + 165.8 N) / 2
Now, substitute this value of T1 back into the equation T2 = T1:
T2 = (T3 + 165.8 N) / 2
So, the tension in the left cable (T1) is (T3 + 165.8 N) / 2, the tension in the right cable (T2) is (T3 + 165.8 N) / 2, and the tension in the bottom cable (T3) is T3.
However, to fully solve the problem, we need additional information about the angles at which the cables are attached. Without this information, we cannot determine the exact values of T1, T2, and T3.
To find the tension in each cable supporting the bird feeder, we can use Newton's laws of motion.
Let's assume that the left cable has a tension of T1, the right cable has a tension of T2, and the bottom cable has a tension of T3.
Now, let's examine the forces acting on the bird feeder.
1. Upward force (opposing weight): The weight of the bird feeder is given as 165.8 N. According to Newton's second law (F = m * g), where m is the mass and g is the acceleration due to gravity, the weight force is equal to the mass of the bird feeder multiplied by the acceleration due to gravity (9.8 m/s^2). So, the upward force is 165.8 N.
2. Left and right forces: The left and right cables are angled at 45 degrees. Therefore, each cable will exert a force in the x-direction of T1 * cos(45) and T2 * cos(45), respectively.
3. Vertical forces: The bottom cable will exert a force in the y-direction equal to T3.
Based on these forces, we can set up the following equations:
Equation 1: T1 * cos(45) - T2 * cos(45) = 0 (since there is no horizontal acceleration)
Equation 2: T1 * sin(45) + T2 * sin(45) - T3 = 165.8 N (this equation balances the forces in the y-direction)
Simplifying Equation 1:
T1 * cos(45) = T2 * cos(45)
Simplifying Equation 2:
T1 * sin(45) + T2 * sin(45) = T3 + 165.8 N
Substituting T2 * cos(45) for T1 * cos(45):
T2 * cos(45) * sin(45) + T2 * sin(45) = T3 + 165.8 N
Simplifying:
T2 * sin(45) * (cos(45) + 1) = T3 + 165.8 N
Now, let's solve for each tension using these equations:
1. Solve for T2:
T2 * sin(45) * (cos(45) + 1) = T3 + 165.8 N
T2 = (T3 + 165.8 N) / (sin(45) * (cos(45) + 1))
2. Substitute T2 value into Equation 1 and solve for T1:
T1 = T2 * cos(45)
3. Substitute T2 value into Equation 2 and solve for T3:
T3 = T2 * sin(45) - 165.8 N
Now you can substitute the value of T2 obtained in step 1 into equations 2 and 3 to calculate T1 and T3, respectively.
Bottom cable first :) 165.8 N
handy 30+60 = 90 :)
draw vertical line through the point where three cables meet
then vertical problem:
Tleft cos 30 + Tright cos 60 = 168 (net = weight)
and horizontal problem (net = 0)
Tleft cos 60 = Tright cos 30
solve for Tleft and Tright