A rope suspended from a ceiling supports an object of weight W at its opposite end. Another rope tied to the first at the middle is pulled horizontally with a force of 30N. The junction P of the ropes is in equilibrium. Calculate the weight W and the tens
To solve this problem, we can set up an equilibrium equation for the forces acting on the junction P. The forces in equilibrium can be represented by two vectors: the vertical force acting downward due to the weight W, and the horizontal force acting to the right due to the tension in the second rope.
Let's break down the problem step by step:
Step 1: Identify the forces acting on the junction P:
- The weight W acting vertically downward.
- The tension in the second rope acting horizontally to the right, with a force of 30N.
Step 2: Express the equilibrium equation:
Since the junction P is in equilibrium, the sum of the forces in the vertical and horizontal directions is zero. Therefore, we can express the equilibrium equation as:
Vertical forces = Horizontal forces
Step 3: Write the equilibrium equation:
The vertical forces are represented by the weight W, which is acting downward. The horizontal forces are represented by the tension in the second rope, which is pulling to the right.
So, we can write the equilibrium equation as:
W = 30N
Step 4: Solve for W:
From the equilibrium equation, we can see that the weight W is equal to 30N. Therefore, the weight of the object hanging from the rope is 30N.
Step 5: Calculate the tension in the rope:
Since the junction P is in equilibrium, the tension in the rope can be determined by considering the two halves of the rope on either side of the junction. Since the forces are balanced, half of the weight is supported by each half of the rope.
Therefore, half of the weight W is equal to half of 30N, which is:
Tension = W/2 = 30N/2 = 15N
The tension in the rope is 15N.
So, the weight of the object W is 30N and the tension in the rope is 15N.
To calculate the weight (W) of the object and the tension in each of the ropes, we can use the principles of equilibrium which state that the sum of the forces acting on an object should be zero, both horizontally and vertically.
First, let's consider the vertical forces. The weight (W) of the object is acting downwards, and the tension in the rope is acting upwards. Since the junction P is in equilibrium, the vertical forces are balanced:
W - tension = 0
This equation tells us that the weight (W) is equal to the tension in the rope. Therefore, we can say:
W = tension
Next, let's consider the horizontal forces. The rope that is being pulled horizontally with a force of 30N is exerting a force on the junction P. This force is balanced by the tension in the rope on the other side. Again, since the junction is in equilibrium, the horizontal forces are balanced:
tension + 30N = 0
Solving this equation for tension, we find:
tension = -30N
Now, substitute this value of tension back into the previous equation:
W = -30N
Since the weight (W) cannot be negative, we know that the magnitude of the weight is equal to 30N.
Therefore, the weight (W) of the object is 30N and the tension in each of the ropes is also 30N.