A weight of 2000 pounds is suspended from two cables as shown in the figure. What is the
tension in the two cables?
What is the tension in the left cable?
pounds
(Round to one decimal place as needed.)
G
20°
50°
2000 pounds

A weight of 2000 pounds is suspended from two cables as shown in the figure What is thetension in the two cablesWhat is the tension in the left cablepoundsRound class=


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: T1cos⁡θ1T1​cosθ1​

   Vertical component: T1sin⁡θ1T1​sinθ1​

For T2T2​:

   Horizontal component: T2cos⁡θ2T2​cosθ2​

   Vertical component: T2sin⁡θ2T2​sinθ2​

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:

T1cos⁡θ1=T2cos⁡θ2T1​cosθ1​=T2​cosθ2​

Vertical Equilibrium:

T1sin⁡θ1+T2sin⁡θ2=WT1​sinθ1​+T2​sinθ2​=W

Given W=2000W=2000 pounds, we have:

T1sin⁡θ1+T2sin⁡θ2=2000T1​sinθ1​+T2​sinθ2​=2000

3. Solve the System of Equations:

We now have two equations with two unknowns (T1T1​ and T2T2​):

   T1cos⁡θ1=T2cos⁡θ2T1​cosθ1​=T2​cosθ2​

   T1sin⁡θ1+T2sin⁡θ2=2000T1​sinθ1​+T2​sinθ2​=2000

From the horizontal equilibrium equation, solve for T1T1​ in terms of T2T2​:

T1=T2cos⁡θ2cos⁡θ1T1​=T2​cosθ1​cosθ2​​

Substitute this expression for T1T1​ into the vertical equilibrium equation:

T2cos⁡θ2cos⁡θ1sin⁡θ1+T2sin⁡θ2=2000T2​cosθ1​cosθ2​​sinθ1​+T2​sinθ2​=2000

Simplify using trigonometric identities:

T2(cos⁡θ2sin⁡θ1cos⁡θ1+sin⁡θ2)=2000T2​(cosθ1​cosθ2​sinθ1​​+sinθ2​)=2000

Factor out T2T2​:

T2(sin⁡θ1cos⁡θ1cos⁡θ2+sin⁡θ2)=2000T2​(cosθ1​sinθ1​​cosθ2​+sinθ2​)=2000

Simplify the fraction sin⁡θ1cos⁡θ1cosθ1​sinθ1​​ to tan⁡θ1tanθ1​:

T2(tan⁡θ1cos⁡θ2+sin⁡θ2)=2000T2​(tanθ1​cosθ2​+sinθ2​)=2000

Solve for T2T2​:

T2=2000tan⁡θ1cos⁡θ2+sin⁡θ2T2​=tanθ1​cosθ2​+sinθ2​2000​

Substitute T2T2​ back into the equation for T1T1​:

T1=T2cos⁡θ2cos⁡θ1T1​=T2​cosθ1​cosθ2​​

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.

   cos⁡30∘=32cos30∘=23

​​

sin⁡30∘=12sin30∘=21​

cos⁡45∘=22cos45∘=22

​​

sin⁡45∘=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=T2cos⁡45∘cos⁡30∘T1​=T2​cos30∘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.