Question 6 (paper Question VI) — Wheel-Crank-Slider: Angular Velocity and Acceleration (Part B · Dynamics, equal value): 04-BS-3, Undated paper, Q6

Question 6 (paper Question VI) — Wheel-Crank-Slider: Angular Velocity and Acceleration (Part B · Dynamics, equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Wheel C rotates about a fixed axis at its centre, radius $r=127$ mm to crank pin A. Connecting rod $AB=508$ mm links A to slider B, which travels along a fixed horizontal guide. At the instant shown, crank $CA$ is vertical (A directly above C); rod AB makes $\approx31.4^\circ$ below horizontal (measured from the printed figure). $v_B=152$ mm/s, $a_B=76$ mm/s$^2$, both directed along the (horizontal) guide.

Given data
Quantity	Value
Wheel radius $r$ (CA)	127 mm
Rod length $AB$	508 mm
$v_B$	152 mm/s (rightward)
$a_B$	76 mm/s² (rightward)

Find. The angular acceleration $\alpha_C$ of wheel C at this instant.

Figure 6. Wheel C (radius 127 mm), rod AB = 508 mm, slider B on a horizontal guide.

Approach. Rigid-body velocity/acceleration equations, $\vec v_B=\vec v_A+\vec\omega_{AB}\times\vec r_{B/A}$ and the acceleration counterpart, with $\vec v_A=\vec\omega_C\times\vec r_{A/C}$; because CA is exactly vertical this instant, the slider's zero vertical velocity forces $\omega_{AB}=0$, which simplifies the acceleration equations considerably.

Set up vectors. $\vec r_{A/C}=(0,127)$ mm. Rod direction: $\vec r_{B/A}=508(\cos31.4^\circ,-\sin31.4^\circ)=(433.5,-264.8)$ mm.
Velocity equation, $y$-component (isolates $\omega_{AB}$). $v_{By}=r_{A/C,x}\,\omega_C+r_{B/A,x}\,\omega_{AB}=0\cdot\omega_C+433.5\,\omega_{AB}=0\ \Rightarrow\ \boxed{\omega_{AB}=0}$ (a consequence of CA being exactly vertical, independent of the rod's angle).
Velocity equation, $x$-component (solves $\omega_C$). $v_{Bx}=-r_{A/C,y}\,\omega_C-r_{B/A,y}\,\omega_{AB}=-127\,\omega_C=152\ \Rightarrow\ \boxed{\omega_C=-1.197\text{ rad/s (i.e. 1.197 rad/s CW)}}$.
Acceleration equation, $y$-component (isolates $\alpha_{AB}$; with $\omega_{AB}=0$ its centripetal term vanishes): $0=r_{A/C,x}\,\alpha_C+r_{B/A,x}\,\alpha_{AB}-\omega_C^2\,r_{A/C,y}=433.5\,\alpha_{AB}-127\,\omega_C^2\Rightarrow\alpha_{AB}=\dfrac{127(1.4326)}{433.5}=\boxed{0.4196\text{ rad/s}^2}$.
Acceleration equation, $x$-component (solves $\alpha_C$). $a_{Bx}=-127\,\alpha_C+264.8\,\alpha_{AB}=76\Rightarrow \alpha_C=\dfrac{264.8(0.4196)-76}{127}=\boxed{0.277\text{ rad/s}^2}$, positive (CCW) — opposite sense to $\omega_C$, so wheel C's clockwise spin is momentarily decelerating.

Final Results — Question 6
Quantity	Value
$\omega_{AB}$	0 (exact, at this instant)
$\omega_C$	1.197 rad/s, clockwise
$\alpha_{AB}$	0.420 rad/s²
$\alpha_C$	0.277 rad/s², counter-clockwise (decelerating the CW spin)

Topic: Relative-motion velocity and acceleration analysis of a rigid-body linkage (crank-rod-slider).

Key relations: $\vec v_B=\vec v_A+\vec\omega\times\vec r_{B/A}$; $\vec a_B=\vec a_A+\vec\alpha\times\vec r_{B/A}-\omega^2\vec r_{B/A}$; for rotation about a fixed axis $\vec v_A=\vec\omega_C\times\vec r_{A/C}$, $\vec a_A=\vec\alpha_C\times\vec r_{A/C}-\omega_C^2\vec r_{A/C}$.

Why this works. With the crank exactly vertical, the slider's velocity constraint (purely horizontal) has no component along the rod's own transverse direction contributed by $r_{A/C}$'s $x$-component (which is zero) — this special instant is why the problem is set up with the crank vertical: it isolates $\omega_{AB}=0$ immediately, then $\omega_C$, before tackling acceleration.

Common pitfall: assuming $\omega_{AB}=0$ implies its whole acceleration contribution vanishes — only the centripetal ($-\omega_{AB}^2\vec r_{B/A}$) term drops out; the tangential term $\vec\alpha_{AB}\times\vec r_{B/A}$ survives and must still be solved for.

Source: Hibbeler, Engineering Mechanics: Dynamics, Ch. 16 (Planar Kinematics of a Rigid Body).

Question 6 of 6: Question 6 (paper Question VI) — Wheel-Crank-Slider: Angular Velocity and Acceleration (Part B · Dynamics, equal value)

Question 6 (paper Question VI) — Wheel-Crank-Slider: Angular Velocity and Acceleration (Part B · Dynamics, equal value)