In: Physics

# Find the moment of inertia Ix of particle a with respect to the x axis

Find the moment of inertia Ix of particle a with respect to the x-axis (that is, if the x-axis is the axis of rotation), the moment of inertia Iy of particle a with respect to the y axis, and the moment of inertia Iz of particle a with respect to the z-axis (the axis that passes through the origin perpendicular to both the x and y axes).
Express your answers in terms of m and r separated by commas.

## Solutions

##### Expert Solution

Concepts and reason

The concept required to solve the problem is the moment of inertia. Use the mass and the distance from $$\mathrm{x}$$ axis to calculate the moment of inertia with respect to $$\mathrm{x}$$ axis. Use the mass and the distance from y axis to calculate the moment of inertia with respect to y axis. Calculate the sum of the moment of inertias with respect to $$\mathrm{x}$$ and $$\mathrm{y}$$ axis to calculate the moment of inertia with respect to $$\mathrm{z}$$ axis.

Fundamentals

The moment of inertia of the object is, $$I=M R^{2}$$

Here, $$M$$ is the mass of the object and $$R$$ is the perpendicular distance.

(C. 1) The moment of inertia of particle is, $$I=M R^{2}$$

The mass of particle a is $$m$$ and the perpendicular distance of particle a from $$\mathrm{x}$$ axis is $$r$$. Substitute $$m$$ for $$M$$ and $$r$$ for $$R$$. The moment of inertia of particle a with respect to $$\mathrm{x}$$ axis is, $$I_{x}=m r^{2}$$

Part C.1 The moment of inertia of particle a with respect to $$\mathrm{x}$$ axis, $$I_{x},$$ is $$\mathrm{mr}^{2}$$

The moment of inertia of the particle a with respect to x axis is the product of mass of particle a and the square of perpendicular distance from $$\mathrm{x}$$ axis.

(C. 2) The moment of inertia of particle is, $$I=M R^{2}$$

The mass of particle a is $$m$$ and the perpendicular distance of particle a from y axis is $$3 r$$. Substitute $$m$$ for $$M$$ and $$3 r$$ for $$R$$. The moment of inertia of particle a with respect to y axis is,

$$I_{y}=m(3 r)^{2}$$

$$=9 m r^{2}$$

Part C.2 The moment of inertia of particle a with respect to $$\mathrm{y}$$ axis, $$I_{y},$$ is $$9 \mathrm{mr}^{2}$$

The moment of inertia of the particle a with respect to $$\mathrm{y}$$ axis is directly proportional to mass of particle $$a$$ and the square of perpendicular distance from y axis.

(C.3) The moment of inertia of particle a with respect to $$z$$ axis is, $$I_{z}=I_{x}+I_{y}$$

Substitute $$m r^{2}$$ for $$I_{x}$$ and $$9 m r^{2}$$ for $$I_{y}$$. The moment of inertia of the particle with respect to $$z$$ axis is,

$$I_{z}=m r^{2}+9 m r^{2}$$

$$=10 m r^{2}$$

Part C.3 The moment of inertia of particle a with respect to $$z$$ axis, $$I_{z},$$ is $$10 m r^{2}$$

The moment of inertia of the particle a with respect to $$z$$ axis is the sum of the moment of inertia of particle a with respect to x axis and the sum of the moment of inertia of particle a with respect to y axis.

Part C.1 The moment of inertia of particle a with respect to $$\mathrm{x}$$ axis, $$I_{x},$$ is $$\mathrm{mr}^{2}$$

Part $$\mathrm{C} .2$$ The moment of inertia of particle a with respect to $$\mathrm{y}$$ axis, $$I_{y},$$ is $$9 \mathrm{mr}^{2}$$

Part $$C .3$$ The moment of inertia of particle a with respect to $$z$$ axis, $$I_{z},$$ is $$10 m r^{2}$$