Power input of a Force acting on a particle moving with Velocity

[_h2tm]

Homework Statement

Find the power input of a force $\vec{F}$ acting on a particle that moves with a velocity $\vec{V}$ for each of the following situations.

• $\vec{F}$ = 4$\hat{i}$ N  + 3$\hat{j}$ N , $\vec{V}$ = 7$\hat{i}$ m/s 
  
• $\vec{F}$ = 7$\hat{i}$ N  - 5$\hat{j}$ N , $\vec{V}$ = -5$\hat{i}$ m/s + 4$\hat{j}$ m/s 
  
• $\vec{F}$ = 2$\hat{i}$ N + 10$\hat{j}$ N , $\vec{V}$ = 2$\hat{i}$ m/s  + 3$\hat{j}$ m/s 

Homework Equations

P = $\frac{dW}{dt}$ = $\vec{F}$ • $\vec{v}$

The Attempt at a Solution

• $\sqrt{4^2 + 3^2}$ $\times$ cos(arctan(3/4)) $\times$ 7
  For this problem, I got the correct answer: 28 W.
  
• $\sqrt{7^2 + (-5)^2}$ $\times$ $\sqrt{(-5)^2 + 4^2}$
  I also got the correct answer for this problem: 55 W, except that the answer is negative, I figure because the velocity vector's $\hat{i}$ and $\hat{j}$ components are in the opposite direction of the force's components (individually). Is this supposition correct?
  
• $\sqrt{2^2 + 10^2}$ $\times$ sin(arctan(10/2)) $\times$ $\sqrt{2^2 + 3^2}$ $\times$ sin(arctan(3/2))
  I cannot figure out what I am doing wrong on this one. (The correct answer is 34.)

I am having a hard time wrapping by head around dot products. I'm guess that is my problem here. Any and all help is greatly appreciated.

 

[frogjg2003]

When you do dot products, you take the sum of the products of each component.
For the first one:
P = 4N*7m/s (the i components) + 3N*0m/s (the j components) = 28 W.
For the second one:
P=7N*-5m/s (i components) + -5N*4m/s (j components) = -55W.
Try the third one out this way.
P= (product of i components) + (product of j components) = ?

 

[frogjg2003]

What you are doing is finding it using magnitudes and angles, which you must first calculate. When calculating angles, you also have to think about the fact that it may be more than 90 degrees, something that arctan will not provide.

 

[_h2tm]

OK, thanks. I completely get how to do this in the future.

Regarding your second comment though -- I understand about arctan. I drew a diagram so I knew where my vectors were. Was this comment just FYI or was my attempted method something that would work, I was just simply doing it incorrectly?

 

[Nickie]

As a scientist, it is important to understand the concepts and equations involved in a problem before attempting to solve it. In this case, the equation for power (P = \frac{dW}{dt} = \vec{F} • \vec{v}) is correct, but your approach to solving the problem is not entirely accurate.

The dot product of two vectors \vec{A} and \vec{B} can be written as \vec{A} • \vec{B} = |\vec{A}| |\vec{B}| cos(\theta), where \theta is the angle between the two vectors. So for the first problem, the correct calculation would be:

P = (\vec{F} • \vec{V}) = (4\hat{i} N  + 3\hat{j} N ) • (7\hat{i} m/s) = (4)(7) cos(0) = 28 W

For the second problem, the velocity vector (-5\hat{i} m/s + 4\hat{j} m/s) is in the opposite direction of the force vector (7\hat{i} N  - 5\hat{j} N ). This means that the angle between the two vectors is 180 degrees, so the calculation would be:

P = (\vec{F} • \vec{V}) = (7\hat{i} N  - 5\hat{j} N ) • (-5\hat{i} m/s + 4\hat{j} m/s) = (7)(-5) cos(180) = -35 W

For the third problem, the correct calculation would be:

P = (\vec{F} • \vec{V}) = (2\hat{i} N + 10\hat{j} N) • (2\hat{i} m/s + 3\hat{j} m/s) = (2)(2) cos(0) + (10)(3) cos(90) = 4 + 0 = 4 W

Remember that power is a scalar quantity, so it does not have a direction. Therefore, the negative sign in the second problem indicates that the power is being dissipated (negative) rather than generated (positive).

In summary, it is important to understand the concepts and equations involved