A bug steps onto the edge of a disk of radius a that is spinning at a constant a
ID: 1880096 • Letter: A
Question
A bug steps onto the edge of a disk of radius a that is spinning at a constant angular speed of . The bug moves toward the center of the disk at constant speed v.
a) Derive a differential equation for the path of the bug using polar coordinates.
b) How many revolutions will the disk make before the bug reaches the corner? (The solution will be in terms of the angular speed and radius of the disk).
c) Referring to part (b), what is the total distance the bug will travel, taking into account the motion of the disk?
JUST COMPLETEE ANSWERS, PLEASE!
Explanation / Answer
given radius of disc = a
angular speed = w
bug moves towards center of disc with constant speed v
then
dr/dt = -v
a. phi = w*t ( assuming initial angle = 0 deg)
then in polar coordinates
r = -vt + ro
phi = wt + phi
at t= 0, t = a
r = -vt + a
phi = wt ( phio = 0)
b. time taken to reach other corner = T
T = t1 + t2
at t = t1
r = 0
t1 = a/v
for t > t1
r = vt + c
0 = a + c
r = vt - a
at t = T
r = a
a = vT - a
T = 2a/v
c. total distance = s
ds = sqrt(dx^2 + dy^2)
x = rcos(theta)
dx = -rsin(theta)d(theta) + cos(theta)dr
y = rsin(theta)
dy = rcos(tehta)d(tehta) + sin(theta)*dr
ds = sqrt(r^2*d^2(theta) + dr^2)
dr = +-vdt
d(theta) = wdt
hence
ds = sqrt(r^2w^2dt^2 + v^2dt^2)
ds = sqrt((vt - a)^2w^2 + v^2)dt
intergrating from t = 0 to t = 2a/v
s = (v^2*arcsinh(aw/v) + aw(a^2w^2 + v^2))/vw
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