ENGR273 2026 Q4 (ENGR5004)

ENGR273 2026 Q4Past paperCurrent specFirst25 marks30 min

ENGR273 Summer 2026 Q4[VALID] official answers

A uniform slender rod AB of mass $m$ has its ends constrained to move in two smooth guides (Figure Q4-1).

End A slides on the horizontal guide $y = 0$ .
End B slides on a straight guide inclined at angle $\beta$ to the $+x$ axis (guide line passes through the origin), i.e. $y = (\tan\beta)\,x$ , $0 < \beta < 90^\circ$ .

At the instant shown:

$A = (x_A, 0)$ with known speed and acceleration to the right: $\mathbf{v}_A = v_A\,\mathbf{i}$ , $\mathbf{a}_A = a_A\,\mathbf{i}$ .
$B = (x_B, x_B\tan\beta)$ with known acceleration magnitude $a_B$ along the inclined guide as shown in figure Q4-1 (direction $+\mathbf{t}$ , where $\mathbf{t}$ is the unit tangent of the guide).

A driving actuator applies an unknown force $\mathbf{P}$ at A along $+x$ (through the slider at A). Neglect gravity. Positive angular sense is counterclockwise $+\mathbf{k}$ . $L = AB$ .

Figure Q4-1: rod AB with A on the horizontal guide and B on the guide y = x tan(beta).

Formulas you may need

Instantaneous centre of zero velocity: intersection of the perpendiculars to two known velocity directions; $v = \omega\, r_{/IC}$ (learn this)
Relative velocity $\mathbf{v}_B = \mathbf{v}_A + \boldsymbol\omega\times\mathbf{r}_{B/A}$ (learn this)
Relative acceleration $\mathbf{a}_B = \mathbf{a}_A + \boldsymbol\alpha\times\mathbf{r}_{B/A} - \omega^2\mathbf{r}_{B/A}$ (learn this)
Equations of motion $\sum\mathbf{F} = m\mathbf{a}_G$ , $\sum M_G = I_G\alpha$ , or about any point P: $\sum M_P = I_G\alpha + (\mathbf{r}_{G/P}\times m\mathbf{a}_G)\cdot\mathbf{k}$ (learn this)
Slender rod $I_G = \dfrac{1}{12} m L^2$ (given in the question)

(a)
Draw the free-body diagram and kinetic diagram of rod AB. Indicate guide reactions at A and B, the actuator force $\mathbf{P}$ , mass centre G, and inertia terms $m\mathbf{a}_G$ and $I_G$ (with $I_G = \frac{1}{12} m L^2$ ).
[8]2:2
(b)
Determine the instantaneous centre of zero velocity (IC) of rod AB and locate it relative to points A and B. Hence determine the instantaneous angular velocity $\boldsymbol{\omega}_{AB}$ and the instantaneous velocity of the mass centre G.
[10]2:1
(c)
Using planar rigid-body acceleration relations, determine the instantaneous angular acceleration $\boldsymbol{\alpha}_{AB}$ and the acceleration of the mass centre $\mathbf{a}_G$ . Hence determine the actuator force $\mathbf{P}$ and the guide reactions at A and B required at this instant (in terms of the given variables).
[7]