13. Rotational Inertia & Energy

A uniform rectangular door of mass $M$ and width $w$ rotates about its hinges along one vertical edge. What is the moment of inertia of the door about the axis of rotation?

For a single particle of mass $m$ located at a distance $r$ from the axis of rotation, what is the moment of inertia $I$ of the particle?

For a particle labeled $a$ with mass $ma$ located at a distance $ra$ from the axis of rotation, what is its moment of inertia $Ia$?

Which of the following expressions gives the moment of inertia of a thin rectangular plate of width $b$ and height $h$ about the x-axis (along its base)?

For a thin rectangular plate of width $b$ and height $h$, determine the moment of inertia of the shaded area about the x-axis passing along its base.

Consider four point masses, each of mass $m$, placed at the corners of a square of side length $a$. What is the moment of inertia of the system about an axis that passes through the masses at corners $b$ and $d$ (i.e., along the diagonal of the square)?

Calculate the moment of inertia of each of the following uniform objects about the axes indicated. Consult Table 9.2 as needed. A thin 2.50-kg rod of length 75.0 cm, about an axis perpendicular to it and passing through (i) one end and (ii) its center, and (iii) about an axis parallel to the rod and passing through it.

Which of the following factors does the $\mathrm{moment} \mathrm{of} \mathrm{inertia}$ of an object depend on?

A system is made of two small masses (M_LEFT = 3 kg, M_RIGHT = 4 kg) attached to the ends of a 5 kg, 2-m long thin rod, as shown. Calculate the moment of inertia of the system if it spins about a perpendicular axis through the mass on the left.

A solid disc 4 m in diameter has a moment of inertia equal to 30 kg m² about an axis through the disc, perpendicular to its face. The disc spins at a constant 120 RPM. Calculate the mass of the disc.

A slender rod with length L has a mass per unit length that varies with distance from the left end, where x = 0, according to dm/dx = γx, where γ has units of kg/m². Use Eq. (9.20) to calculate the moment of inertia of the rod for an axis at the left end, perpendicular to the rod. Use the expression you derived in part (a) to express I in terms of M and L. How does your result compare to that for a uniform rod? Explain.

You are a project manager for a manufacturing company. One of the machine parts on the assembly line is a thin, uniform rod that is 60.0 cm long and has mass 0.400 kg. What is the moment of inertia of this rod for an axis at its center, perpendicular to the rod?

If we multiply all the design dimensions of an object by a scaling factor f, its volume and mass will be multiplied by f³. By what factor will its moment of inertia be multiplied?