. Suppose an ant crawls onto a turntable spinning at one revolution per unit tim

Question

. Suppose an ant crawls onto a turntable spinning at one revolution per unit time. A stationary observer looking from above will see its path as motion in the ry-plane (a) Suppose the ant stops on the edge of the spinning platter (radis R) Find its position, velocity and acceleration relative to the observer as vector-valued finctions of time t. (b) Now imagine the ant walks down a radius of the platter toward the center of the turntable. From its perspective, it walks at constant speed and in a straight line, arriving at the center at time t-1. Write the ant's position as a vector-valued function of time t. (c) At any given time t, let b be the unit vector pointing from the ant's position toward the center of the turntable, and let lvecc be the unit vector in the direction of the spinning turntable (i.e., perpendicular to b). Write the acceleration of the ant as a linear combination of b and c

Explanation / Answer

at any time t, position of ant wrt to the turntable is (x,y)

then wrt to ground frame it is (Rcos(theta), Rsin(theta))

where theta = wt where w is angular speed of the disc

a. at distance from the center R

as relative to observer r = R(cos(wt)i + sin(wt)j) where i and j are unit vectors in x and y directions

so, v(t) = r' = -Rw(sin(wt)i - cos(wt)j)

a(t) = r" = -Rw^2(cos(wt)i + sin(wt)j)

b. now given, dr/dt = v ( constnat speed of the ant)

and, v = R/1

v = R

so dr/dt = R

r = Rt + c

at t = 0, r = -R

at t = 1, r = 0

r = Rt - R

r = R(t - 1)

hence

position vector rho(t) = r(cos(wt)i + sin(wt)j)

rho(t) = R(t - 1)(cos(wt)i + sin(wt)j)

c. b is the radial unit vector poiting towards the center of the turntable

c be the unit vector perpendivular to it

then acceleration a(t) = w^2*R(t - 1) b

thereis no tangential acceleration as there is no change in tangential speed

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