Answer :
Answer:
Explanation:
To solve for the tensions in the two cables suspending a weight of 2000 pounds, we need to analyze the forces in the system using principles of static equilibrium. Let's denote the tensions in the two cables as T1T1 and T2T2, and we'll assume the angles each cable makes with the horizontal are given or can be determined.
For the sake of explanation, let's assume the angles the cables make with the horizontal are θ1θ1 for T1T1 and θ2θ2 for T2T2.
Assumptions and Setup:
The weight W=2000W=2000 pounds acts downward.
The system is in static equilibrium, meaning the sum of all forces in both the horizontal and vertical directions is zero.
T1T1 and T2T2 are the tensions in the cables making angles θ1θ1 and θ2θ2 with the horizontal, respectively.
Step-by-Step Solution:
1. Resolve Forces into Components:
The tension forces have both horizontal and vertical components. These components can be resolved as follows:
For T1T1:
Horizontal component: T1cosθ1T1cosθ1
Vertical component: T1sinθ1T1sinθ1
For T2T2:
Horizontal component: T2cosθ2T2cosθ2
Vertical component: T2sinθ2T2sinθ2
2. Apply Equilibrium Conditions:
In static equilibrium, the sum of the forces in both the xx- (horizontal) and yy- (vertical) directions must be zero.
Horizontal Equilibrium:
T1cosθ1=T2cosθ2T1cosθ1=T2cosθ2
Vertical Equilibrium:
T1sinθ1+T2sinθ2=WT1sinθ1+T2sinθ2=W
Given W=2000W=2000 pounds, we have:
T1sinθ1+T2sinθ2=2000T1sinθ1+T2sinθ2=2000
3. Solve the System of Equations:
We now have two equations with two unknowns (T1T1 and T2T2):
T1cosθ1=T2cosθ2T1cosθ1=T2cosθ2
T1sinθ1+T2sinθ2=2000T1sinθ1+T2sinθ2=2000
From the horizontal equilibrium equation, solve for T1T1 in terms of T2T2:
T1=T2cosθ2cosθ1T1=T2cosθ1cosθ2
Substitute this expression for T1T1 into the vertical equilibrium equation:
T2cosθ2cosθ1sinθ1+T2sinθ2=2000T2cosθ1cosθ2sinθ1+T2sinθ2=2000
Simplify using trigonometric identities:
T2(cosθ2sinθ1cosθ1+sinθ2)=2000T2(cosθ1cosθ2sinθ1+sinθ2)=2000
Factor out T2T2:
T2(sinθ1cosθ1cosθ2+sinθ2)=2000T2(cosθ1sinθ1cosθ2+sinθ2)=2000
Simplify the fraction sinθ1cosθ1cosθ1sinθ1 to tanθ1tanθ1:
T2(tanθ1cosθ2+sinθ2)=2000T2(tanθ1cosθ2+sinθ2)=2000
Solve for T2T2:
T2=2000tanθ1cosθ2+sinθ2T2=tanθ1cosθ2+sinθ22000
Substitute T2T2 back into the equation for T1T1:
T1=T2cosθ2cosθ1T1=T2cosθ1cosθ2
Example with Specific Angles:
Let's say the angles are θ1=30∘θ1=30∘ and θ2=45∘θ2=45∘.
Convert the angles to radians if necessary or use the trigonometric values directly.
cos30∘=32cos30∘=23
sin30∘=12sin30∘=21
cos45∘=22cos45∘=22
sin45∘=22sin45∘=22
Substitute these into the equations:
T2=2000(13⋅22+22)T2=(3
1⋅22
+22
)2000
Simplify the expression in the denominator:
T2=2000223+22T2=23
2
+22
2000
Factor out 2222
from the denominator:
T2=200022(13+1)T2=22
(3
1+1)2000
Simplify further:
T2=2000⋅22(1+13)T2=2
(1+3
1)2000⋅2
T2=40002(3+13)T2=2
(3
3
+1)4000
Multiply by the conjugate or simplify:
T2=400032(3+1)T2=2
(3
+1)40003
T2=400032(3+1)×3−13−1T2=2
(3
+1)40003
×3
−13
−1
T2=40003(3−1)2(3−1)T2=2(3−1)40003
(3
−1)
T2=40003(3−1)4T2=440003
(3
−1)
T2=10003(3−1)T2=10003
(3
−1)
Approximate 3≈1.7323
≈1.732:
T2≈1000⋅1.732⋅(1.732−1)T2≈1000⋅1.732⋅(1.732−1)
T2≈1000⋅1.732⋅0.732T2≈1000⋅1.732⋅0.732
T2≈1267.0 poundsT2≈1267.0pounds
Now calculate T1T1:
T1=T2cos45∘cos30∘T1=T2cos30∘cos45∘
T1=1267.0⋅2232T1=1267.0⋅23
22
T1=1267.0⋅23T1=1267.0⋅3
2
T1≈1267.0⋅1.4141.732T1≈1267.0⋅1.7321.414
T1≈1035.0 poundsT1≈1035.0pounds
Final Answer:
For the angles θ1=30∘θ1=30∘ and θ2=45∘θ2=45∘:
The tension in cable 1 (T1T1) is approximately 1035.01035.0 pounds.
The tension in cable 2 (T2T2) is approximately 1267.01267.0 pounds.
These values are derived based on the given angles. Adjust the angles as necessary if different values are provided in the actual problem statement.
