A uniform 100-N pipe is used as lever. Where the fulcrum (the support point) must be placed if a 500-N weight at one end is to balance a 200-N weight at the other end? What is the reaction force exerted by the support on the pipe?
Let the fulcrum be placed at a distance x from the 500-N weight and (100 - x) from the 200-N weight. We can set up the following equation to satisfy the balance condition:
500 * x = 200 * (100 - x)
Solving for x, we get:
500x = 20000 - 200x
700x = 20000
x ≈ 28.57 cm
So, the fulcrum must be placed approximately 28.57 cm from the 500-N weight (and 71.43 cm from the 200-N weight) to balance the lever.
Now let's find the reaction force exerted by the support on the pipe. To do this, we can analyze the forces acting on the pipe:
The total weight on the pipe = 500 + 200 + 100 = 800 N
Since the pipe is in equilibrium, the reaction force (R) exerted by the support must be equal to the total weight acting on the pipe:
R = 800 N
To balance the lever, the sum of the moments about the fulcrum must be equal to zero. The moment is the product of the force and the distance from the fulcrum.
Let's denote the distance from the fulcrum to the 500-N weight as x and the distance from the fulcrum to the 200-N weight as y.
The moment equation can be written as:
Moment of 500-N weight = Moment of 200-N weight
(500 N) * x = (200 N) * y
Now, we can solve for x in terms of y:
x = (200 N * y) / 500 N
Simplifying the expression, we get:
x = 2y/5
The fulcrum needs to be placed at a distance of 2y/5 from the 500-N weight. This means the fulcrum should be closer to the 500-N weight.
To find the reaction force exerted by the support on the pipe, we need to consider the vertical equilibrium. The sum of the upward forces must be equal to the sum of the downward forces.
Let's denote the reaction force exerted by the support on the pipe as R.
R + 500 N + 200 N = Weight of the pipe (100 N)
R = 100 N - 700 N
R = -600 N
The reaction force exerted by the support on the pipe is -600 N, indicating that it is acting downward.
To determine the position of the fulcrum, let's use the principle of moments or torque. Torque is calculated by multiplying the force applied to an object by its distance from the fulcrum.
In this case, we have a 500-N weight acting at one end of the pipe and a 200-N weight acting at the other end. Let's refer to the distance from the fulcrum to the 500-N weight as "d1" and the distance from the fulcrum to the 200-N weight as "d2". We need to find the position of the fulcrum, which we'll call "x".
According to the principle of moments, the total torque on the pipe must be zero for it to be in equilibrium. Mathematically, this can be expressed as:
Total torque = (Force1 × distance1) + (Force2 × distance2) = 0
Since the total torque is zero, we can say:
(500 N) × (d1) + (-200 N) × (d2) = 0
Simplifying the equation:
500 × d1 = 200 × d2
Now we have a relationship between the distances. We can use this equation to determine the position of the fulcrum.
Next, let's consider the reaction force exerted by the support on the pipe. At equilibrium, the sum of the vertical forces must be zero. The reaction force is equal in magnitude but opposite in direction to the sum of the weight forces acting on the pipe.
In this case, the total weight on the pipe is 500 N (from the 500-N weight) plus 200 N (from the 200-N weight). Therefore, the reaction force exerted by the support on the pipe is -700 N (-700 N upward to balance the downward forces).
To summarize:
- We use the principle of moments to determine the position of the fulcrum by equating the torques on both sides of the pipe.
- The reaction force exerted by the support on the pipe is -700 N.