the three weights are hanging in static equilibrium. The two weights left and right of center have a mass of 100 Kg. The weight in the center has a mass of 100 Kg. Treat the strings as massless and the pulleys as frictionless. If the sin of the angle theta in the figure is sin theta = 1/n3 , what is n3?
To solve this question, we can start by analyzing the forces acting on each weight:
1. The weight on the left has a tension force pulling it towards the left, and the weight in the center has two equal and opposite tension forces acting on it, because the string is passing through the pulley.
2. The weight on the right also has a tension force pulling it towards the right.
Since the system is in static equilibrium, the sum of the forces in the horizontal direction and the sum of the forces in the vertical direction must both be zero.
Let's consider the vertical forces first:
1. The weight on the left is balanced by the vertical component of the tension force in the left string, which is equal to (100 kg) * g, where g is the acceleration due to gravity.
2. The weight in the center is balanced by the sum of the vertical components of the tension forces in both strings. Each component is (100 kg) * g/2 because the tension is split equally on both sides.
3. The weight on the right is balanced by the vertical component of the tension force in the right string, which is also (100 kg) * g.
So we have the equation:
(100 kg) * g - 2 * (100 kg) * g/2 + (100 kg) * g = 0
Simplifying the equation, we get:
100 kg * g = 100 kg * g
Now, let's consider the horizontal forces:
1. The weight on the left is balanced by the horizontal component of the tension force in the left string, which is equal to T1 * sin(theta).
2. The weight on the right is balanced by the horizontal component of the tension force in the right string, which is equal to T2 * sin(theta).
Since the system is in static equilibrium, the horizontal forces must also be balanced:
T1 * sin(theta) = T2 * sin(theta)
Dividing both sides of the equation by sin(theta), we get:
T1 = T2
Now, since the pulleys are frictionless, the tensions in both strings must be equal. Therefore, we can conclude that T1 = T2.
Finally, we can substitute T1 = T2 into the horizontal forces equation:
T1 * sin(theta) = T2 * sin(theta)
T1 * sin(theta) = T1 * sin(theta)
Now we can see that the sin(theta) cancels out on both sides of the equation.
Therefore, sin(theta) = 1/n3, where n3 is the value of sin(theta).