Find the work done by a force F of 30 pounds acting in the direction (2,3) in moving an object 3 feet from (0,0) to the point in the first quadrant along the line y = (1/2)x.
The answer is 85.38 foot-pounds.
I just need to figure out how to get the answer.
F1 -> 30 lbs, direction <2, 2>
magnitude of <2, 2> = 2sqrt(2)
Find unit vector of F1 (divide original components by magnitude)
-> <30/sqrt(2), 30/sqrt(2)>
F2 -> 3 feet, direction y = 1/2x
3 = sqrt(x^2 + y^2)
y = 1/2x
3 = sqrt(x^2 + (1/2x)^2)
9 = 5/4* x^2
x^2 = 36/5
x = 6/sqrt(5)
y = 2/sqrt(5)
F1 = <30/sqrt(2), 30/sqrt(2)>
F2 = <6/sqrt(5), 3/sqrt(5)>
Find dot product of F1 and F2
(30/sqrt(2))*(6/sqrt(5))+(30/sqrt(2))*(3/sqrt(5)) = 85.38
Answer in the back of the book is 85.38
Find the angle of the Force F
cos T = 2/sqrt13 = .555
T = 56.3 deg above x axis
or
tan T = 2/3
so T = 56.3
Find the direction of motion from slope of line
tan slope = 1/2
slope angle = 26.6 deg
angle between force and motion = 56.3-26.6 = 29.7 degrees
so
component of force in direction of motion = 30 cos 29.7
= 26.06 pounds
26.06*3 = 78.2 ft lbs
beats me how to get 85.38
It would be much closer if the force vector were direction(3,2)
the vector is actually <2,2>
and the line angle is 64.3
the vector is indeed <2,2>, but the line angle (angle formed by y=1/2x and x-axis) is as Damon posted, 26.6 degree. As for why it isn't 64.3 degree is because slope is change in y over the change in x, thus this means horizontal change is 2 and vertical change is 1. When finding the line angle, simply use arc tan(opp/adj) or in this case arc tan(1/2) which will result in 26.6 degree.
As for the answer, its actually 95.36 ft lb if the vector is <2,2>
First step : Using distance formula, find the point lying on the line y= 1/2(x) with given distance = 3 ft. Find distance vector.
Second step : Find Force vector in the direction of <2,3>
Third step : use formula W= Force * distance, get the work done in ft pound.
To find the work done by a force, you can use the formula:
Work = Force * Distance * Cos(theta)
Where:
- Force is the magnitude of the force vector
- Distance is the length of the path
- Theta is the angle between the force vector and the direction of motion
In this case, the force F is given as 30 pounds and it is acting in the direction (2,3). To find the magnitude, or the length, of this force vector, you can use the Pythagorean theorem:
Magnitude of F = sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13)
The distance is given as 3 feet, as the object moves from (0,0) to a point on the line y = (1/2)x. To find the angle theta, you can use the slope of the line, which is 1/2. Remember that the cosine of an angle is equal to the adjacent side divided by the hypotenuse. In this case, the adjacent side is 2.
Cos(theta) = adjacent/hypotenuse = 2/sqrt(13)
Plugging in the values into the work formula:
Work = 30 * 3 * 2/sqrt(13)
Now, we need to calculate and approximate the answer. Using a calculator, we have:
Work ≈ 85.384
Therefore, the work done by the force F is approximately 85.38 foot-pounds.