A 14 N force with a fixed orientation does work on a particle as the particle moves through displacement d = (2i - 4j + 3k) m. What is the angle between the force and the displacement if the change in the particle's kinetic energy is (a) +7.54 J and (b) -7.54 J?

f*d=fd

To find the angle between the force and the displacement, we can use the formula:

Work Done = Force * Displacement * cos(theta)

where theta is the angle between the force and the displacement.

For Case (a) - when the change in the particle's kinetic energy is +7.54 J:

Given:
Force (F) = 14 N
Displacement (d) = (2i - 4j + 3k) m
Change in Kinetic Energy (ΔKE) = +7.54 J

Let's calculate the angle between the force and the displacement.

Work Done = Force * Displacement * cos(theta)
+7.54 J = 14 N * (2i - 4j + 3k) m * cos(theta)

The dot product of the force and displacement vectors is given by:
(F*d) = |F| * |d| * cos(theta)

Substituting the magnitudes of the force and displacement vectors:
14 N * (2i - 4j + 3k) m = 14 * sqrt((2)^2 + (-4)^2 + 3^2) * sqrt(2^2 + (-4)^2 + 3^2) * cos(theta)

Simplifying:
14 * sqrt(29) = sqrt(29) * sqrt(29) * cos(theta)

Dividing by sqrt(29):
14 = cos(theta)

Taking the inverse cosine of both sides:
theta = acos(14)

The angle between the force and displacement when the change in kinetic energy is +7.54 J is theta = acos(14).

For Case (b) - when the change in the particle's kinetic energy is -7.54 J:

Given:
Force (F) = 14 N
Displacement (d) = (2i - 4j + 3k) m
Change in Kinetic Energy (ΔKE) = -7.54 J

Using the same steps as above, we will find the angle theta between the force and displacement.

Work Done = Force * Displacement * cos(theta)
-7.54 J = 14 N * (2i - 4j + 3k) m * cos(theta)

14 * sqrt(29) = sqrt(29) * sqrt(29) * cos(theta)

14 = cos(theta)

theta = acos(14)

The angle between the force and displacement when the change in kinetic energy is -7.54 J is theta = acos(14).

Please note that the value of cos(theta) cannot be greater than 1 or less than -1, so the angle theta = acos(14) does not have a real value.

To find the angle between the force and the displacement, we can use the work-energy theorem, which states that the work done on an object equals the change in its kinetic energy. The work done (W) is given by the dot product of the force (F) and displacement (d).

(a) Since the change in kinetic energy is positive, we have:
Change in kinetic energy = +7.54 J

Using the formula: W = F ⋅ d
W = |F| ⋅ |d| ⋅ cosθ

Here, |F| is the magnitude of the force, |d| is the magnitude of displacement, and θ is the angle between the force and displacement.

We are given |F| = 14 N, and |d| = √(2^2 + (-4)^2 + 3^2) = √(4 + 16 + 9) = √(29).

Substituting these values and solving for cosθ:
7.54 = 14 ⋅ √(29) ⋅ cosθ
cosθ = 7.54 / (14 ⋅ √(29))

To find θ, we take the inverse cosine (cos^-1) of both sides:
θ = cos^-1(7.54 / (14 ⋅ √(29)))

(b) Since the change in kinetic energy is negative, we have:
Change in kinetic energy = -7.54 J

Using the same formula as above, we substitute the negative value and solve for cosθ:
-7.54 = 14 ⋅ √(29) ⋅ cosθ
cosθ = -7.54 / (14 ⋅ √(29))

Taking the inverse cosine (cos^-1) of both sides:
θ = cos^-1(-7.54 / (14 ⋅ √(29)))

This will give us the angle between the force and displacement for both cases.

The scalar "dot" product of the Force F and the displacement d,

F*d = |F||d| cos theta.
equals the work done, which equals the kinetic energy increase.

Use that fact to compute the cosine of theta, the angle between the two vectors.

The magnitude of F, represented by |F|, is 14N.

The magnitude of d is sqrt29 = 5.39 m