Find the resultant of the system of coplanar forces shown in the figure giving it's magnitude and the angle it makes with the 400N force.
F1=300N
F2=400N
F3=200N;60°
To find the resultant of the system of coplanar forces, we can resolve each force into its horizontal and vertical components and then add up the corresponding components.
Let's start by resolving F1 = 300N into its horizontal and vertical components.
The horizontal component of F1 (F1x) can be found using the formula:
F1x = F1 * cos(theta1)
where theta1 is the angle F1 makes with the horizontal axis, which is not given. Therefore, we cannot find the exact value of F1x.
The vertical component of F1 (F1y) can be found using the formula:
F1y = F1 * sin(theta1)
where theta1 is the angle F1 makes with the horizontal axis, which is not given. Therefore, we cannot find the exact value of F1y.
Next, let's resolve F2 = 400N into its horizontal and vertical components.
The horizontal component of F2 (F2x) can be found using the formula:
F2x = F2 * cos(theta2)
where theta2 is the angle F2 makes with the horizontal axis, which is not given. Therefore, we cannot find the exact value of F2x.
The vertical component of F2 (F2y) can be found using the formula:
F2y = F2 * sin(theta2)
where theta2 is the angle F2 makes with the horizontal axis, which is not given. Therefore, we cannot find the exact value of F2y.
Finally, let's resolve F3 = 200N; 60° into its horizontal and vertical components.
The horizontal component of F3 (F3x) can be found using the formula:
F3x = F3 * cos(theta3)
where theta3 is the angle F3 makes with the horizontal axis, which is given as 60°.
F3x = 200N * cos(60°)
F3x = 200N * 0.5
F3x = 100N
The vertical component of F3 (F3y) can be found using the formula:
F3y = F3 * sin(theta3)
where theta3 is the angle F3 makes with the horizontal axis, which is given as 60°.
F3y = 200N * sin(60°)
F3y = 200N * 0.866
F3y = 173.2N
Now, we can add up the corresponding components to find the resultant components.
Horizontal component of the resultant (R_x) = F1x + F2x + F3x
= (unknown value) + (unknown value) + 100N
Vertical component of the resultant (R_y) = F1y + F2y + F3y
= (unknown value) + (unknown value) + 173.2N
Since the values of F1x, F1y, F2x, F2y are unknown, we cannot find the exact values of R_x and R_y. Therefore, we cannot find the magnitude and angle made by the resultant with the 400N force without knowing the values of the unknown components.
To find the resultant of the system of coplanar forces, we need to add the vector components of all the forces. The resultant force can be found using the formula:
Resultant force = √(Rx² + Ry²)
where Rx represents the sum of the horizontal components and Ry represents the sum of the vertical components.
Let's start by calculating the horizontal and vertical components of each force:
Force F1:
Fx1 = F1 * cos(0°)
= 300N * cos(0°)
= 300N
Fy1 = F1 * sin(0°)
= 300N * sin(0°)
= 0N
Force F2:
Fx2 = F2 * cos(180°)
= 400N * cos(180°)
= -400N
Fy2 = F2 * sin(180°)
= 400N * sin(180°)
= 0N
Force F3:
Fx3 = F3 * cos(60°)
= 200N * cos(60°)
= 100N
Fy3 = F3 * sin(60°)
= 200N * sin(60°)
= 173.2N
Now, let's calculate the sum of the horizontal components (Rx) and the sum of the vertical components (Ry):
Rx = Fx1 + Fx2 + Fx3
= 300N + (-400N) + 100N
= 0N
Ry = Fy1 + Fy2 + Fy3
= 0N + 0N + 173.2N
= 173.2N
Next, we can calculate the magnitude of the resultant force using the formula mentioned earlier:
Resultant force = √(Rx² + Ry²)
= √(0N² + 173.2N²)
= √(29998.24N²)
= 173.2N
The magnitude of the resultant force is 173.2N.
Finally, we can calculate the angle that the resultant force makes with the 400N force:
Angle = arctan(Ry/Rx)
= arctan(173.2N/0N)
= 90°
Therefore, the resultant force makes an angle of 90° with the 400N force.