The resultant of two forces ,1 and 20 Nos 27.5 N at 160° to the 20N force .find the magnitude of force Q an the direction it makes with 20 N

To find the resultant of two forces, we need to break them down into their components and then combine those components. In this case, we have two forces: one with a magnitude of 1 N and another with a magnitude of 20 N, with an angle of 160° between them. Let's go through the steps to calculate the magnitude of the resultant force and the angle it makes with the 20 N force.

Step-by-Step Breakdown

1. Identify the Forces

We have:

• Force F1 = 1 N
• Force F2 = 20 N
• Angle θ = 160° (between F1 and F2)

2. Resolve Forces into Components

To find the resultant force, we first resolve each force into its horizontal (x) and vertical (y) components.

For the 20 N force (F2):

• F2x = 20 N * cos(0°) = 20 N (since it's along the x-axis)
• F2y = 20 N * sin(0°) = 0 N

For the 1 N force (F1) at 160°:

• F1x = 1 N * cos(160°) = 1 N * (-0.9397) ≈ -0.9397 N
• F1y = 1 N * sin(160°) = 1 N * (0.3420) ≈ 0.3420 N

3. Combine the Components

Now, we can find the total components of the resultant force (R).

• Rx = F2x + F1x = 20 N - 0.9397 N ≈ 19.0603 N
• Ry = F2y + F1y = 0 N + 0.3420 N ≈ 0.3420 N

4. Calculate the Magnitude of the Resultant Force

The magnitude of the resultant force R can be found using the Pythagorean theorem:

R = √(Rx² + Ry²)

Substituting the values:

R = √((19.0603 N)² + (0.3420 N)²) ≈ √(363.25 + 0.1165) ≈ √(363.3665) ≈ 19.06 N

5. Determine the Direction of the Resultant Force

The angle θ with respect to the 20 N force can be calculated using the tangent function:

tan(θ) = Ry / Rx

Substituting the values:

tan(θ) = 0.3420 N / 19.0603 N

θ = arctan(0.0179) ≈ 1.02°

Final Results

Thus, the magnitude of the resultant force is approximately 19.06 N, and it makes an angle of about 1.02° with the 20 N force.

This method of resolving forces into components is crucial in physics, especially when dealing with forces that are not aligned along the same line. It allows us to analyze complex situations in a more manageable way.