1.128: Particle P moves on a circle with an arclength given as a function of tim

Question

1.128: Particle P moves on a circle with an arclength given as a function of time as shown. Find the time(s) ant the angle(s) when the tangential and normal components are equal.

1.138: Find the radius of curvature of the "Witch of Agnesi" curve at x=0. A point P moves from left to right along the curve defined in the precedig problem with a constant x component (x0) of velocity. FInd the acceleration of P when it reaches the point (x,y)= (0,2a).

1.128 Particle P moves on a circle (Figure P1·128) with an arclength given as a function of time as shown. Find the time(s) and the angle(s) when the tangential and normal acceleration components are equal 27 ft seft "s = 0 here at t-0 1.138 Find the radius of curvature of the "Witch of Ag- nesi" curve at x-0. (See Figure P1.138.) 8a3 4a2 +x2 Figure P1.138 A point P moves from left to right along the curve defined in the preceding problem with a constant x com- ponent (o) of velocity. Find the acceleration of P when it reaches the point (x, y)-(0, 2a)

Explanation / Answer

1.128

tangential acceleration = d^2/dt^2 = 3t^2

tangential speed= ds/dt = t^3

normal acceleration = v^2/r = t^6/27

t^6/27 = 3t^2

t^4 = 81

t = 3sec

s = 3^4/4 = 81/4

angle = (81/4)/27 = 0.75 radians = 0.75*180/3.142 degrees = 42.96 degrees

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