Question 4 - A-Level Maths

Edexcel M5 2016 June — Question 4 10 marks

Exam Board	Edexcel
Module	M5 (Mechanics 5)
Year	2016
Session	June
Marks	10
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Moments of inertia
Type	Prove MI by integration
Difficulty	Challenging +1.8 This is a challenging 3D moment of inertia problem requiring setup of a double integral (or perpendicular axis theorem application), careful coordinate geometry to express distances from an axis through a diameter, and algebraic manipulation. It's significantly harder than routine 2D MOI questions but follows standard M5 techniques for students who studied this now-removed topic.
Spec	6.05b Circular motion: v=r*omega and a=v^2/r

4. Find, using integration, the moment of inertia of a uniform cylindrical shell of radius \(r\), height \(h\) and mass \(M\), about a diameter of one end.
(10)

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Question 4:

Answer	Marks
Ring element at height \(z\), mass \(dm = \frac{M}{h}dz\) (shell, radius \(r\) constant)	B1 M1
MOI of ring about diameter at height \(z\): by perpendicular axes: \(I_{ring,diam} = \frac{1}{2}r^2\,dm\)	M1 A1
Parallel axis theorem to diameter of one end: distance \(= z\), so \(dI = \frac{1}{2}r^2\,dm + z^2\,dm\)	M1 A1
Integrate: \(I = \int_0^h \left(\frac{r^2}{2} + z^2\right)\frac{M}{h}\,dz = \frac{M}{h}\left[\frac{r^2 z}{2}+\frac{z^3}{3}\right]_0^h\)	M1 A1
\(= \frac{M}{h}\left(\frac{r^2 h}{2}+\frac{h^3}{3}\right) = M\left(\frac{r^2}{2}+\frac{h^2}{3}\right)\)	A1 A1

## Question 4:

**Ring element** at height $z$, mass $dm = \frac{M}{h}dz$ (shell, radius $r$ constant) | B1 M1 |

**MOI of ring about diameter at height $z$:** by perpendicular axes: $I_{ring,diam} = \frac{1}{2}r^2\,dm$ | M1 A1 |

**Parallel axis theorem** to diameter of one end: distance $= z$, so $dI = \frac{1}{2}r^2\,dm + z^2\,dm$ | M1 A1 |

**Integrate:** $I = \int_0^h \left(\frac{r^2}{2} + z^2\right)\frac{M}{h}\,dz = \frac{M}{h}\left[\frac{r^2 z}{2}+\frac{z^3}{3}\right]_0^h$ | M1 A1 |

$= \frac{M}{h}\left(\frac{r^2 h}{2}+\frac{h^3}{3}\right) = M\left(\frac{r^2}{2}+\frac{h^2}{3}\right)$ | A1 A1 |

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4. Find, using integration, the moment of inertia of a uniform cylindrical shell of radius $r$, height $h$ and mass $M$, about a diameter of one end.\\
(10)

\hfill \mbox{\textit{Edexcel M5 2016 Q4 [10]}}

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