3 forces act concurrently on an object. First force acts 75 N north. 2nd acts 100 N east. 3rd acts 89 N at 15 degrees south of west. What is resultant force?
b) what is equilibrant?
To find the resultant force, we need to add the three forces together. Since forces have both magnitude and direction, we can represent them as vectors.
Step 1: Draw a coordinate system:
Draw a 2-dimensional coordinate system with a horizontal x-axis and a vertical y-axis. You can label the positive direction as "North" for the y-axis and the positive direction as "East" for the x-axis.
Step 2: Represent the forces as vectors:
Represent each force as a vector with its magnitude and direction.
- The first force of 75 N north is represented as a vector with a length of 75 units pointing upwards along the positive y-axis.
- The second force of 100 N east is represented as a vector with a length of 100 units pointing to the right along the positive x-axis.
- The third force of 89 N at 15 degrees south of west can be broken down into its x and y components:
- The x-component is calculated as 89 N * cos(15) and is represented as a vector pointing to the left along the negative x-axis.
- The y-component is calculated as 89 N * sin(15) and is represented as a vector pointing downwards along the negative y-axis.
Step 3: Add the vectors:
Add all the vectors together by placing them head-to-tail. The resultant vector is the straight line connecting the tail of the first vector to the head of the last vector.
Step 4: Find the magnitude and direction of the resultant:
Measure the length of the resultant vector, which represents its magnitude. The direction of the resultant can be found using trigonometry. You can calculate the angle using the inverse tangent function (arctan) of the y-component divided by the x-component.
Now let's calculate the values step by step:
- First force: 75 N north (0 degrees)
- Second force: 100 N east (90 degrees)
- Third force: 89 N at 15 degrees south of west (-105 degrees)
Step 1: Draw the coordinate system.
Step 2: Represent the forces as vectors.
- First force: A vector pointing 75 units upwards along the positive y-axis.
- Second force: A vector pointing 100 units to the right along the positive x-axis.
- Third force: The x-component is -89 N * cos(15) and the y-component is -89 N * sin(15).
Step 3: Add the vectors.
Connect the tail of the first vector (75 N) to the head of the second vector (100 N). Then connect the tail of the resulting vector to the head of the third vector (-89 N).
Step 4: Find the magnitude and direction of the resultant.
- Measure the length of the resultant vector.
- Calculate the angle using the arctan of the y-component divided by the x-component.
Now you have both the magnitude and direction of the resultant force.
To find the resultant force, we will first resolve the forces into their respective x and y components and then add them together.
Given:
- First force (F1) = 75 N north
- Second force (F2) = 100 N east
- Third force (F3) = 89 N at 15 degrees south of west
Step 1: Resolve F1 into its x and y components:
F1x = 0 N (since it acts in the north direction)
F1y = 75 N
Step 2: Resolve F2 into its x and y components:
F2x = 100 N
F2y = 0 N (since it acts in the east direction)
Step 3: Resolve F3 into its x and y components:
F3x = 89 N * cos(15°) = 84.98 N (approximately)
F3y = -89 N * sin(15°) = -22.99 N (approximately) [negative because it acts south of west]
Step 4: Add up the x and y components:
Rx = F1x + F2x + F3x = 0 N + 100 N + 84.98 N = 184.98 N (approximately, positive because F3x is to the right)
Ry = F1y + F2y + F3y = 75 N + 0 N + (-22.99 N) = 52.01 N (approximately, positive because F1y is upward)
Step 5: Calculate the magnitude and direction of the resultant force:
Resultant force (R) = √(Rx^2 + Ry^2)
= √(184.98^2 + 52.01^2)
= √(34224 + 2704)
= √(36928)
= 192 N (approximately)
To find the direction of the resultant force, we use the tangent function:
θ = tan^(-1)(|Ry| / |Rx|)
θ = tan^(-1)(52.01 N / 184.98 N)
θ = tan^(-1)(0.2811)
θ = 15.21° (approximately)
Therefore, the resultant force is approximately 192 N in magnitude and makes an angle of 15.21° with the positive x-axis.
b) The equilibrant is a force that, when added to the given forces, results in a net force of zero. In other words, it is a force equal in magnitude but opposite in direction to the resultant force. In this case, the equilibrant will have a magnitude of 192 N and will be in the direction opposite to the resultant force.