If a ball is given an initial speed of 15 m/s on a horizontal, frictionless surface, how fast will the ball be rolling in 5 seco
1 answer:
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a. I've attached a plot of the surface. Each face is parameterized by
• with
and
;
• with
and
;
• with
and
;
• with
and
; and
• with
and
.
b. Assuming you want outward flux, first compute the outward-facing normal vectors for each face.
Then integrate the dot product of <em>f</em> with each normal vector over the corresponding face.
c. You can get the total flux by summing all the fluxes found in part b; you end up with 42π - 56/3.
Alternatively, since <em>S</em> is closed, we can find the total flux by applying the divergence theorem.
where <em>R</em> is the interior of <em>S</em>. We have
The integral is easily computed in cylindrical coordinates:
as expected.