Mathematically determine
the resultant force if two
forces, A and B, act on an
object. Force A is 85 N and
is at an angle of 20° to the
horizontal. Force B is 125 N
and is at an angle of 60° to
the horizontal.
104 N
85cis20° + 125cis60° = 142.4 + 137.3i = 197.8 @ 44°
To determine the resultant force when two forces act on an object, we can use vector addition. First, we need to resolve each force into their horizontal and vertical components.
Let's start with Force A:
Horizontal component of Force A = A * cos(angle A)
= 85 N * cos(20°)
≈ 80.186 N
Vertical component of Force A = A * sin(angle A)
= 85 N * sin(20°)
≈ 29.175 N
Now let's move on to Force B:
Horizontal component of Force B = B * cos(angle B)
= 125 N * cos(60°)
≈ 62.5 N
Vertical component of Force B = B * sin(angle B)
= 125 N * sin(60°)
≈ 108.253 N
Next, we can sum up the horizontal and vertical components separately:
Horizontal component of the resultant force = Horizontal component of Force A + Horizontal component of Force B
= 80.186 N + 62.5 N
≈ 142.686 N
Vertical component of the resultant force = Vertical component of Force A + Vertical component of Force B
= 29.175 N + 108.253 N
≈ 137.428 N
Finally, we can find the magnitude and direction of the resultant force using the Pythagorean theorem and trigonometry:
Magnitude of the resultant force = sqrt((Horizontal component)^2 + (Vertical component)^2)
= sqrt((142.686 N)^2 + (137.428 N)^2)
≈ 196.52 N
Direction of the resultant force = arctan(Vertical component / Horizontal component)
= arctan(137.428 N / 142.686 N)
≈ 44.41°
Therefore, the resultant force acting on the object is approximately 196.52 N at an angle of 44.41° to the horizontal.
To mathematically determine the resultant force of two forces, A and B, acting on an object, we can use vector addition. Here are the steps to calculate it:
Step 1: Resolve the forces into their horizontal and vertical components.
Force A:
- The magnitude of Force A is 85 N.
- The angle it makes with the horizontal is 20°.
So, the horizontal component of Force A (A_h) can be calculated as:
A_h = A * cos(angle)
A_h = 85 N * cos(20°)
And the vertical component of Force A (A_v) can be calculated as:
A_v = A * sin(angle)
A_v = 85 N * sin(20°)
Force B:
- The magnitude of Force B is 125 N.
- The angle it makes with the horizontal is 60°.
Similarly, the horizontal component of Force B (B_h) can be calculated as:
B_h = B * cos(angle)
B_h = 125 N * cos(60°)
And the vertical component of Force B (B_v) can be calculated as:
B_v = B * sin(angle)
B_v = 125 N * sin(60°)
Step 2: Add the horizontal and vertical components separately.
Resultant horizontal component (R_h) = A_h + B_h
Resultant vertical component (R_v) = A_v + B_v
Step 3: Use the Pythagorean theorem to calculate the magnitude of the resultant force.
The magnitude of the resultant force (R) can be calculated as:
R = sqrt(R_h^2 + R_v^2)
Step 4: Determine the angle of the resultant force with the horizontal.
The angle (θ) the resultant force makes with the horizontal can be calculated as:
θ = arctan(R_v/R_h)
Now, let's plug the given values into the formulas to find the answer.
A_h = 85 N * cos(20°) ≈ 81.32 N
A_v = 85 N * sin(20°) ≈ 29.02 N
B_h = 125 N * cos(60°) ≈ 62.5 N
B_v = 125 N * sin(60°) ≈ 108.25 N
R_h = 81.32 N + 62.5 N ≈ 143.82 N
R_v = 29.02 N + 108.25 N ≈ 137.27 N
R = sqrt((143.82 N)^2 + (137.27 N)^2) ≈ 193.66 N
θ = arctan(137.27 N/143.82 N) ≈ 44.16°
Therefore, the resultant force is approximately 193.66 N and it makes an angle of approximately 44.16° with the horizontal.