An object of weight W hangs by three ropes as shown above. What is the value of the angle @ if T1 = T2 = W?
____________
\@ @/
T2\ /T1
\__/
|W|
a) 15
b) 30
c) 40
d) 45
e) 60
To find the value of the angle @, we can analyze the forces acting on the object.
First, let's consider the vertical forces. The weight of the object (W) pulls it downwards. Since the object is in equilibrium, the sum of the vertical forces must be zero. Therefore, the upward force provided by the tension in the ropes must balance the weight.
Now, let's look at the forces horizontally. Since both T1 and T2 have the same tension and are symmetrically placed, they cancel each other out. So, the horizontal forces are balanced as well.
From the information given, we can assume that the angles at T1 and T2 are equal. Let's call this angle θ.
Using trigonometry, we can find the relationship between the forces. The vertical component of each tension force (T1 and T2) can be found using the equation:
T_vertical = T * sin(θ)
Since T1 is equal to T2, we have:
T1_vertical = T2_vertical = T * sin(θ)
The sum of these vertical components should balance the weight, W:
2T * sin(θ) = W
Since T1 = T2 = W, we can substitute W for T:
2W * sin(θ) = W
Simplifying the equation, we have:
2 * sin(θ) = 1
Dividing both sides by 2, we get:
sin(θ) = 1/2
To find the value of θ, we can use the inverse sine function, or arcsin:
θ = arcsin(1/2)
Now we need to find the value of θ. Evaluating the arcsin(1/2), we get:
θ = 30 degrees
Therefore, the value of the angle @ is 30 degrees.
So the answer is option b) 30.