A constant force of 60,30− pounds moves an object in a straight line from the point )0,0( to the point)10,10(−. Compute the work done.
Take the inner product of (60,30) with the displacement vector of (10,10)
60*10 + 30*10 = ...
And don't forget to insert the correct units...
To compute the work done by a constant force moving an object, we can use the formula:
Work = Force * Distance * Cos(θ)
Where:
- Force is the magnitude of the constant force
- Distance is the distance traveled by the object
- θ is the angle between the force and the direction of motion
In this case, the force is given as 60,30− pounds, and the object moves from the point (0,0) to (10,10). The distance can be calculated using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distance first:
Distance = √((10 - 0)^2 + (10 - 0)^2)
= √(100 + 100)
= √200
= 10√2
Next, we need to calculate the angle θ. The angle can be obtained using the formula:
θ = arctan((y2 - y1)/(x2 - x1))
θ = arctan((10 - 0)/(10 - 0))
= arctan(10/10)
= arctan(1)
= π/4 (in radians) or 45° (in degrees)
Now, we can calculate the work done:
Work = Force * Distance * Cos(θ)
= (60,30−) * (10√2) * Cos(π/4)
= (60,30−) * (10√2) * √2/2
= (60,30−) * 10 * √2
= (600√2, 300√2−) pounds
Therefore, the work done by the constant force is (600√2, 300√2−) pounds.
To compute the work done by a constant force when moving an object, we use the formula:
Work = Force * Distance * cos(theta),
where:
- Force is the magnitude of the force applied,
- Distance is the displacement vector between the initial and final points,
- theta is the angle between the force vector and the displacement vector.
Given that the force is 60.30 pounds and the distance is from (0,0) to (10,10), we can calculate the work done.
First, let's find the displacement vector (distance) from (0,0) to (10,10):
- From (0,0) to (10,10), the change in x-coordinate is 10 - 0 = 10, and the change in y-coordinate is 10 - 0 = 10.
- Therefore, the displacement vector is (10,10).
Next, we need to find the angle theta between the force vector and the displacement vector.
- Since the force is constant and moves in a straight line, the angle theta is 0 degrees. (Both the force vector and the displacement vector are parallel.)
Now, we can calculate the work done using the formula:
Work = Force * Distance * cos(theta).
Plugging in the values:
Work = 60.30 * (10,10) * cos(0)
= 60.30 * (10,10) * 1 (cos(0) is 1)
= 60.30 * (10,10)
= 603.
Therefore, the work done by the constant force is 603 pounds.