A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of 152 cm and makes an angle of 125° with the positive x axis. The resultant displacement has a magnitude of 150 cm and is directed at an angle of 35.0° to the positive x axis. Find the magnitude and direction of the second displacement.
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To find the magnitude and direction of the second displacement, we can use vector addition.
Let's break down the given information into components.
The first displacement has a magnitude of 152 cm and makes an angle of 125° with the positive x-axis. We can represent this as a vector with its x and y components.
Using trigonometry, we can find:
x-component = magnitude * cos(angle)
x1 = 152 cm * cos(125°)
y-component = magnitude * sin(angle)
y1 = 152 cm * sin(125°)
Similarly, for the resultant displacement, the magnitude is 150 cm, and it makes an angle of 35° with the positive x-axis.
Using trigonometry again, we can find:
x-component = magnitude * cos(angle)
x_resultant = 150 cm * cos(35°)
y-component = magnitude * sin(angle)
y_resultant = 150 cm * sin(35°)
Now that we have both x and y components, we can find the x and y components of the second displacement.
x2 = x_resultant - x1
y2 = y_resultant - y1
To find the magnitude of the second displacement, we can use the Pythagorean theorem:
magnitude = sqrt(x2^2 + y2^2)
Finally, to find the direction of the second displacement, we can use trigonometry to find the angle it makes with the positive x-axis:
angle = atan(y2 / x2)
Plugging in the values, we can calculate the magnitude and direction of the second displacement.