WM1AC practice questions

Test-style questions covering Weeks 1–4, with my own numbers. Try each one before opening the answer. Use radians unless the question says otherwise.

Before you start

Decide first: differentiate, integrate, or constant-acceleration equations? (See the table in Calculus for kinematics.)

Maths

1. $\mathbf{a}=3\mathbf{i}-2\mathbf{j}+\mathbf{k}$ and $\mathbf{b}=\mathbf{i}+4\mathbf{j}-2\mathbf{k}$. Find $\mathbf{a}\cdot \mathbf{b}$, the angle between them, and $\mathbf{a}\times \mathbf{b}$.

Answer

$\mathbf{a}\cdot \mathbf{b}=3-8-2=-7$ · $\cos \theta =-7/(\sqrt{14}\sqrt{21})$ → $\theta =114°$ $\mathbf{a}\times \mathbf{b}=(4-4)\mathbf{i}-(-6-1)\mathbf{j}+(12+2)\mathbf{k}=7\mathbf{j}+14\mathbf{k}$

2. A 40 N force acts along +z at the point $(0.2,0.1,0)$ m. Find its moment about the origin.

Answer

$\mathbf{M}=\mathbf{r}\times \mathbf{F}=(0.1\times 40)\mathbf{i}-(0.2\times 40)\mathbf{j}=4\mathbf{i}-8\mathbf{j}$ N m

Particles

3. A slider’s position is $s=4t^3-6t^2+1$ m. Find its velocity and acceleration at $t=2$ s, and when it’s momentarily at rest.

Answer

$v=12t^2-12t$ → $v(2)=24$ m/s · $a=24t-12$ → $a(2)=36$ m/s² $v=12t(t-1)=0$ at t = 0 and t = 1 s

4. A cart speeds up uniformly from 2 m/s to 8 m/s over 15 m. Find its acceleration and how long it takes.

Answer

$64=4+2a(15)$ → $a=2$ m/s² · $t=(8-2)/2=3$ s

5. A particle has $a=2t$ m/s², starting at $v=1$ m/s and $s=0$. Find $v$ and $s$ at $t=3$ s.

Answer

$v=1+t^2=10$ m/s · $s=t+t^3/3=3+9=12$ m

6. A wheel of radius 0.15 m starts from rest with $\alpha =4$ rad/s². After 2 s, find the speed and total acceleration of a point on its rim.

Answer

$\omega =8$ rad/s · $v=1.2$ m/s · $a_t=0.6$, $a_n=64\times 0.15=9.6$ m/s² · $a=9.62$ m/s²

Rigid bodies and mechanisms

7. A motor at 1200 rpm brakes uniformly to rest in 5 s. Find α and the number of revolutions it makes while stopping.

Answer

${\omega }_0=125.7$ rad/s · $\alpha =-25.1$ rad/s² · $\theta =\frac{1}{2}\times 125.7\times 5=314$ rad $=50$ rev

8. A motor at 3000 rpm has a 12 mm (radius) pinion driving a 60 mm gear. A 40 mm pulley on the same shaft drives a 100 mm pulley by belt. Find the output speed.

Answer

Gear: $3000\times 12/60=600$ rpm · Belt: $600\times 40/100=$ 240 rpm

9. A hydraulic cylinder forms the third side of a triangle whose other two sides are fixed at 1.2 m and 0.8 m. The angle between them is 60° and opening at 0.5 rad/s. Find the cylinder’s length and how fast it’s extending.

Answer

$s^2=1.44+0.64-1.92\cos 60°=1.12$ → $s=1.06$ m $2s\dot{s}=2(1.2)(0.8)\sin \theta \dot{\theta }$ → $\dot{s}=\frac{0.96\times 0.866\times 0.5}{1.06}=0.393$ m/s

10. A slider-crank has crank $OA=0.15$ m turning at 10 rad/s anticlockwise and rod $AB=0.4$ m, with slider B on the horizontal line through O. At the instant the crank is vertical (A directly above O), find the rod’s angular velocity and B’s velocity.

Answer

${\mathbf{v}}_A=\mathbf{\omega }\times {\mathbf{r}}_{A/O}=10\mathbf{k}\times 0.15\mathbf{j}=-1.5\mathbf{i}$ m/s ${\mathbf{r}}_{B/A}=0.371\mathbf{i}-0.15\mathbf{j}$ · $v_B\mathbf{i}=-1.5\mathbf{i}+{\omega }_{AB}(0.15\mathbf{i}+0.371\mathbf{j})$ $\mathbf{j}$: $0=0.371{\omega }_{AB}$ → ${\omega }_{AB}=0$ · $\mathbf{i}$: $v_B=-1.5$ m/s At this instant the rod is only translating, so B moves at A’s speed, 1.5 m/s to the left.

Short written answers

11. Give one example each of translation, fixed-axis rotation and general plane motion in a slider-crank.

Answer

Slider: translation · crank: fixed-axis rotation · connecting rod: general plane motion.

12. Why do two meshing gears share a tangential velocity but not an angular velocity?

Answer

Their teeth move together at the contact point, so the rim speeds $r\omega$ are equal. Different radii then need different ω.

Week 4: statics and kinetics

13. A 60 N force acts at the end of a 0.5 m horizontal bar pinned at O, pointing up and away from O at 25° above the horizontal. Find its moment about O.

Answer

Only the vertical component has a moment arm (the horizontal one acts along the bar, through O). $M_O=60\sin 25°\times 0.5=25.4\times 0.5=$ 12.7 N·m anticlockwise

14. A 3 m beam is pinned at A (its left end) and rests on a roller at B, 2.5 m from A. It carries 400 N at 1 m from A and 200 N at the free end (3 m from A). Find the reactions.

Answer

$\sum M_A=0$: $B(2.5)=400(1)+200(3)=1000$ → $B=400$ N ↑ $\sum F_y=0$: $A_y=600-400=$ 200 N ↑ · $A_x=0$ See Static equilibrium and free body diagrams.

15. An 8 kg box sits on a 30° ramp with ${\mu }_s=0.65$. Does it slide? What friction force acts?

Answer

$\tan 30°=0.577<0.65$, so it stays put. Friction is only as big as needed: $F=mg\sin 30°=$ 39.2 N (below the limit ${\mu }_{s}N=0.65\times 68.0=44.2$ N).

16. A 25 kg robot must accelerate at 1.5 m/s² on a flat floor with $C_{rr}=0.02$. What total driving force do its wheels need?

Answer

$F=ma+C_{rr}mg=37.5+4.9=$ 42.4 N

17. A 6 kg block on a table (${\mu }_k=0.2$) is pulled by a rope over a smooth pulley by a hanging 5 kg mass. Find the acceleration and the tension.

Answer

Hanging: $5g-T=5a$ · block: $T-0.2(6g)=6a$. Adding: $5g-1.2g=11a$ → $a=3.39$ m/s² $T=5(9.81-3.39)=$ 32.1 N

18. A T-shaped section has a 120 × 20 mm flange on top of a 20 × 80 mm web (total height 100 mm), centred. Find $\bar{y}$ from the bottom.

Answer

Flange: 2400 mm² at y = 90 · web: 1600 mm² at y = 40 $\bar{y}=(2400\times 90+1600\times 40)/4000=$ 70 mm (and $\bar{x}$ is on the centreline by symmetry)

19. What’s a two-force member, and why does it make an equilibrium problem easier?

Answer

A link pinned at both ends with no load in between. It can only carry force along its length (tension or compression), so its unknown force has a known direction: one unknown instead of two.

20. Why does a slider-crank need more torque to start than to keep running?

Answer

Static friction exceeds kinetic friction; the motor must accelerate all the moving masses from rest ($\sum F=ma$, $\sum M=I\alpha$); and near dead centre the crank has very little leverage on the rod.