Hard +2.3 This M5 question requires setting up and evaluating a non-trivial integral for a 3D solid of revolution, applying the perpendicular axis theorem correctly to discs, then using symmetry arguments for part (b). It demands strong integration skills, spatial reasoning about 3D geometry, and careful algebraic manipulation—significantly harder than standard A-level calculus questions but within reach of well-prepared Further Maths students.
7. (a) Find, using integration, the moment of inertia of a uniform solid hemisphere, of mass \(m\) and radius \(a\), about a diameter of its plane face. [0pt]
[You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about a diameter is \(\frac { 1 } { 4 } m r ^ { 2 }\).]
(b) Hence find the moment of inertia of a uniform solid sphere, of mass \(M\) and radius \(a\), about a diameter.
The images you've shared show only blank answer pages (continuation sheets for Question 7) from what appears to be a Physics and Maths Tutor exam paper. These pages contain no mark scheme content — they are simply lined writing spaces for student responses.
To extract mark scheme content, I would need images of the actual mark scheme document rather than the blank answer booklet pages.
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## Question 7:
### Part (a): Moment of inertia of a uniform solid hemisphere about a diameter of its plane face (10 marks)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Consider a disc at height $x$ from the plane face, with radius $r = \sqrt{a^2 - x^2}$ | M1 | Setting up a typical element as a disc |
| Mass of disc element: $dm = \rho \pi r^2 \, dx = \rho \pi (a^2 - x^2) \, dx$ | B1 | Correct mass element |
| $\rho = \frac{m}{\frac{2}{3}\pi a^3} = \frac{3m}{2\pi a^3}$ | B1 | Correct density |
| Moment of inertia of disc element about diameter: $dI = \frac{1}{4}r^2 \, dm$ | M1 | Using given result for disc |
| But axis passes through centre of plane face, so parallel axis theorem needed: $dI = \frac{1}{4}r^2 \, dm + x^2 \, dm$ | M1 | Applying parallel axis theorem correctly |
| $I = \int_0^a \left(\frac{1}{4}(a^2-x^2) + x^2\right)\rho\pi(a^2-x^2)\,dx$ | A1 | Correct integral set up |
| $= \rho\pi \int_0^a \left(\frac{1}{4}(a^2-x^2)^2 + x^2(a^2-x^2)\right)dx$ | M1 | Expanding correctly |
| $= \rho\pi \int_0^a \left(\frac{1}{4}a^4 - \frac{1}{2}a^2x^2 + \frac{1}{4}x^4 + a^2x^2 - x^4\right)dx$ | A1 | Correct expansion |
| $= \rho\pi \int_0^a \left(\frac{1}{4}a^4 + \frac{1}{2}a^2x^2 - \frac{3}{4}x^4\right)dx$ | A1 | Simplified integrand |
| $= \rho\pi \left[\frac{1}{4}a^4x + \frac{1}{6}a^2x^3 - \frac{3}{20}x^5\right]_0^a = \rho\pi a^5\left(\frac{1}{4}+\frac{1}{6}-\frac{3}{20}\right) = \rho\pi a^5 \cdot \frac{4}{15}$ | A1 | Correct integration and evaluation |
| $I = \frac{3m}{2\pi a^3} \cdot \pi a^5 \cdot \frac{4}{15} = \frac{2}{5}ma^2$ | A1 | $I = \frac{2}{5}ma^2$ |
### Part (b): Moment of inertia of a uniform solid sphere about a diameter (2 marks)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sphere = two hemispheres, each of mass $\frac{M}{2}$ and radius $a$ | M1 | Using result from (a) with mass $\frac{M}{2}$ and adding |
| $I = 2 \times \frac{2}{5} \cdot \frac{M}{2} \cdot a^2 = \frac{2}{5}Ma^2$ | A1 | $I = \frac{2}{5}Ma^2$ |
The images you've shared show only blank answer pages (continuation sheets for Question 7) from what appears to be a Physics and Maths Tutor exam paper. These pages contain no mark scheme content — they are simply lined writing spaces for student responses.
To extract mark scheme content, I would need images of the actual **mark scheme document** rather than the blank answer booklet pages.
Could you share the mark scheme pages directly?
7. (a) Find, using integration, the moment of inertia of a uniform solid hemisphere, of mass $m$ and radius $a$, about a diameter of its plane face.\\[0pt]
[You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass $m$ and radius $r$, about a diameter is $\frac { 1 } { 4 } m r ^ { 2 }$.]\\
(b) Hence find the moment of inertia of a uniform solid sphere, of mass $M$ and radius $a$, about a diameter.
\hfill \mbox{\textit{Edexcel M5 2015 Q7 [12]}}