obtain the resultant and direction of each pair of the forces
1.20N on top of a triangle and, 15N3 under the triangle and 45 as the degree of the triangle.
2.20N on top of right angle triangle and 15N under the right angle triangle.
To find the resultant and direction of each pair of forces, we can use vector addition.
1. Forces on a Triangle:
- Given force on top of the triangle: 20N
- Given force under the triangle: 15N
- Angle of the triangle: 45 degrees
To find the resultant, we first need to resolve each force into its horizontal and vertical components.
- Resolving the force on top of the triangle:
Horizontal component: 20N * cos(45°) = 14.1N
Vertical component: 20N * sin(45°) = 14.1N
- Resolving the force under the triangle:
Horizontal component: 15N * cos(180°) = -15N
Vertical component: 15N * sin(180°) = 0N
Next, we add the horizontal and vertical components separately to find the resultant:
Horizontal component: 14.1N + (-15N) = -0.9N
Vertical component: 14.1N + 0N = 14.1N
Finally, we can find the magnitude and direction of the resultant using the Pythagorean theorem and inverse tangent function:
Magnitude of the resultant:
Magnitude = √((-0.9N)^2 + (14.1N)^2) = √(0.81N^2 + 198.81N^2) = √(199.62N^2) = 14.12N
Direction of the resultant:
Direction = tan^(-1)(Vertical component / Horizontal component) = tan^(-1)(14.1N / -0.9N) = -87.91°
Therefore, the resultant of the forces on the triangle is 14.12N at an angle of -87.91°.
2. Forces on a Right Angle Triangle:
- Given force on top of the right angle triangle: 20N
- Given force under the right angle triangle: 15N
To find the resultant, we need to resolve each force into its horizontal and vertical components.
- Resolving the force on top of the right angle triangle:
Horizontal component: 20N * cos(90°) = 0N
Vertical component: 20N * sin(90°) = 20N
- Resolving the force under the right angle triangle:
Horizontal component: 15N * cos(0°) = 15N
Vertical component: 15N * sin(0°) = 0N
Adding the horizontal and vertical components, we find:
Horizontal component: 0N + 15N = 15N
Vertical component: 20N + 0N = 20N
Using the same steps as above, we find:
Magnitude of the resultant:
Magnitude = √((15N)^2 + (20N)^2) = √(225N^2 + 400N^2) = √(625N^2) = 25N
Direction of the resultant:
Direction = tan^(-1)(Vertical component / Horizontal component) = tan^(-1)(20N / 15N) = 53.13°
Therefore, the resultant of the forces on the right angle triangle is 25N at an angle of 53.13°.
To obtain the resultant and direction of each pair of forces, we can use vector addition. This involves combining the forces as vectors, taking into account their magnitudes and directions.
1. For the first pair of forces:
- The 20N force on top of the triangle can be represented as a vector pointing downwards. Let's call it F1.
- The 15N force under the triangle can be represented as a vector pointing upwards. Let's call it F2.
- The 45-degree angle of the triangle means that the resultant force will form an angle of 45 degrees with the horizontal axis.
To find the resultant force, we need to add F1 and F2 as vectors. First, we need to break down the forces into their horizontal and vertical components.
- F1 has a vertical component equal to 20N * sin(45°) = 14.14N pointing downwards.
- F2 has a vertical component equal to 15N * cos(45°) = 10.61N pointing upwards.
- The horizontal components of both forces cancel each other out since they are equal and opposite.
To find the resultant force, we add the vertical components together:
Resultant vertical component = F1 vertical component - F2 vertical component
= 14.14N - 10.61N
= 3.53N
The direction of the resultant force can be found by using the tangent of the angle:
Tangent of resultant angle = Resultant vertical component / Resultant horizontal component
= 3.53N / 0N (since the horizontal components cancel each other out)
= Undefined (division by zero)
Since the horizontal component is zero, we cannot determine the direction of the resultant force in this case.
2. For the second pair of forces:
- The 20N force on top of the right-angle triangle can be represented as a vector pointing downwards. Let's call it F3.
- The 15N force under the right-angle triangle can be represented as a vector pointing towards the right. Let's call it F4.
To find the resultant force, we need to add F3 and F4 as vectors. Again, we break down the forces into their horizontal and vertical components.
- F3 has a vertical component equal to 20N * sin(90°) = 20N pointing downwards.
- F4 has a horizontal component equal to 15N * cos(90°) = 15N pointing towards the right.
- The vertical components of both forces cancel each other out since they are equal and opposite.
To find the resultant force, we add the horizontal components together:
Resultant horizontal component = F4 horizontal component
= 15N
The resultant force has only a horizontal component of 15N, pointing towards the right. Since there is no vertical component, we can say that the direction of the resultant force is along the horizontal axis, towards the right.