The engine of a car is being lifted out of the car by using a pulley system. The weight of the engine is 8000N. Use a scale diagram to determine the horizontal force, F1 that a mechanic has to use to hold the engine in such a position that the rope makes an angle of 30degrees with the vertical. Also determine the magnitude of the force, F2 that the rope exerts on the engine. Determine the magnitude of F1 and F2 both graphically and algebraically.
To determine the magnitude of the horizontal force, F1, graphically and algebraically, we can use the concept of equilibrium in a pulley system. In this scenario, there are three forces acting on the engine: the weight of the engine (8000N) acting vertically downwards, the force exerted by the rope (F2), and the horizontal force exerted by the mechanic (F1).
1. Graphical Method:
To construct a scale diagram, follow these steps:
- Start by drawing a vertical line to represent the vertical direction (acting downwards due to the weight of the engine).
- From the bottom end of the vertical line, draw a line at a 30-degree angle to represent the rope.
- Measure the length of the vertical line and scale it based on the force magnitude (8000N).
- Now, draw a horizontal line from the top end of the angled line to complete the triangle representing the forces.
- Determine the horizontal length of the line, and the measured length represents the magnitude of force F1.
2. Algebraic Method:
To solve the forces algebraically, we'll use the concept of vector components:
- Decompose the weight of the engine (8000N) into two components: one acting vertically downwards (F2), and the other horizontally (F1).
- The vertical component, F2, can be found using the formula F2 = weight * sin(30).
- The horizontal component, F1, can be found using the formula F1 = weight * cos(30).
By substituting the given values into the formulas using a calculator:
F2 = 8000N * sin(30) = 4000N
F1 = 8000N * cos(30) = 6928N (rounded to the nearest whole number)
Therefore, using both the graphical and algebraic methods, the magnitude of force F1 is approximately 6928N, and the magnitude of force F2 is approximately 4000N.