Question #187239

The rod OA in Figure rotates in the horizontal plane such that θ = sin(t) rad. At the same time,

the collar B is sliding outward along OA so that r = 10t mm. If in both cases t is in seconds,

determine the velocity and acceleration of the collar when t = 1 s.

Expert's answer

Solution

coordinate system-Since time-parametric equation of the are given,it is not necessary to relate r to theta.

Velocity and acceleration- determining the time derivatives and evaluating them when t = 1s, we have

r = 100t^{2} |_{t=1s} = 100 mm theta = t^{3} |_{t=1s} = 1 rad = 57.3^{0}

r' = 200t |_{t=1s }= 200 mm/s theta = 3 t^{3} |_{t=1s} = 3 rad/s

r'' = 200 |_{t=1s }= 200 mm/s^{2} theta = 6 t |_{t=1s} = 6 rad/s^{2}

v = ru_{r }+ r thetha u_{ theta }= { 200 u_{r }+ 300u_{theta} } mm/s

the magnitude of v is

v = sqrt 200^{2} + 300^{2} = 361 mm/s

sigma = tan^{-1} (300/200) = 56.3^{0} sigma + 57.3^{0} = 114^{0}

also

a = [200-100(3)^{2}]u_{r} + [100(6) + 2(200) 3]utheta

the magnitude of a is

a = sqrt 700^{2} + 1800^{2} = 1930 mm/s^{2}

delta = tan^{-1} (1800/700)=68.7^{0} (180^{0}-delta) +57.3^{0} = 169^{0}

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