Question

I need help with B, C, D. These are Calc 3 problems

32. Suppose a particle of mass m has position given by r(0) =< 1,0,0 >, and velocity given by v(0)0,1,-1 > at time t = 0. Also, assume that for every time t 20 the particle experiences only the force given by the vector function F(t) = m < -cos(t), 0, sin(t) >. Disregard units in this problem a) Use Newton's Second Law, F(t) = ma(t) (where a(t) is the acceleration vector of the particle at time t) in order to show that r(t) =< cost,t,-sin t > gives the particle's position at time t, for t 2 0. Show all the necessary work to find this position vector function, do not simply check that it is correct b) Sketch the particle's path for t 2 0 c) Determine the curvature of the particle's path at time t d) Assume a second particle is also present, and has a position given by the vector equation r2(t)sin(t)-1,n-t,cos(t) -1>, for t 2 0. Determine at which times (if any) the particles collide. Determine at which points in space (if any) the paths of the particles intersect

Answer 1

The path uaill be spiral the Combined mohen alandz ondrater Conshtule crcle ahle axisislinean ene Becaue

Fore) VCt)Sint +ost k at)-cast 1+sint Cose

cose sint tCos ositsint 1-Sint tcost K1-Cost 4int Cas & 10-12

Racs af cusnature v2 2 R- Ainormal) R-SnttCost anet Sin9a 2 2 R (1+Sint+ cat) aper apet

at any tme af colln Loll same - yy their Coadile =2) AT Sint Cost-D CaStSint-1) parthcle ill collide t2 sec put tr2 pennk allde at this pent
when we are calculating the radius of curvature we considered that component of acceleration which is perpendicular to Velocity.. thats why I calculate theta first.