What work is done by a force (in newtons) F = 3.1xi + 3.1j, with x in meters, that moves a particle from a position r1 = 2.1i + 2.5j to r2 = - -4.9i -3.9j?
Work is the dot product of Force and displacement.
Change in displacement is r2 minus r1. I will be happy to critique your work on this.
Rememeber dot product is a scalar, the sum of i component times i component plus j component times j component etc.
To find the work done by a force, we can use the formula:
Work (W) = Force (F) * Displacement (d) * cos(θ),
where F is the force vector, d is the displacement vector, and θ is the angle between the force and the displacement vectors.
In this case, we have:
F = 3.1xi + 3.1j (Force vector),
d = r2 - r1 = (-4.9i - 3.9j) - (2.1i + 2.5j) (Displacement vector).
To find the angle θ, we can use the dot product formula:
θ = arccos(F · d / (|F| * |d|)).
Let's calculate all the required quantities step by step.
1. Calculation of the displacement vector:
d = (-4.9i - 3.9j) - (2.1i + 2.5j)
= -4.9i - 3.9j - 2.1i - 2.5j
= -7i - 6.4j.
2. Calculation of the dot product between the force and displacement vectors:
F · d = (3.1xi + 3.1j) · (-7i - 6.4j)
= 3.1 * -7 + 3.1 * -6.4.
To get the magnitude (|F|) and magnitude (|d|) of the vectors, we can use the Pythagorean theorem:
|F| = sqrt((3.1)^2 + (3.1)^2) = sqrt(19.21 + 9.61),
|d| = sqrt((-7)^2 + (-6.4)^2) = sqrt(49 + 40.96).
3. Calculation of the angle θ:
θ = arccos((3.1 * -7 + 3.1 * -6.4) / (sqrt(19.21 + 9.61) * sqrt(49 + 40.96))).
Now, with all the required values computed, we can substitute them into the work formula:
W = (3.1xi + 3.1j) * (-7i - 6.4j) * cos(θ).
After evaluating this expression, you will obtain the work done by the force.