The resultant of two forces ,Q and 20 N is 27.5N at 160° to the 20N forces .Find the magnetude of force Q and the angle it make with 20 N

To solve the problem of finding the magnitude of force Q and the angle it makes with the 20 N force, we can use the principles of vector addition. The resultant force is given as 27.5 N at an angle of 160° relative to the 20 N force. Let's break this down step by step.

Understanding the Forces

We have two forces:

• Force 1 (F1) = 20 N
• Force 2 (F2) = Q (unknown magnitude)

Setting Up the Problem

To visualize this, we can represent the forces as vectors. The 20 N force can be placed along the positive x-axis. The angle of 160° indicates that the resultant force is directed 160° from the 20 N force, which means it is in the second quadrant of the coordinate system.

Breaking Down the Forces into Components

We can express the forces in terms of their x and y components:

• For the 20 N force:
  • F1x = 20 N (cos 0°) = 20 N
  • F1y = 20 N (sin 0°) = 0 N
• For the resultant force (R):
  • Rx = 27.5 N (cos 160°)
  • Ry = 27.5 N (sin 160°)

Calculating the Components of the Resultant Force

Now, let's calculate the components of the resultant force:

• Rx = 27.5 N * cos(160°) ≈ 27.5 N * (-0.9397) ≈ -25.8 N
• Ry = 27.5 N * sin(160°) ≈ 27.5 N * 0.3420 ≈ 9.4 N

Setting Up the Equations

Since we know the components of the forces, we can set up the following equations based on the x and y components:

• For the x-components: F1x + F2x = Rx
  • 20 N + Q * cos(θ) = -25.8 N
• For the y-components: F1y + F2y = Ry
  • 0 N + Q * sin(θ) = 9.4 N

Solving the Equations

From the x-component equation, we can express Q * cos(θ):

• Q * cos(θ) = -25.8 N - 20 N = -45.8 N

From the y-component equation, we can express Q * sin(θ):

• Q * sin(θ) = 9.4 N

Finding the Magnitude of Q

Now, we can use the Pythagorean theorem to find the magnitude of Q:

• Q² = (Q * cos(θ))² + (Q * sin(θ))²
• Substituting the values:
  • Q² = (-45.8 N)² + (9.4 N)²
  • Q² = 2096.64 + 88.36 = 2185
  • Q = √2185 ≈ 46.7 N

Calculating the Angle θ

To find the angle θ, we can use the tangent function:

• tan(θ) = (Q * sin(θ)) / (Q * cos(θ))
• tan(θ) = 9.4 N / -45.8 N
• θ = arctan(-0.205) ≈ -11.5°

Since the angle is negative, we can adjust it to find the angle with respect to the 20 N force, which is in the second quadrant. Thus, the angle θ is approximately 180° - 11.5° = 168.5°.

Final Results

In summary, the magnitude of force Q is approximately 46.7 N, and it makes an angle of approximately 168.5° with the 20 N force.