| Exam Board | OCR |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2012 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Rotation about fixed axis: angular acceleration and velocity |
| Difficulty | Challenging +1.3 This is a multi-part M4 rotation question requiring moment of inertia (I = ml²/3 for rod about end), torque equations, and energy methods. While it involves several steps and integration of rotational dynamics concepts, each part follows standard M4 techniques without requiring novel insight. The verification in part (iv) adds mild complexity but is routine for this level. |
| Spec | 3.04a Calculate moments: about a point3.04b Equilibrium: zero resultant moment and force6.02d Mechanical energy: KE and PE concepts6.02e Calculate KE and PE: using formulae6.02i Conservation of energy: mechanical energy principle6.04e Rigid body equilibrium: coplanar forces6.05a Angular velocity: definitions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(I = \frac{4}{3}(4)(1.2^2)\) \((= 7.68)\) | B1 | |
| \(4 \times 9.8 \times 1.2 - 20 = I\alpha\) | M1 | Equation of angular motion (three terms) |
| Angular acceleration is \(3.52\) rad s\(^{-2}\) (3 sf) | A1, A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4 \times 9.8 \times 1.2\cos\theta - 20 = 0\) | M1 | Moment of weight in terms of \(\theta\) |
| Angle is \(1.13\) rad \((64.8°)\) (3 sf) | A1, A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| WD is \(20 \times 1.132\) | M1, A1 | Using \(C\theta\); FT |
| \(-20\theta = \frac{1}{2}I\omega^2 - 4\times9.8\times1.2\sin\theta\) | M1 | Equation involving KE \(\frac{1}{2}I\omega^2\) and PE |
| \(-20\times1.132 = \frac{1}{2}(7.68)\omega^2 - 4\times9.8\times1.2\sin(1.132)\) | A1 | FT if values of WD and/or \(\theta\) used |
| Maximum angular speed is \(2.28\) rad s\(^{-1}\) (3 sf) | A1 [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(20\theta = 4\times9.8\times1.2\sin\theta\) | M1 | Equation involving WD and PE |
| Let \(f(\theta) = 20\theta - 47.04\sin\theta\) | A1 | |
| \(f(2.0) = -2.77\), \(f(2.1) = 1.39\) | ||
| Sign change shows that \(2.0 < \theta < 2.1\) | A1 [3] | AG Correctly shown |
# Question 5:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $I = \frac{4}{3}(4)(1.2^2)$ $(= 7.68)$ | B1 | |
| $4 \times 9.8 \times 1.2 - 20 = I\alpha$ | M1 | Equation of angular motion (three terms) |
| Angular acceleration is $3.52$ rad s$^{-2}$ (3 sf) | A1, A1 [4] | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4 \times 9.8 \times 1.2\cos\theta - 20 = 0$ | M1 | Moment of weight in terms of $\theta$ |
| Angle is $1.13$ rad $(64.8°)$ (3 sf) | A1, A1 [3] | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| WD is $20 \times 1.132$ | M1, A1 | Using $C\theta$; FT |
| $-20\theta = \frac{1}{2}I\omega^2 - 4\times9.8\times1.2\sin\theta$ | M1 | Equation involving KE $\frac{1}{2}I\omega^2$ and PE |
| $-20\times1.132 = \frac{1}{2}(7.68)\omega^2 - 4\times9.8\times1.2\sin(1.132)$ | A1 | FT if values of WD and/or $\theta$ used |
| Maximum angular speed is $2.28$ rad s$^{-1}$ (3 sf) | A1 [5] | |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $20\theta = 4\times9.8\times1.2\sin\theta$ | M1 | Equation involving WD and PE |
| Let $f(\theta) = 20\theta - 47.04\sin\theta$ | A1 | |
| $f(2.0) = -2.77$, $f(2.1) = 1.39$ | | |
| Sign change shows that $2.0 < \theta < 2.1$ | A1 [3] | AG Correctly shown |
---5 A uniform rod of mass 4 kg and length 2.4 m can rotate in a vertical plane about a fixed horizontal axis through one end of the rod. The rod is released from rest in a horizontal position and a frictional couple of constant moment 20 Nm opposes the motion.\\
(i) Find the angular acceleration of the rod immediately after it is released.\\
(ii) Find the angle that the rod makes with the horizontal when its angular acceleration is zero.\\
(iii) Find the maximum angular speed of the rod.\\
(iv) The rod first comes to instantaneous rest after rotating through an angle $\theta$ radians from its initial position. Find an equation for $\theta$, and verify that $2.0 < \theta < 2.1$.
\hfill \mbox{\textit{OCR M4 2012 Q5 [15]}}