Calculating the Angle Between a Force and Displacement

[TomFoolery]

Homework Statement

A 13 N force with a fixed orientation does work on a particle as the particle moves through displacement = (3i - 5j + 3k) m. What is the angle between the force and the displacement if the change in the particle's kinetic energy is (a) +25.6 J and (b) -25.6 J?

Homework Equations

Angle = √(x^2 + y^2 + z^2)
Work = mad or F (dot) d

The Attempt at a Solution

I'm not sure how to approach this. I know that the dot product really just means to multiply the x parts by x parts, y parts by y parts, and z parts by z parts (or $\hat{i}$, $\hat{j}$, and $\hat{k}$ parts, if you prefer).

If the particle starts at (0, 0, 0), then the force would have caused it to go to its current position (3, -5, 3). I'm still not sure that is what is being said though.

 

[SammyS]

There is another way to express the dot product, $\vec{F}\cdot\vec{d}\,.$

It involves the magnitude of each of the vectors as well as the cosine of the angle between their directions.

 

[kjohnson]

Yes, work is equal to Fx*dx+Fy*dy+Fz*dz where where Fx,Fy,Fz are the force components and dx,dy,dz are the displacement components...but there is another common way to write the the dot product of two vectors that should make quick work of this problem.

 

[TomFoolery]

Got it, thanks for your help.

 

[rwisz]

I would approach this problem by first understanding the given information and identifying the relevant equations. In this case, we are given a force of 13 N with a fixed orientation and a displacement vector of (3i - 5j + 3k) m. The problem also mentions a change in kinetic energy of +25.6 J and -25.6 J, indicating that the force is doing work on the particle.

To calculate the angle between the force and displacement, we can use the dot product equation: F (dot) d = |F||d|cosθ. This tells us that the work done by the force is equal to the magnitude of the force multiplied by the magnitude of the displacement, multiplied by the cosine of the angle between them.

Using the given information, we can calculate the magnitude of the force and displacement as follows:

|F| = 13 N
|d| = √(3^2 + (-5)^2 + 3^2) = √43 ≈ 6.56 m

We can then plug these values into the dot product equation and solve for the angle (θ):

+25.6 J = (13 N)(6.56 m)cosθ
cosθ = +25.6 J / (13 N)(6.56 m) ≈ 0.304
θ ≈ 71.8°

So, the angle between the force and displacement is approximately 71.8° when the change in kinetic energy is +25.6 J.

For part (b), where the change in kinetic energy is -25.6 J, we can follow the same steps and solve for the angle:

-25.6 J = (13 N)(6.56 m)cosθ
cosθ = -25.6 J / (13 N)(6.56 m) ≈ -0.304
θ ≈ 108.2°

Therefore, the angle between the force and displacement is approximately 108.2° when the change in kinetic energy is -25.6 J.

In conclusion, the angle between the force and displacement can be calculated using the dot product equation and the given values of force, displacement, and change in kinetic energy.