Vector Displacement Problem

[singingblonde]

Homework Statement

A man pushing a mop across a floor causes the mop to undergo two displacements. The first has a magnitude of 150 cm and makes an angle of 120 degrees with the positive x axis. The resultant displacement has a magnitude of 140 cm and is directed at an angle of 35 degrees to the positive x axis. Find the magnitude and direction of the second displacement (second vector)

Homework Equations

Vector Component Addition

The Attempt at a Solution

I realize that somehow, we know the resultant vector and one of the vectors and need to solve for the second vector. Because nothing is known about the second vector, I don't know how to go about that. I've been working on this problem all day and truly need some help.

 

[CompuChip]

Hi singingblonde.

Suppose for a moment that the problem was reversed: the two vectors were given (for example: 150 cm with an angle of 120 degrees with the x-axis, and 140 cm at an angle of 35 degrees to the x-axis) and you were asked to calculate the resultant vector. Would you know how to solve that?

 

[tomcue]

As a scientist, it is important to approach problems systematically and use known equations to solve them. In this case, we can use vector component addition to solve for the second displacement.

First, let's draw a diagram to visualize the problem. We know that the first displacement has a magnitude of 150 cm and an angle of 120 degrees with the positive x axis. This can be represented by vector A. The resultant displacement has a magnitude of 140 cm and an angle of 35 degrees with the positive x axis. This can be represented by vector R. Now, we need to find the magnitude and direction of the second displacement, which we can represent by vector B.

Using vector component addition, we can break down the resultant vector R into its x and y components. The x component of R can be found using the formula Rx = Rcosθ, where θ is the angle between R and the positive x axis. Similarly, the y component of R can be found using the formula Ry = Rsinθ. Plugging in the values given in the problem, we get Rx = 140cos35 = 114.3 cm and Ry = 140sin35 = 80.3 cm.

Now, we can use the same approach to break down vector A into its x and y components. We know that the x component of A is 150cos120 = -75 cm (since the angle is in the third quadrant) and the y component of A is 150sin120 = 129.9 cm.

To find the x and y components of vector B, we can use the fact that the sum of the x components of A and B must equal the x component of R, and the sum of the y components of A and B must equal the y component of R. So, we can set up the following equations:

Bx + (-75) = 114.3
By + 129.9 = 80.3

Solving for Bx and By, we get Bx = 189.3 cm and By = -49.6 cm. Now, we can find the magnitude and direction of vector B using the Pythagorean theorem and inverse tangent function, respectively.

The magnitude of vector B is given by B = √(Bx^2 + By^2) = √(189.3^2 + (-49.6)^2) = 195.5 cm.

The direction of vector