If the slotted arm AB rotates counterclockwise with a constant angular velocity of θ = 2 rad/s , determine the magnitudes of the velocity and acceleration of peg P at θ = 30° . The peg is constrained to move in the slots of the fixed bar CD and rotating bar AB .

Question

If the slotted arm AB rotates counterclockwise with a constant angular velocity of [latex]\dot { heta}[/latex] = 2 rad/s, determine the magnitudes of the velocity and acceleration of peg P at [latex] heta[/latex] = 30°. The peg is constrained to move in the slots of the fixed bar CD and rotating bar AB.

Accepted Answer

Time Derivatives:

[latex]\begin{array}{rlrl}r & =4 \sec heta & \ \dot{r} & =(4 \sec heta( an heta) \dot{ heta}) \mathrm{ft} / \mathrm{s} & & \dot{ heta}=2 \,\mathrm{rad} / \mathrm{s} \ \ddot{r} & =4\left[\sec heta( an heta) \ddot{ heta}+\dot{ heta}\left(\sec heta\left(\sec ^2 heta ight) \dot{ heta}+ an heta \sec heta( an heta) \dot{ heta} ight) ight] & & \ddot{ heta}=0 \ & =4\left[\sec heta( an heta) \dot{ heta}+\dot{ heta}^2\left(\sec 3 heta+ an ^2 heta \sec heta ight) ight] \mathrm{ft} / \mathrm{s}^2 & \end{array}[/latex]

When [latex] heta=30^{\circ}[/latex],

[latex]\begin{aligned}& \left.r ight|_{ heta=30^{\circ}}=4 \sec 30^{\circ}=4.619 \,\mathrm{ft} \& \left.\dot{r} ight|_{ heta=30^{\circ}}=\left(4 \sec 30^{\circ} an 30^{\circ} ight)(2)=5.333 \,\mathrm{ft} / \mathrm{s} \& \left.\ddot{r} ight|_{ heta=30^{\circ}}=4\left[0+2^2\left(\sec ^3 30^{\circ}+ an ^2 30^{\circ} \sec 30^{\circ} ight) ight]=30.79 \,\mathrm{ft} / \mathrm{s}^2\end{aligned}[/latex]

Velocity:

[latex]v_r=\dot{r}=5.333 \,\mathrm{ft} / \mathrm{s} \quad v_ heta=r \dot{ heta}=4.619(2)=9.238 \,\mathrm{ft} / \mathrm{s}[/latex]

Thus, the magnitude of the peg's velocity is

[latex]v=\sqrt{v_r^2+v_ heta^2}=\sqrt{5.333^2+9.238^2}=10.7 \,\mathrm{ft} / \mathrm{s}[/latex]

Acceleration:

[latex]\begin{aligned}& a_r=\ddot{r}-r \dot{ heta}^2=30.79-4.619\left(2^2 ight)=12.32 \,\mathrm{ft} / \mathrm{s}^2 \& a_ heta=r \ddot{ heta}+2 \dot{r} \dot{ heta}=0+2(5.333)(2)=21.23 \,\mathrm{ft} / \mathrm{s}^2\end{aligned}[/latex]

Thus, the magnitude of the peg's acceleration is

[latex]a=\sqrt{a_r^2+a_ heta^2}=\sqrt{12.32^2+21.23^2}=24.6 \,\mathrm{ft} / \mathrm{s}^2[/latex]

Question 12.181: If the slotted arm AB rotates counterclockwise with a consta...

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