13. Rotational Inertia & Energy

Which of the following factors does the moment of inertia $I$ of an object depend on?

Given a thin rectangular plate of width $b$ and height $h$, with its base along the $x$-axis and one side along the $y$-axis, determine the moment of inertia of the plate about the $y$-axis using integration.

A thin rectangular plate of width $b$ and height $h$ has its left edge along the y-axis and extends from $x=0$ to $x=b$. If the plate has a uniform surface mass density $\rho$, what is the moment of inertia of the plate about the y-axis?

A uniform DVD has mass $m$ and radius $R$. What is its moment of inertia for rotation about an axis perpendicular to its plane and passing through its center?

A thin rectangular plate of width $b$ and height $h$ has its left edge along the y-axis and extends from $x=0$ to $x=b$. If the plate has a uniform mass per unit area $\rho$, what is the moment of inertia of the plate about the y-axis?

Which of the following integrals correctly represents the moment of inertia of a planar area about the x-axis, where $y$ is the distance from the x-axis and $dA$ is an infinitesimal area element?

A thin composite area consists of a rectangle of width $b$ and height $h$ attached to the top of a semicircle of radius $r$, with both shapes centered on the y-axis. Which of the following expressions gives the total moment of inertia of the composite area about the y-axis?

A flywheel has a moment of inertia of $1.10kg\cdot m^2$ about its rotation axis. If a constant torque of $5.50N\cdot m$ is applied, what is the angular acceleration of the flywheel?

Which of the following integrals correctly represents the moment of inertia of a planar area about the $x$-axis?

Consider four uniform objects of equal mass and radius: (A) a solid sphere, (B) a solid cylinder, (C) a thin spherical shell, and (D) a thin ring, all rotating about their central axis. Which moment of inertia is the smallest?

(III) Determine the moment of inertia of a uniform solid cone whose base has radius R₀, height L and mass M. The axis of rotation (𝒵) is the symmetry axis perpendicular to the base, Fig. 10–66. [Hint: Think of the cone as a stack of infinitesimally thin disks of mass dm, radius R, and thickness dz.]

A 25 kg solid door is 220 cm tall, 91 cm wide. What is the door's moment of inertia for rotation on its hinges?

A 12-cm-diameter DVD has a mass of 21 g. What is the DVD's moment of inertia for rotation about a perpendicular axis through the edge of the disk?

A rod of length L and mass M has a nonuniform mass distribution. The linear mass density (mass per length) is λ = cx², where x is measured from the center of the rod and c is a constant. Find an expression for c in terms of L and M.

Calculate by direct integration the moment of inertia for a thin rod of mass M and length L about an axis located distance d from one end. Confirm that your answer agrees with Table 12.2 when d=0 and when d = L/2.