A particle undergoes two displacements. The first has a magnitude of 165 cm and makes an angle of 120.0° with the positive x axis. The resultant of the two displacements is 140 cm directed at an angle of 41.5° to the positive x axis. Find the magnitude and direction of the second displacement.1 cm at 2° counter-clockwise from the positive x axis

To find the magnitude and direction of the second displacement, we can use vector addition. We start by breaking down the first displacement into its components, then we can determine the second displacement by using the resultant vector and the first displacement.

Step 1: Break Down the First Displacement

The first displacement has a magnitude of 165 cm and is directed at an angle of 120.0° from the positive x-axis. We can find its x and y components using trigonometric functions:

• X-component:
  D1x = D1 * cos(θ) = 165 * cos(120°)
  Since cos(120°) = -0.5, we have:
  D1x = 165 * (-0.5) = -82.5 cm
• Y-component:
  D1y = D1 * sin(θ) = 165 * sin(120°)
  Since sin(120°) = √3/2 ≈ 0.866, we have:
  D1y = 165 * 0.866 ≈ 142.9 cm

So, the components of the first displacement are approximately:

• D1x ≈ -82.5 cm
• D1y ≈ 142.9 cm

Step 2: Analyze the Resultant Displacement

The resultant displacement is given as 140 cm at an angle of 41.5° from the positive x-axis. We can also break this down into its components:

• X-component:
  R_x = R * cos(θ) = 140 * cos(41.5°)
  Using cos(41.5°) ≈ 0.7547, we find:
  R_x ≈ 140 * 0.7547 ≈ 105.7 cm
• Y-component:
  R_y = R * sin(θ) = 140 * sin(41.5°)
  Using sin(41.5°) ≈ 0.6561, we find:
  R_y ≈ 140 * 0.6561 ≈ 91.8 cm

Thus, the components of the resultant displacement are:

• R_x ≈ 105.7 cm
• R_y ≈ 91.8 cm

Step 3: Calculate the Second Displacement Components

Let’s denote the second displacement as D2. The components of D2 can be expressed as:

• X-component:
  D2x = R_x - D1x = 105.7 - (-82.5) = 105.7 + 82.5 = 188.2 cm
• Y-component:
  D2y = R_y - D1y = 91.8 - 142.9 = -51.1 cm

Step 4: Determine the Magnitude and Direction of D2

Now we can find the magnitude of the second displacement using the Pythagorean theorem:

Calculating this gives:

Next, we find the direction (angle) of D2 using the arctangent function:

Calculating this gives:

This angle is measured counter-clockwise from the positive x-axis. To express it in a standard format, we can convert it to a positive angle:

Final Results

Thus, the magnitude of the second displacement is approximately 194.8 cm, and its direction is approximately 344.8° counter-clockwise from the positive x-axis.