ON is the vertical plumb line.A horizontal force F applies at M to push the piston at O and N,with the help of rods MO,MN.The rods MO,MN have the same length,when the distance of ON is 200cm,the distance of M to the plumb line ON is 10cm.Assuming the weight of the piston and the rods are negligible,ehat is the total compressive force acting on the object W?
To determine the total compressive force acting on the object, we need to analyze the equilibrium of forces in the system.
First, let's break down the forces acting on the system:
1. Vertical force at O (Plumb line ON): This force acts downward and can be expressed as W, where W represents the weight of the object.
2. Horizontal force applied at M: This force, denoted as F, pushes the piston horizontally.
3. Reaction forces at O and N: These forces act vertically in response to the applied forces and prevent the object from moving vertically.
To solve for the compressive force, we need to consider the equilibrium condition of the system. Since the weight of the piston and rods is negligible, the only significant forces are F and the vertical reaction forces at O and N.
Let's denote the compressive force as C. Since the rods MO and MN have equal length, the vertical reaction forces at O and N will be equal in magnitude.
Now, let's apply the equations of equilibrium:
Sum of forces in the x-direction: F = 0 (since there are no other horizontal forces)
Sum of forces in the y-direction: C - W = 0 (since the vertical reaction forces at O and N cancel out the weight of the object)
We are given that the distance between M and the plumb line ON is 10 cm. This distance is known as the lever arm or moment arm, and it determines the torque caused by the force F. The torque depends on the lever arm length and the applied force.
Since the lever arm is perpendicular to the line of action of the force F, it creates a moment (torque) equal to F multiplied by the lever arm distance.
To calculate the distance MN and apply the principle of moments, we can use the concept of similar triangles. We have the ratio of the lengths of the corresponding sides as:
MN / ON = MO / M plumb line
Since MN = MO, we can write:
MN / 200 cm = 10 cm / M plumb line
M plumb line = 200 cm × (10 cm / MN)
Now, we can substitute this value in the equation for the moment (torque):
C × 10 cm = F × M plumb line
Finally, solving the two equilibrium equations simultaneously:
F = 0 (from the sum of forces in the x-direction)
C - W = 0 (from the sum of forces in the y-direction)
C × 10 cm = F × M plumb line
Since F = 0, we can solve for C:
C = W
Therefore, the total compressive force acting on the object is equal to its weight, denoted as W.