Given that,
Mass of object = 9.90 kg
Time =5.40 s
Suppose the force is (2.00i + 9.00j + 5.30k) N, initial position is (2.70i - 2.90j + 5.50k) m and final position is (-4.10i + 3.30j + 5.40k) m.
We need to calculate the displacement
Using formula of displacement
![s=r_{2}-r_{1}](https://tex.z-dn.net/?f=s%3Dr_%7B2%7D-r_%7B1%7D)
Where,
= initial position
= final position
Put the value into the formula
![s= (-4.10i + 3.30j + 5.40k)-(2.70i - 2.90j + 5.50k)](https://tex.z-dn.net/?f=s%3D%20%28-4.10i%20%2B%203.30j%20%2B%205.40k%29-%282.70i%20-%202.90j%20%2B%205.50k%29)
![s= -6.80i+6.20j-0.1k](https://tex.z-dn.net/?f=s%3D%20-6.80i%2B6.20j-0.1k)
(a). We need to calculate the work done on the object
Using formula of work done
![W=F\cdot s](https://tex.z-dn.net/?f=W%3DF%5Ccdot%20s)
Put the value into the formula
![W=(2.00i + 9.00j + 5.30k)\cdot (-6.80i+6.20j-0.1k)](https://tex.z-dn.net/?f=W%3D%282.00i%20%2B%209.00j%20%2B%205.30k%29%5Ccdot%20%28-6.80i%2B6.20j-0.1k%29)
![W=-13.6+55.8-0.53](https://tex.z-dn.net/?f=W%3D-13.6%2B55.8-0.53)
![W=41.67\ J](https://tex.z-dn.net/?f=W%3D41.67%5C%20J)
(b). We need to calculate the average power due to the force during that interval
Using formula of power
![P=\dfrac{W}{t}](https://tex.z-dn.net/?f=P%3D%5Cdfrac%7BW%7D%7Bt%7D)
Where, P = power
W = work
t = time
Put the value into the formula
![P=\dfrac{41.67}{5.40}](https://tex.z-dn.net/?f=P%3D%5Cdfrac%7B41.67%7D%7B5.40%7D)
![P=7.71\ Watt](https://tex.z-dn.net/?f=P%3D7.71%5C%20Watt)
(c). We need to calculate the angle between vectors
Using formula of angle
![\theta=\cos^{-1}(\dfrac{r_{1}r_{2}}{|r_{1}||r_{2}|})](https://tex.z-dn.net/?f=%5Ctheta%3D%5Ccos%5E%7B-1%7D%28%5Cdfrac%7Br_%7B1%7Dr_%7B2%7D%7D%7B%7Cr_%7B1%7D%7C%7Cr_%7B2%7D%7C%7D%29)
Put the value into the formula
![\theta=\cos^{-1}\dfrac{(-4.10i + 3.30j + 5.40k)\cdot(2.70i - 2.90j + 5.50k)}{7.54\times6.778})](https://tex.z-dn.net/?f=%5Ctheta%3D%5Ccos%5E%7B-1%7D%5Cdfrac%7B%28-4.10i%20%2B%203.30j%20%2B%205.40k%29%5Ccdot%282.70i%20-%202.90j%20%2B%205.50k%29%7D%7B7.54%5Ctimes6.778%7D%29)
![\theta=79.7^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%3D79.7%5E%7B%5Ccirc%7D)
Hence, (a). The work done on the object by the force in the 5.40 s interval is 41.67 J.
(b). The average power due to the force during that interval is 7.71 Watt.
(c). The angle between vectors is 79.7°