the force F = (3.16 N)i + (4.21 N)j acts on a particle as it moves along the x axis from x = 2.81 m to x = 6.52 m.
what is the work done, in joules, by the force?
To find the work done by a force, we use the formula:
Work = Force * Displacement * cos(θ)
where:
- Force is the magnitude of the force vector,
- Displacement is the magnitude of the displacement vector, and
- θ is the angle between the force and displacement vectors.
In this case, the force is given as F = (3.16 N)i + (4.21 N)j. Since the particle is moving along the x-axis, the displacement vector will be in the x-direction.
Displacement = Δx = xf - xi = 6.52 m - 2.81 m = 3.71 m
To find the angle θ between the force and displacement vectors, we need to calculate the dot product of the force vector F and the unit vector in the x-direction, i.
F · i = (3.16 N)(1) + (4.21 N)(0) = 3.16 N
Since F · i is equal to the magnitude of the force vector F multiplied by the magnitude of the displacement vector cos(θ), we can rewrite the equation as:
3.16 N = |F| * (3.71 m) * cos(θ)
Solving for |F|:
|F| = 3.16 N / (3.71 m * cos(θ))
Since cos(θ) = 1 (since the force and displacement vectors are in the same direction), the magnitude of the force vector |F| is:
|F| = 3.16 N / (3.71 m)
Now we can calculate the work done:
Work = 3.16 N * 3.71 m * cos(0°)
Since cos(0°) = 1, the work done is:
Work = 3.16 N * 3.71 m = 11.7276 J
Therefore, the work done by the force is approximately 11.73 Joules.
To calculate the work done by a force along a straight line, you can use the formula:
Work = Force * Distance * cos(θ)
In this case, the force is given as F = (3.16 N)i + (4.21 N)j, and the particle moves along the x-axis, so the angle between the force and the displacement is 0 degrees or cos(0) = 1.
The distance is the difference in positions along the x-axis, which is Δx = final position - initial position = x_f - x_i = 6.52 m - 2.81 m = 3.71 m.
Now, substitute the values into the formula:
Work = (3.16 N * 3.71 m) * 1 = 11.7276 joules
Therefore, the work done by the force is 11.7276 joules.
work = force * distance
= √(3.16^2 + 4.21^2)N * (6.52-2.81)m = ____ J