| Exam Board | OCR |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2004 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments of inertia |
| Type | Force at pivot/axis |
| Difficulty | Standard +0.3 This is a standard M4 rotation question with straightforward applications of center of mass, moment of inertia formulas, and rotational dynamics equations. Parts (i)-(iii) involve routine calculations using standard formulas, while part (iv) requires resolving forces but follows a predictable method. The question is slightly above average difficulty due to the multi-step nature and the need to coordinate several concepts, but all techniques are standard textbook exercises for M4 students. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids6.05f Vertical circle: motion including free fall |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{x} = 1.75\) | M1 A1 | [2] Centre of mass calculation |
| \(I = \frac{4}{3}ml^2 + 5 \cdot 2l^2 = \frac{4}{3} \times 15 \times 1.4^2 + 20 \times 1.4^2 = 78.4\) kg m² | M1 A1 | [2] Moment of inertia about pivot |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin\theta = 0.8\) (show) | M1 A1 | [3] Torque equation; verification that \(\sin\theta = 0.8\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H = 20 \times 9.8 \times 0.6 + 20 \times 1.2^2 \times 1.75 = 168\) N | M1 A1 | Radial component of equation; correct substitution |
| Answer | Marks | Guidance |
|---|---|---|
| \(P = 20 \times 9.8 \times 0.8 - 20 \times 1.75 \times 3.5 = 34.3\) | M1 A1 | [6] Tangential component; correct substitution |
$M\bar{x} = \sum m_i x_i$
$20\bar{x} = 15 \times 1.4 + 5 \times 2.8$
$\bar{x} = 1.75$ | M1 A1 | [2] Centre of mass calculation
$I = \frac{4}{3}ml^2 + 5 \cdot 2l^2 = \frac{4}{3} \times 15 \times 1.4^2 + 20 \times 1.4^2 = 78.4$ kg m² | M1 A1 | [2] Moment of inertia about pivot
$C = I\alpha$
$196 \times 1.75\sin\theta = 78.4 \times 3.5$
$\sin\theta = 0.8$ (show) | M1 A1 | [3] Torque equation; verification that $\sin\theta = 0.8$
**N2 radially BA:**
$H - mg\cos\theta = m\omega^2 r$
$H = 20 \times 9.8 \times 0.6 + 20 \times 1.2^2 \times 1.75 = 168$ N | M1 A1 | Radial component of equation; correct substitution
**N2 normal:**
$mg\sin\theta - P = mr\omega$
$P = 20 \times 9.8 \times 0.8 - 20 \times 1.75 \times 3.5 = 34.3$ | M1 A1 | [6] Tangential component; correct substitution
---6 A rigid body consists of a uniform rod $A B$, of mass 15 kg and length 2.8 m , with a particle of mass 5 kg attached at $B$. The body rotates without resistance in a vertical plane about a fixed horizontal axis through $A$.\\
(i) Find the distance of the centre of mass of the body from $A$.\\
(ii) Find the moment of inertia of the body about the axis.\\
\includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-3_475_682_680_719}
At one instant, $A B$ makes an acute angle $\theta$ with the downward vertical, the angular speed of the body is $1.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$ and the angular acceleration of the body is $3.5 \mathrm { rad } \mathrm { s } ^ { - 2 }$ (see diagram).\\
(iii) Show that $\sin \theta = 0.8$.\\
(iv) Find the components, parallel and perpendicular to $B A$, of the force acting on the body at $A$.\\[0pt]
[Question 7 is printed overleaf.]\\
\includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-4_949_1112_281_550}
A small bead $B$, of mass $m$, slides on a smooth circular hoop of radius $a$ and centre $O$ which is fixed in a vertical plane. A light elastic string has natural length $2 a$ and modulus of elasticity $m g$; one end is attached to $B$, and the other end is attached to a light ring $R$ which slides along a smooth horizontal wire. The wire is in the same vertical plane as the hoop, and at a distance $2 a$ above $O$. The elastic string $B R$ is always vertical, and $O B$ makes an angle $\theta$ with the downward vertical (see diagram).\\
(i) Show that $\theta = 0$ is a position of stable equilibrium.\\
(ii) Find the approximate period of small oscillations about the equilibrium position $\theta = 0$.
\hfill \mbox{\textit{OCR M4 2004 Q6 [13]}}