Prove MI by integration

A question is this type if and only if it asks the student to derive a moment of inertia formula using integration (e.g. for discs, spheres, cones, rods, laminae, solids of revolution), where the primary task is the integration proof itself.

32 questions · Challenging +1.5

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Edexcel M5 2014 June Q4
8 marks Challenging +1.8
4. A uniform solid sphere has mass \(M\) and radius \(a\). Prove, using integration, that the moment of inertia of the sphere about a diameter is \(\frac { 2 M a ^ { 2 } } { 5 }\)
[0pt] [You may assume without proof that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about an axis through its centre and perpendicular to its plane is \(\frac { 1 } { 2 } m r ^ { 2 }\).]
Edexcel M5 2015 June Q7
12 marks Hard +2.3
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.
Edexcel M5 2016 June Q4
10 marks Challenging +1.8
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)
Edexcel M5 2018 June Q4
6 marks Challenging +1.2
4. A uniform lamina of mass \(M \mathrm {~kg}\) is modelled as the region which is bounded by the curve with equation \(y = x ^ { 2 }\), the positive \(x\)-axis and the line with equation \(x = 2\). The unit of length on both axes is the metre. Find the moment of inertia of the lamina about the \(x\)-axis.
(6)
Edexcel M5 Specimen Q1
5 marks Standard +0.3
  1. A bead of mass 0.125 kg is threaded on a smooth straight horizontal wire. The bead moves from rest at the point \(A\) with position vector ( \(2 \mathbf { i } + \mathbf { j } - \mathbf { k }\) ) m relative to a fixed origin \(O\) to a point \(B\) with position vector ( \(3 \mathbf { i } - 4 \mathbf { j } - \mathbf { k }\) ) m relative to \(O\) under the action of a force \(\mathbf { F } = ( 14 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\) N. Find
    1. the work done by \(\mathbf { F }\) as the bead moves from \(A\) to \(B\),
    2. the speed of the bead at \(B\).
    3. (a) Prove, using integration, that the moment of inertia of a uniform rod, of mass \(m\) and length \(2 a\), about an axis perpendicular to the rod through its centre is \(\frac { 1 } { 3 } m a ^ { 2 }\).
      (3)
    A uniform wire of mass \(4 m\) and length \(8 a\) is bent into the shape of a square.
  2. Find the moment of inertia of the square about the axis through the centre of the square perpendicular to its plane.
    (4)
Edexcel M5 Q4
15 marks Challenging +1.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{19c5a621-c175-4d58-9002-4bcdefd02b71-08_515_417_210_758} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform lamina of mass \(M\) is in the shape of a right-angled triangle \(O A B\). The angle \(O A B\) is \(90 ^ { \circ } , O A = a\) and \(A B = 2 a\), as shown in Figure 1.
  1. Prove, using integration, that the moment of inertia of the lamina \(O A B\) about the edge \(O A\) is \(\frac { 2 } { 3 } M a ^ { 2 }\).
    (You may assume without proof that the moment of inertia of a uniform rod of mass \(m\) and length \(2 l\) about an axis through one end and perpendicular to the rod is \(\frac { 4 } { 3 } m l ^ { 2 }\).) The lamina \(O A B\) is free to rotate about a fixed smooth horizontal axis along the edge \(O A\) and hangs at rest with \(B\) vertically below \(A\). The lamina is then given a horizontal impulse of magnitude \(J\). The impulse is applied to the lamina at the point \(B\), in a direction which is perpendicular to the plane of the lamina. Given that the lamina first comes to instantaneous rest after rotating through an angle of \(120 ^ { \circ }\),
  2. find an expression for \(J\), in terms of \(M , a\) and \(g\).
Edexcel M5 Q7
16 marks Challenging +1.8
7. Prove, using integration, that the moment of inertia of a uniform solid right circular cone, of mass \(M\) and base radius \(a\), about its axis is \(\frac { 3 } { 10 } M a ^ { 2 }\).
[0pt] [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about an axis through its centre and perpendicular to its plane is \(\frac { 1 } { 2 } m r ^ { 2 }\).]