A 520 N cat burglar is supported by cables in the figure below. Assume the angle θ of the inclined cable is 32.0°.
Find the tension in each cable. (Inclined, vertical, and horizontal)
Inclined Cable: 517.2 N
Vertical Cable: 437.2 N
Horizontal Cable: 437.2 N
To find the tension in each cable, we can start by resolving the forces acting on the cat burglar along the vertical and horizontal directions.
1. Resolving forces along the vertical direction:
Since the cat burglar is not accelerating in the vertical direction (assuming, we neglect air resistance), the sum of the vertical forces acting on the cat burglar must be zero.
The vertical forces acting on the cat burglar are the weight (mg) and the vertical component of tension in the inclined cable (T⊥).
The weight (mg) can be calculated as the product of the mass (m) and the acceleration due to gravity (g).
mg = 520 N
The vertical component of tension (T⊥) can be found using trigonometry. It is given by:
T⊥ = T * sin(θ)
where T is the tension in the inclined cable and θ is the angle of the inclined cable.
2. Resolving forces along the horizontal direction:
There are two horizontal forces acting on the cat burglar - the horizontal component of tension in the inclined cable (T∥) and the tension in the vertical cable (Tvertical).
The horizontal component of tension (T∥) can also be found using trigonometry. It is given by:
T∥ = T * cos(θ)
Now we have resolved the forces in both the vertical and horizontal directions. We can use these equations to solve for the tensions in each cable.
Tvertica + T⊥ = mg (Vertical forces must add up to zero)
T∥ = Tvertical (Horizontal forces must add up to zero)
Substituting the values, we can solve for T and Tvertical.
T + T * sin(θ) = mg (Equation 1)
T∥ = Tvertical (Equation 2)
Using Equation 2, we can substitute T with T∥ in Equation 1:
T∥ + T∥ * sin(θ) = mg
Now we can solve for T∥:
T∥ * (1 + sin(θ)) = mg
T∥ = mg / (1 + sin(θ))
Similarly, we can solve for T using T⊥ = T * sin(θ):
T * sin(θ) = mg - T⊥
T * sin(θ) = mg - T * sin(θ)
T * sin(θ) + T * sin(θ) = mg
T * (sin(θ) + sin(θ)) = mg
T = mg / (2 * sin(θ))
Finally, we have the values for T and T∥ which represent the tension in the inclined cable and the vertical cable, respectively.
To find the tension in each cable, we can use trigonometry and the concept of forces in equilibrium. Let's denote the tension in the inclined cable as T1, the tension in the vertical cable as T2, and the tension in the horizontal cable as T3.
Given:
Weight (force due to gravity) acting on the cat burglar = 520 N
Angle of the inclined cable, θ = 32.0°
Step 1: Resolve the weight into its components.
Since the weight acts vertically downwards, we can resolve it into perpendicular components.
Vertical component (weight): Wv = Weight * cos(θ)
Wv = 520 N * cos(32.0°)
Horizontal component (weight): Wh = Weight * sin(θ)
Wh = 520 N * sin(32.0°)
Step 2: Apply forces in equilibrium.
For the cat burglar to be in equilibrium, the sum of the forces in the horizontal and vertical directions must equal zero.
In the horizontal direction:
T3 - Wh = 0
T3 = Wh
In the vertical direction:
T1 - Wv - T2 = 0
T1 - Wv = T2
Substituting the values of Wv and Wh from Step 1, we have:
T1 - 520 N * cos(32.0°) = T2
T3 = 520 N * sin(32.0°)
Step 3: Calculate the values of T1, T2, and T3.
Now, we have two equations:
T1 - 520 N * cos(32.0°) = T2 ...(Equation 1)
T3 = 520 N * sin(32.0°) ...(Equation 2)
We can solve these equations simultaneously to find the values of T1, T2, and T3.
To simplify the calculations, let's assume the value of sin(32.0°) is approximately 0.53 and cos(32.0°) is approximately 0.85.
Substituting these values into Equation 2:
T3 = 520 N * 0.53
T3 ≈ 275.6 N
Substituting the value of T3 into Equation 1:
T1 - 520 N * 0.85 = T2
T1 ≈ T2 + 442 N
Therefore, the tension in each cable is approximately:
T1 ≈ T2 + 442 N
T2 ≈ T1 - 442 N
T3 ≈ 275.6 N
Please note that these are approximate values.