Rod OA rotates counterclockwise with a constant angular velocity of ..=6 rad/s.

Question

Rod OA rotates counterclockwise with a constant angular velocity of ..=6 rad/s. Through mechanical means collar B moves along the rod with a spred of r..=(4t^2)m/s, where t is in seconds. If r=0 when t=0, determine the magnitude of velocity and acceleration of the collar when t=0.75s.

es counterclockwise with a constant angular velocity of -6 rad/s. Through moves along the rod with a speed of t (4tt)m/s, where t is in Rod OA rotat mechanical means c seconds. If r = 0 w of the collar when t- 0.75 s ollar B hen t = 0, determi

Explanation / Answer

form the given data

theta' = 6 rad/s
hence
theta = 6t ( because theta = 0 at t = 0)
r' = 4t^2 m/s
hecne
r = 4t^3/3 ( because r = 0 when theta = 0)
hence

coordinates of collar at time t in cartesian coordinates are
x = rcos(theta) = 4t^2*cos(6t)/3
y = x*sin(theta) = 4t^2*sin(6t)/3

hence
x' = -8t^2*sin(6t) + 8t*cos(6t)/3
y' = 8t^2*cos(6t) + 8t*sin(6t)/3

x" = -48t^2*cos(6t) - 16t*sin(6t) - 16t*sin(6t) + 8*cos(6t)/3
y" = -48t^2*sin(6t) + 16*t*cos(6t) + 16t*sin(6t) + 8sin(6t)/3

for t= 0.75 s
6t = 4.5 rad

x' = 3.9772 m/s
y' = -2.9036 m/s

x" = 28.59 m/s/s
y" = 18.727 m/s/s

hecne velocity magnitude = sqroot(x'^2 + y'^2) = 4.9243 m/s
acceleration magnitude, a = sqroot(x"^2 + y"^2) = 34.177 m/s/s

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