a window with a smooth hinge at the top is held open at an angle 30 degrees to the vertical by a horizontal force applied to the center of it lower edge. the force of gravity on the window is 3kgf acting through its mid point. draw a scale diagram of the arrangement and indicate on it a triangle of forces by the letter A B C. hence either by measurement or calculation.find the magnitude of the force exerted by its hinge.
To draw a scale diagram of the arrangement and solve for the magnitude of the force exerted by the hinge, we will start by understanding the given information:
1. The window has a smooth hinge at the top.
2. The window is held open at an angle of 30 degrees to the vertical.
3. A horizontal force is applied to the center of its lower edge.
4. The force of gravity on the window is 3 kgf, acting through its midpoint.
To draw the scale diagram:
1. Begin by drawing a vertical line to represent the vertical direction, which will be the reference for angles and forces.
2. At the top of the vertical line, draw a small horizontal line segment to represent the smooth hinge.
3. Draw a line segment from the lower end of the hinge, inclined at an angle of 30 degrees with the vertical line. This represents the window held open at an angle of 30 degrees.
4. Draw a horizontal line segment from the lower end of the inclined line to represent the application of the horizontal force at the center of the lower edge.
5. Add arrowheads to the line segments to indicate the direction of the forces. Label this line segment as Force AB.
The scale diagram should now represent the given arrangement of the window and the force applied.
To find the magnitude of the force exerted by the hinge (force BC), we need to use the triangle of forces (triangle ABC) formed in the diagram.
Let's consider the equilibrium of forces:
1. Force AB represents the horizontal force applied to the center of the window's lower edge.
2. Force BC represents the force exerted by the hinge.
3. The force of gravity (3 kgf) acts vertically downwards through the midpoint of the window.
Since the window is in equilibrium, the vertical components of forces AB and BC are equal and opposite to the force of gravity. Similarly, the horizontal components of forces AB and BC balance each other out.
Now, let's calculate the magnitude of force BC:
1. Determine the vertical component of force AB using trigonometry.
Vertical component of AB (BC) = Force AB * sin(30°)
2. Set the vertical component of force BC to equal the force of gravity (3 kgf).
Force BC * sin(30°) = 3 kgf
3. Solve for the magnitude of force BC:
Force BC = 3 kgf / sin(30°)
You can calculate the value of force BC either by using a calculator or by referring to a trigonometric table.
By following these steps, you can draw a scale diagram of the arrangement and find the magnitude of the force exerted by the hinge.