| Exam Board | OCR |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Angular kinematics – constant angular acceleration/deceleration |
| Difficulty | Moderate -0.3 This is a straightforward application of constant angular acceleration equations (rotational analogues of SUVAT). Part (i) requires substituting into θ = ω₀t + ½αt², part (ii) needs finding when the wheel stops then calculating the angle for a time interval, and part (iii) uses ω² = ω₀² + 2αθ. While it's a multi-part question requiring careful arithmetic and sign conventions, it involves direct formula application with no conceptual challenges or novel problem-solving, making it slightly easier than average. |
| Spec | 6.05a Angular velocity: definitions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Using \(\theta = \omega_1 t + \frac{1}{2}\alpha t^2\), \(1020 = 80 \times 15 + \frac{1}{2}\alpha \times 15^2\) | M1 | |
| \(\alpha = -1.6\) | A1 | |
| Angular deceleration is \(1.6 \text{ rad s}^{-2}\) | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Using \(\theta = \omega_2 t - \frac{1}{2}\alpha t^2\), \(\theta = 0 - \frac{1}{2} \times (-1.6) \times 5^2\) | M1 | |
| Angle is 20 rad | A1 ft [2] | ft is \(12.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Using \(\omega_2^2 = \omega_1^2 + 2\alpha\theta\) | M1 | |
| \(0 = 80^2 + 2 \times (-1.6)\theta\), \(\theta = 2000\) | A1 ft | |
| Number of revolutions is 318 (3 sf) | A1 [3] | Accept \(\frac{1000}{\pi}\) |
# Question 1:
## Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Using $\theta = \omega_1 t + \frac{1}{2}\alpha t^2$, $1020 = 80 \times 15 + \frac{1}{2}\alpha \times 15^2$ | M1 | |
| $\alpha = -1.6$ | A1 | |
| Angular deceleration is $1.6 \text{ rad s}^{-2}$ | [2] | |
## Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Using $\theta = \omega_2 t - \frac{1}{2}\alpha t^2$, $\theta = 0 - \frac{1}{2} \times (-1.6) \times 5^2$ | M1 | |
| Angle is 20 rad | A1 ft [2] | ft is $12.5|\alpha|$ |
## Part (iii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Using $\omega_2^2 = \omega_1^2 + 2\alpha\theta$ | M1 | |
| $0 = 80^2 + 2 \times (-1.6)\theta$, $\theta = 2000$ | A1 ft | |
| Number of revolutions is 318 (3 sf) | A1 [3] | Accept $\frac{1000}{\pi}$ |
---1 A wheel is rotating and is slowing down with constant angular deceleration. The initial angular speed is $80 \mathrm { rad } \mathrm { s } ^ { - 1 }$, and after 15 s the wheel has turned through 1020 radians.\\
(i) Find the angular deceleration of the wheel.\\
(ii) Find the angle through which the wheel turns in the last 5 s before it comes to rest.\\
(iii) Find the total number of revolutions made by the wheel from the start until it comes to rest.
\hfill \mbox{\textit{OCR M4 2010 Q1 [7]}}