Find analytically the resultant of the forces shown below:
Where: F1 = 20 kg (4,4)
F2 = 15 kg (-5,5)
F3 = 5 kg (-2,-2)
F4 = 10 kg (3,-3)
Fr = F1 + F2 + F3 + F4.
Fr = 20[45o] + 15[135o] + 5[[225o] + 10[315o]
Fr = 14.14+14.14i + 10.61-10.61i -3.54-3.54i
+7.07-7.07i,
Fr = 28.28 - 7.08i = 29.16kg[-14.1o] = 29.16kg[14.1o] S. of E.
To find the resultant of the forces, we can start by finding the x-component and y-component of each force, and then summing up the components to obtain the resultant.
Let's calculate the components of each force:
F1 = 20 kg (4,4)
F1x = 20 kg * 4 = 80 kg
F1y = 20 kg * 4 = 80 kg
F2 = 15 kg (-5,5)
F2x = 15 kg * -5 = -75 kg
F2y = 15 kg * 5 = 75 kg
F3 = 5 kg (-2,-2)
F3x = 5 kg * -2 = -10 kg
F3y = 5 kg * -2 = -10 kg
F4 = 10 kg (3,-3)
F4x = 10 kg * 3 = 30 kg
F4y = 10 kg * -3 = -30 kg
Now, we can sum up the x-components and y-components:
Resultant x-component (Rx) = F1x + F2x + F3x + F4x
= 80 kg + (-75 kg) + (-10 kg) + 30 kg
= 25 kg
Resultant y-component (Ry) = F1y + F2y + F3y + F4y
= 80 kg + 75 kg + (-10 kg) + (-30 kg)
= 115 kg
Therefore, the resultant of the given forces is 25 kg in the x-direction and 115 kg in the y-direction. So, the resultant vector can be represented as R = (25, 115) kg.
To find the resultant of the forces, we need to calculate the vector sum of all the individual forces. This can be done by adding up the components of each force vector.
Step 1: Convert the forces to vector form
Given:
F1 = 20 kg (4,4)
F2 = 15 kg (-5,5)
F3 = 5 kg (-2,-2)
F4 = 10 kg (3,-3)
The forces are already given in vector form, where the first number represents the x-component and the second number represents the y-component.
Step 2: Add up the x-components and the y-components separately
To find the x-component of the resultant force, add up the x-components of all the forces:
Fx = F1x + F2x + F3x + F4x = 4 + (-5) + (-2) + 3
Fx = 0
To find the y-component of the resultant force, add up the y-components of all the forces:
Fy = F1y + F2y + F3y + F4y = 4 + 5 + (-2) + (-3)
Fy = 4
Step 3: Combine the x-component and y-component to find the resultant force
The resultant force can be represented as a vector with the x-component and y-component calculated above:
Resultant force = (Fx, Fy) = (0, 4)
So, analytically, the resultant of the given forces is a force of 4 kg in the positive y-direction.