Answer to Question #177178 in Classical Mechanics for Joseph Se

Question

The final answer should be in the format of "tolerance of ± in the third significant digit."

1.The piston of the hydraulic cylinder gives pin A a constant velocity v = 3.6 ft/sec in the direction shown for an interval of its motion. For the instant when θ = 67°, determine r˙, r¨, θ˙, and θ¨ where r = line OA. The distance of the line A to line B is 6.8”.

r˙ = _ ft/sec

r¨ = _ ft/sec²

θ˙ = _ rad/sec

θ¨ = _ rad/sec²

Answer 1

$\dot{r}=v_r=v cos \theta=14.0663~\frac {ft}{s},$

$v_{\theta}=-v sin \theta=-3.3138~\frac {ft}{s},$

$v_{\theta}=r\dot{\theta}\implies \dot{\theta}=\frac{v_{\theta}}{r}=-0.4873~\frac {rad}{s},$

$a_r=acos \theta=0.3907a~\frac{ft}{s^2},$

$a_{\theta}=-asin \theta=-0.9205~\frac{ft}{s^2},$

$a_r=\ddot{r}-r\dot{\theta}^2,\implies \ddot{r}=a_r+r\dot{\theta}^2=0.3907a+1.6147~\frac{ft}{s^2},$

$a_{\theta}=r\ddot{\theta}+2\dot{r}\dot{\theta}\implies \ddot{\theta}=\frac{a_{\theta}-2\dot{r}\dot{\theta}}{r}=1.8807~\frac{rad}{s^2}.$