A particle moves from A to D in FIGURE EX10.36 while experiencing...

Question

A particle moves from A to D in FIGURE EX10.36 while experiencing force F = (6i + 8j) N. How much work does the force do if the particle follows path. (b) ACD. Is this a conservative force? Explain.

Accepted Answer

Hi, everyone in this practice problem, we are asked to determine the work done by the force on the car following a path of P QR based on the figure and also classify whether the force acting upon it is considered conservative or non conservative. So we're having a car traveling from point B to point R along the path of B QR where point B is located at 0.0.0.0.0, comma zero, point Q is located at uh 00.0.5 comma zero and point R is located at 00.0.5 comma four at A X and Y coordinate, the car is experiencing a force of F equals to 12 I plus six J Newton. And we're asked to determine the work done and also classify the force, whether it's conservative or not. The options are 67 jewels and the force to be non conservative, 48 jewels for the work and the force to be non conservative, 84 jewels and the force to be conservative and 96 Jews for the work and the force to be conservative. So uh in order for us to calculate the work done by the force, the work done is actually defined by the dot product of the force and the displacement, namely the product of the displacement and the component of the force acting in the direction of the displacement itself. So the work done W is going to equals to the force, the dot product to the distance or the displacement. So the total work along the path of P QR can be found as the sum of the individual work done by the force when the car actually moves through the path of B Q and through the path of QR. So W P QR which is the one that we are interested at will equals to W P Q plus W QR. So um substituting the work equation here where work equals to force dot product displacement. We can find that the total work P QR will equals to the force dot product. The distance of P Q plus the force dot product, the distance of QR from the figure and the location of the points, we can determine that the distance or the displacement of P Q will be 5 m which is the distance from 0.5 comma 0 to 0, comma zero. And the displacement from point or the displacement for QR is going to equals to four m because that is from 0.5 comma 4 to 40.5 comma zero in the Y direction. Um From this information, we can actually start plugging in all the values that we know because we are already given the like the total force or the force in the factor form. So the total P QR we done equals to the force which is 12 I plus six J. Newton dot product D P Q which is five m in the eye direction. So 5 m plus divorce 12 I plus six J Newton dot four J because the D QR is in the J direction four J meter. So doing the stock product, we can get W P QR two equals to 12 multiplied by five, multiply it by I multiply it by I Joel plus six multiplied by five, multiplied by I or multiplied by J multiplied by I Joel plus 12 multiplied by four multiplied by I multiplied by J Joel. And the last term will be six multiplied by four J multiplied by J ju just like. So, so the one thing that we need to remember when we're doing a dot Product equation is that I multiplied by I will equals to J multiplied by J and equals to one while I multiplied by J will equals to J multiplied by I will equals to zero. So in this case, using this equation right here or using this known fact, we can delete or in essentially consider that every I multiplied by J and J multiplied by I components can just be neglected. Therefore, W P QR will then just be 12 multiplied by five Jews plus um six multiplied by four jos which is going to be 60 Joles plus jules, which will then be 84 jules just like. So, so we found the total work done by the force on the car following the path P QR. So in order for us to see whether the force is conservative or not, what we need to do is to also calculate the work done along the path of Pr. So we want to imagine the car going directly from point B to point R instead of going from point B to point R to Q. So the way we want to tackle this is to find W pr and W pr considering that work is going to equals to force dot product displacement, then it's going to equals to force dot product, the displacement from point R to point Q or basically, oh I'm sorry, I mean the displacement from point P to point R or R two P and the displacement from point B to point RDP R will actually equals to five I plus four J meter because we're moving from 0.0 comma 0 to 00.5 comma four, which will then be moving five m in the I component or in the X direction and four m in the J component or in the Y direction. Ok. So we can actually substitute the force which is 12 I plus six J Newton dot Product +25 I plus four J meter. And this will actually give us 12 multiplied by five I I knew or jewels plus uh six multiplied by five J dot I Jules plus 12 multiplied by four, uh I J Jules. And the last term is going to be six multiplied by four. Jo uh J dot JJ multiplied by J Jules. Again, we know that I I JJ will equals to one and I JJ, I will equals to zero. So we can essentially neglect every term that has an J multiplied by I or I multiplied by J or J dot I or I dot J. So this will essentially give us a value of 12 times five jules plus six times four jules, which will then also be 60 jules plus Jews, which will be a total of 84 jules. So apparently the total work done when we are calculating the total work going from point B to point R either through either going through point Q or not. So either going through the path of P QR or going directly from P two R. The work done along the path of QR is the same as the work done along the path of B R. Namely, the work done is actually independent of the path followed because of that. So W Pr equals W P QR, hence, the force can actually be considered as conservative. So with uh total work done of 84 Jews and the force being a conservative force. We can determine that option C is going to be the answer to our problem with total work done being 84 jewel and the force being conservative force. And that will essentially be it for this problem. If you guys still have any sort of confusion, please make sure to check out our adolescent videos on similar topics and that will be all for this one. Thank you.

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A particle moves from A to D in FIGURE EX10.36 while experiencing force F = (6i + 8j) N. How much work does the force do if the particle follows path. (b) ACD. Is this a conservative force? Explain.

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