Question

A rigid body can be hinged about any point on the x-axis. When it is hinged such that the hinge is at x, the moment of inertia is given by $I = 2x^2 - 12x + 27$. The x-coordinate of centre of mass is

• A. x = 2
• B. x = 0
• C. a parabola
• D. x = 1
• E. a straight line
• F. x=3

Answer 1

Answer: (F)

x=3

Answer 2

The moment of inertia of a rigid body about an axis is given by the parallel axis theorem:

$I = I_{CM} + Md^2$

where $I_{CM}$ is the moment of inertia about a parallel axis passing through the center of mass, $M$ is the mass of the body, and $d$ is the perpendicular distance between the two parallel axes.

In this problem, the hinge is at x, and let the x-coordinate of the center of mass be $x_{CM}$. The distance $d$ between the hinge and the center of mass is $|x - x_{CM}|$. So, $d^2 = (x - x_{CM})^2$.

Substituting this into the parallel axis theorem: $I = I_{CM} + M(x - x_{CM})^2$ Expand the term $(x - x_{CM})^2$: $I = I_{CM} + M(x^2 - 2x \cdot x_{CM} + x_{CM}^2)$ Rearrange the terms to match the form of a quadratic equation in x: $I = M x^2 - (2M x_{CM}) x + (I_{CM} + M x_{CM}^2)$

We are given the moment of inertia as: $I = 2x^2 - 12x + 27$

Now, we compare the coefficients of the given equation with the expanded form from the parallel axis theorem:

1. Comparing the coefficient of $x^2$: From the general form: Coefficient of $x^2$ is $M$. From the given equation: Coefficient of $x^2$ is $2$. Therefore, $M = 2$.

2. Comparing the coefficient of $x$: From the general form: Coefficient of $x$ is $-2M x_{CM}$. From the given equation: Coefficient of $x$ is $-12$. Therefore, $-2M x_{CM} = -12$. Substitute the value of $M=2$: $-2(2) x_{CM} = -12$ $-4 x_{CM} = -12$ $x_{CM} = \frac{-12}{-4}$ $x_{CM} = 3$

3. Comparing the constant term: From the general form: Constant term is $(I_{CM} + M x_{CM}^2)$. From the given equation: Constant term is $27$. Therefore, $I_{CM} + M x_{CM}^2 = 27$. Substitute the values of $M=2$ and $x_{CM}=3$: $I_{CM} + 2(3)^2 = 27$ $I_{CM} + 2(9) = 27$ $I_{CM} + 18 = 27$ $I_{CM} = 27 - 18$ $I_{CM} = 9$

From the comparison, the x-coordinate of the center of mass is $x_{CM} = 3$.