Question

The moment of inertia of a uniform semicircular wire of mass \[m\]and radius\[r\], about an axis passing through its centre of mass and perpendicular to its plane is \[m{r^2}\left( {1 - \dfrac{{{k^2}}}{{{\pi ^2}}}} \right)\]. Find the value of k.
A.\[\dfrac{{m{r^2}}}{2}\]
B. \[m{r^2}\]
C. \[m{r^2}\left( {1 - \dfrac{4}{{{\pi ^2}}}} \right)\]
D. \[m{r^2}\left( {1 + \dfrac{4}{{{\pi ^2}}}} \right)\]

Answer 1

Hint: The moment of inertia of the circular wire about the axis passing through the centre of mass and perpendicular to its plane is known. Using the symmetry of the geometry, the complete circular wire shape can be obtained by adding two similar semicircular wires.

Formula used:
The moment of inertia of a circular wire about the centre of mass and perpendicular to its plane is
\[I = M{R^2}\]
where M is the mass of the circular wire and R is the radius of the circular wire.
\[{I_0} = {I_{cm}} + m{d^2}\]
Here, \[{I_{cm}}\] is the moment of inertia of the centre of mass and d is the distance between the axis of rotation and the centre of mass of an object of mass m.

Complete step by step solution:
The mass of the semicircular wire is m and the radius is given as r. The distance of the centre of mass of semicircular from the centre is,
\[d = \dfrac{{2r}}{\pi }\]
Using symmetry, the moment of inertia of circular wire of mass 2m and radius r is,
\[I = \left( {2m} \right){r^2}\]

It is made of two semicircular wires, if the moment of inertia of semicircular about the centre is \[{I_0}\].
\[2{I_0} = I\]
\[\Rightarrow {I_0} = \dfrac{{2m{r^2}}}{2} = m{r^2}\]
Using the parallel axis theorem,
\[{I_0} = {I_{cm}} + m{d^2}\]
\[\Rightarrow {I_{cm}} = {I_0} - m{d^2}\]

Putting the expressions, we get
\[m{r^2}\left( {1 - \dfrac{{{k^2}}}{{{\pi ^2}}}} \right) = m{r^2} - m{\left( {\dfrac{{2r}}{\pi }} \right)^2}\]
\[\Rightarrow m{r^2}\left( {1 - \dfrac{{{k^2}}}{{{\pi ^2}}}} \right) = m{r^2}\left( {1 - \dfrac{4}{{{\pi ^2}}}} \right)\]
On solving the equation, we get
\[{k^2} = 4\]
Hence, the moment of inertia of the semicircular wire about the centre of mass and perpendicular to its plane is \[m{r^2}\left( {1 - \dfrac{4}{{{\pi ^2}}}} \right)\].

Therefore, the correct option is C.

Note: In the question the asked variable was k and in options the complete expression of moment of inertia is given by substituting the numerical value of k. In physics, a moment of inertia is a quantitative measure of a body's rotational inertia, or the resistance that the body shows to having its speed of rotation along an axis altered by the application of a torque (turning force). The axis might be internal or exterior, and it can be fixed or not.