A particle undergoes three successive displacements in a plane as follows:4.0 m southwest ,5.0 m east , and 6.0 m in a direction 60 degrees north of east. Choose the y axis pointing north and the x axis pointing east and find a) the components of each displacement,b) the components of the resultant displacement. c) the magnitude and direction of the resultant displacement ?

To solve this problem, we need to break down each displacement into its components along the x (east) and y (north) axes. After that, we can find the resultant displacement by summing these components. Let's go through this step by step.

Displacement Components

We have three displacements to analyze:

• 4.0 m southwest
• 5.0 m east
• 6.0 m at 60 degrees north of east

1. First Displacement: 4.0 m Southwest

Southwest means the angle is 45 degrees from both the south and west directions. To find the components:

• Angle from the positive x-axis (east) = 180° + 45° = 225°
• Component in x-direction: 4.0 m * cos(225°) = 4.0 m * (-0.7071) ≈ -2.83 m
• Component in y-direction: 4.0 m * sin(225°) = 4.0 m * (-0.7071) ≈ -2.83 m

2. Second Displacement: 5.0 m East

This displacement is straightforward since it lies entirely along the x-axis:

• x-component: 5.0 m
• y-component: 0 m

3. Third Displacement: 6.0 m at 60 Degrees North of East

For this displacement, we can directly find the components:

• x-component: 6.0 m * cos(60°) = 6.0 m * 0.5 = 3.0 m
• y-component: 6.0 m * sin(60°) = 6.0 m * (√3/2) ≈ 5.20 m

Summing the Components

Now, let's sum the components of all three displacements:

• Total x-component: -2.83 m + 5.0 m + 3.0 m = 5.17 m
• Total y-component: -2.83 m + 0 m + 5.20 m = 2.37 m

Resultant Displacement

Now that we have the total components, we can find the magnitude and direction of the resultant displacement.

Magnitude

Using the Pythagorean theorem:

• Magnitude = √((5.17 m)² + (2.37 m)²) ≈ √(26.73 + 5.62) ≈ √32.35 ≈ 5.68 m

Direction

To find the direction, we use the arctangent function:

• Angle θ = tan⁻¹(y/x) = tan⁻¹(2.37 m / 5.17 m) ≈ 24.6°

This angle is measured from the positive x-axis (east) towards the north.

Summary of Results

In summary, we have:

• Components of each displacement:
• 4.0 m southwest: x = -2.83 m, y = -2.83 m
• 5.0 m east: x = 5.0 m, y = 0 m
• 6.0 m at 60° north of east: x = 3.0 m, y ≈ 5.20 m
• Resultant displacement components: x ≈ 5.17 m, y ≈ 2.37 m
• Magnitude of resultant displacement: ≈ 5.68 m
• Direction of resultant displacement: ≈ 24.6° north of east

This structured approach allows us to clearly understand how to break down and analyze vector displacements in a plane. If you have any further questions or need clarification on any part, feel free to ask!