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pin C through the spiral groove described by the equation r = a\theta . If the angular velocity is constant at \theta , determine the radial and transverse components of velocity and acceleration of the pin.

Step-by-step

Time Derivatives: Since \dot {\theta} is constant, then \ddot{\theta} =0

r = a\theta  \dot {r} = a\dot{\theta}    \ddot {r} = a\ddot{\theta}=0

{v}_{r} = \dot {r} = a\dot{\theta}

{v}_{\theta}  =  r\dot{\theta} =a\theta\dot{\theta}

{a}_{r} = \ddot {r} – r{\dot{\theta}}^{2} = 0 – a\theta{\dot{\theta}}^{2} = -a\theta{\dot{\theta}}^{2}

{a}_{r} = r\ddot{\theta} + 2\dot{r}\dot{\theta} = 0 + 2(a\dot{\theta})(\dot{\theta}) = 2a\dot{\theta}

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