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.
To find the work done by a force, we can use the formula:
Work = force * displacement * cos(theta)
where:
- force represents the magnitude of the force applied,
- displacement represents the distance traveled, and
- theta represents the angle between the force vector and the displacement vector.
In this case, the force F is given as 30 pounds. The displacement is the distance traveled from (0,0) to the point in the first quadrant along the line y = (1/2)x. Let's break down the problem into steps:
Step 1: Calculate the displacement
To find the displacement, we need to determine the distance traveled. In this case, we are moving from (0,0) to a point along the line y = (1/2)x. Since the line starts at the origin (0,0), we need to find the x and y coordinates of the point where the line intersects the x-axis.
Substitute y = 0 in the equation y = (1/2)x:
0 = (1/2)x
0 = x
So, the intersection point is (0,0).
Now, we need to find the other point where the line intersects the x-axis. Substitute y = 0 in the equation y = (1/2)x:
0 = (1/2)x
0 = x
So, the second intersection point is (2,0).
The displacement is the distance between these two points. In this case, it is 2 units.
Step 2: Calculate the angle between the force vector and the displacement vector (theta)
The direction of the force is given to be (2,3). To calculate the angle between the force vector and the displacement vector, we need to find the dot product of the force vector and displacement vector.
The displacement vector can be determined from the points (0,0) and (2,0) as (2-0, 0-0) = (2,0).
The dot product of the force vector (2,3) and displacement vector (2,0) is given by:
Dot product = (2 * 2) + (3 * 0)
Dot product = 4 + 0
Dot product = 4
Next, we need to find the magnitudes of the force vector and displacement vector.
The magnitude of the force vector = sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13) pounds
The magnitude of the displacement vector = sqrt(2^2 + 0^2) = sqrt(4) = 2 feet
Now, we can calculate the angle between the force vector and displacement vector using the dot product formula:
cos(theta) = (Dot product) / (Magnitude of force * Magnitude of displacement)
cos(theta) = 4 / (sqrt(13) * 2)
Now, we can calculate the value of cos(theta) using a calculator.
Step 3: Calculate the work done
Finally, we can calculate the work done using the work formula:
Work = force * displacement * cos(theta)
Work = 30 pounds * 2 feet * cos(theta)
Now, substitute the value of cos(theta) you obtained from the calculator to calculate the work done.