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The main lesson of this sample problem is this: It isperfectly all right to choose an easy path instead of a hard path. Figure 8-22a shows a 2.0 kg block of slippery cheese that slides along a frictionless track from point a to point b. The cheese travels through a total distance of 2.0 m along the track, and a net vertical distance of 0.80 m . How much work is done on the cheese by the gravitational force during the slide ? |
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Answer» Solution :KEY IDEAS (1) We cannot calculate the work by using Eq. 8-12 `(W_(g)= mgd cos phi)`. The reason is that the angle `phi` between the directions of the gravitational force `vec(F)_(g)` and the displacement `vec(d)` VARIES along the trackin an unknown way. (Even if we did know the shape of the track and could calculate `phi` along it, the calculation could be very DIFFICULT. ) (2) Because `vec(F)_(g)` is a conservation force, we can find the work by choosing some other path beween a and b - one that makes the calculation easy. Calculations:Let us choose the dashed path in Fig. 8-22b, it consists of two straight segments. Along the horizontal segment, the angle `phi` is a constant `90^(@)`. Even though we do not know the displacement along that horizontal segment, Eq. 8-12 tells us that the work `W_(h)` done there is `W_(h)=mgd cos 90^(@)=0`. Along the vertical segment, the displacement d is 0.80 m and , with `vec(F)_(g)` and `vec(d)` both downward, the angle `phi` is  Figure 8-22 (a) A block of cheese slides along a frictionless track from POINT a to point b. (b) Finding the work done on the cheese by the gravitational force is easier along the dashed path than along the actualpath taken by the cheese, the result is the same for both paths. a constant `0^(@)`. Thus, Eq.8-12 GIVES us, for the work `W_(b)` done along the vertical part of the dashed path, `W_(v) = mgd cos 90^(@)`. ` = (2.0kg) (9.8 m//s^(2))(0.80 m)(1)` `=15.7` J. The total work done on the cheese by `vec(F)_(g)` as the cheese moves from point a to point b along the dashed path is then `W=W_(h)+W_(v)=0+15.7 J ~~ 16 J`. This is also the work done as the cheese slides along the track from a to b. |
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