# Rectangular Resolution and Polygon Theorem

• wellgrin
In summary, this conversation covers various problems involving forces and angles. These include finding the resultant of two forces at an angle, resolving forces by making them lie on the x-axis, determining tension on ropes supporting a weight, and finding the tension of each part of a wire with a weight suspended from it.
wellgrin
1. Find the resultant of two forces of 40 lbs. and 50 lbs. acting at an angle of 60○ between them.
2. Three forces of 30 gms, 50 gms, and 60 gms respectively act at an angle of 120○ from each other. Find the resultant by rectangular-resolution (a) by making the 30-gm force lie on the x-axis, (b) By making the 60-gm force lie on the x-axis.
3. A weight of 100 lbs is supported vertically by two ropes, one making 60○ above the horizontal to the right and the other making an angle of 37○ above the horizontal to the left. Find the tension on the ropes.
4. By rectangular resolution of forces, find the resultant of the following forces: P=40 gms along the x-axis toward the right. Q=50 gms, 30○ above the horizontal toward the left; S = 20 gms vertically up; and T = 60 gms acting downward 30○ to the left of the vertical.
5. Each end of a wire 17 inches long is fastened, respectively, to two hooks lying on the same horizontal line. The hooks are 13 inches apart. A 26-lb weight is suspended in the wire at a point 5 inches from one hook. Find the tension of each part of the wire.

wellgrin said:
1. Find the resultant of two forces of 40 lbs. and 50 lbs. acting at an angle of 60○ between them.
2. Three forces of 30 gms, 50 gms, and 60 gms respectively act at an angle of 120○ from each other. Find the resultant by rectangular-resolution (a) by making the 30-gm force lie on the x-axis, (b) By making the 60-gm force lie on the x-axis.
3. A weight of 100 lbs is supported vertically by two ropes, one making 60○ above the horizontal to the right and the other making an angle of 37○ above the horizontal to the left. Find the tension on the ropes.
4. By rectangular resolution of forces, find the resultant of the following forces: P=40 gms along the x-axis toward the right. Q=50 gms, 30○ above the horizontal toward the left; S = 20 gms vertically up; and T = 60 gms acting downward 30○ to the left of the vertical.
5. Each end of a wire 17 inches long is fastened, respectively, to two hooks lying on the same horizontal line. The hooks are 13 inches apart. A 26-lb weight is suspended in the wire at a point 5 inches from one hook. Find the tension of each part of the wire.

This is a nice set of questions. But it strikes me that this is your home work and not ours!

I believe that the rules of this forum state that you must show your attempts at solution when asking for help.

By the way there are several hints for getting started right in the questions.

1. To find the resultant of two forces of 40 lbs and 50 lbs acting at an angle of 60○ between them, we can use the polygon theorem. This theorem states that if two or more forces are acting on an object, the resultant force can be found by connecting the forces head-to-tail and drawing a line from the starting point to the end point. The length of this line represents the magnitude of the resultant force, and the angle it makes with the horizontal represents the direction of the resultant force. In this case, the resultant force would be approximately 73.6 lbs at an angle of 43.6○ above the horizontal.

2. The rectangular resolution method can also be used to find the resultant of three forces at different angles. By making the 30-gm force lie on the x-axis, we can find the x-component of the resultant force by using the formula Fx = Fcosθ. Similarly, by making the 60-gm force lie on the x-axis, we can find the y-component of the resultant force by using the formula Fy = Fsinθ. The resultant force can then be found using the Pythagorean theorem, where the magnitude is given by √(Fx^2 + Fy^2) and the direction is given by tan^-1(Fy/Fx).

3. In this scenario, we can use the polygon theorem to find the resultant force acting on the weight. By connecting the two ropes head-to-tail, we can see that the resultant force is acting at an angle of 23○ above the horizontal. The magnitude of this force can be found using the law of sines, where (sin 23○)/100 = (sin 60○)/T, where T is the tension in the ropes. Solving for T, we get a tension of approximately 109.8 lbs for each rope.

4. To find the resultant of the given forces using rectangular resolution, we can use the same method as in question 2. By breaking down each force into its x and y components and then adding them together, we can find the resultant force. In this case, the resultant force would have a magnitude of approximately 59.2 gms and would be at an angle of 18.4○ above the horizontal.

5. The tension in each part of the wire can be found by using the polygon theorem. By connecting

## 1. What is the Rectangular Resolution Theorem?

The Rectangular Resolution Theorem is a mathematical principle that states that any line or curve drawn on a rectangular grid can be broken down into a series of straight lines or curves.

## 2. How does the Rectangular Resolution Theorem relate to polygons?

The Rectangular Resolution Theorem is closely related to the Polygon Theorem, which states that any polygon can be broken down into a series of triangles. This is because polygons can be drawn on a rectangular grid, and therefore the Rectangular Resolution Theorem can be applied to them.

## 3. What is the purpose of the Rectangular Resolution Theorem?

The Rectangular Resolution Theorem has many practical applications, such as in computer graphics, where it is used to create smooth curves and lines on a rectangular pixel grid. It also has applications in engineering and architecture, where it can be used to accurately draw and measure angles and curves on a rectangular grid.

## 4. Can the Rectangular Resolution Theorem be applied to non-rectangular grids?

While the Rectangular Resolution Theorem is specifically designed for rectangular grids, similar principles can be applied to non-rectangular grids. For example, the Polar Resolution Theorem applies to grids with a polar coordinate system.

## 5. Are there any limitations to the Rectangular Resolution Theorem?

The Rectangular Resolution Theorem is a powerful mathematical tool, but it does have some limitations. It is designed for 2D grids, so it cannot be applied to 3D objects. It also assumes that the grid is infinitely small, so it may not be completely accurate in real-world situations where the grid is finite.

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