91Ó°ÊÓ

Two thin rods (each of mass$\mathbf{\text{0.20 k²µ}}$) are joined together to form a rigid body as shown in Fig.$\mathbf{\text{10-60}}$ . One of the rods has length$\mathbf{L}\mathbf{1}\mathbf{\text{=0.40″¾}}$ , and the other has length$\mathbf{L}\mathbf{\text{2=0.50m}}$. What is the rotational inertia of this rigid body about (a) an axis that is perpendicular to the plane of the paper and passes through the center of the shorter rod and (b) an axis that is perpendicular to the plane of the paper and passes through the center of the longer rod?

In Fig$\mathbf{\text{10-61}}$., four pulleys are connected by two belts. Pulley A (radius$\mathbf{\text{15 c³¾}}$) is the drive pulley, and it rotates at.$\mathbf{\text{10 r²¹»å/²õ}}$Pulley B (radius$\mathbf{\text{10 c³¾}}$) is connected by belt $\mathbf{1}$to pulley A. Pulley B’ (radius$\mathbf{\text{5 c³¾}}$) is concentric with pulley B and is rigidly attached to it. Pulley C (radius$\mathbf{\text{25 c³¾}}$) is connected by belt $\mathbf{2}$to pulley B’. Calculate (a) the linear speed of a point on belt $\mathbf{1}$, (b) the angularspeed of pulley B, (c) the angular speed of pulley B’, (d) the linear speed of a point on belt$\mathbf{2}$, and (e) the angular speed of pulley C. (Hint: If the belt between two pulleys does not slip, the linear speeds at the rims of the two pulleys must be equal.)

In Fig$\mathbf{\text{10-63}}$., a thin uniform rod (mass, $\mathbf{\text{3.0 k²µ}}$length$\mathbf{\text{4.0″¾}}$) rotates freely about a horizontal axis A that is perpendicular to the rod and passes through a point at distance$\mathbf{d}\mathbf{=}\mathbf{\text{1.0″¾}}$from the end of the rod. The kinetic energy of the rod as it passes through the vertical position is$\mathbf{\text{20J}}$. (a) What is the rotational inertia of the rod about axis A? (b) What is the (linear) speed of the end B of the rod as the rod passes through the vertical position? (c) At what angle u will the rod momentarily stop in its upward swing?

Four particles, each of mass, $\text{0.20 k²µ}$, are placed at the vertices of a square with sides of length$\mathbf{\text{0.50m}}$. The particles are connected by rods of negligible mass. This rigid body can rotate in a vertical plane about a horizontal axis A that passes through one of the particles. The body is released from rest with rod AB horizontal (Fig$\mathbf{\text{10-64}}$.). (a) What is the rotational inertia of the body about axis A? (b) What is the angular speed of the body about axis A when rod AB swings through the vertical position?

Cheetahs running at top speed have been reported at an astounding$\text{114 k³¾/³ó}$(about$\text{71″¾¾±/³ó}$) by observers driving alongside the animals. Imagine trying to measure a cheetah’s speed by keeping your vehicle abreast of the animal while also glancing at your speedometer, which is registering$\text{114 k³¾/³ó}$. You keep the vehicle a constantfrom the cheetah, but the noise of the vehicle causes the cheetah to continuously veer away from you along a circular path of radius$\text{92″¾}$. Thus, you travel along a circular path of radius.$\text{100″¾}$ (a) What is the angular speed of you and the cheetah around the circular paths? (b) What is the linear speed of the cheetah along its path? (If you did not account for the circular motion, you would conclude erroneously that the cheetah’s speed is$\text{114 k³¾/³ó}$, and that type of error was apparently made in the published reports.)

Starting from rest, a disk rotates about its central axis with constant angular acceleration. In $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{s}$, it rotates $\mathbf{25}\mathbf{}\mathbf{rad}$. During that time, what are the magnitudes of

(a) the angular acceleration and

(b) the average angular velocity?

(c) What is the instantaneous angular velocity of the disk at the end of the $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{s}$?

(d) With the angular acceleration unchanged, through what additional angle will the disk turn during the next $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{s}$?

Figure 10 - 27shows three flat disks (of the same radius) that can rotate about their centers like merry-go-rounds. Each disk consists of the same two materials, one denser than the other (density is mass per unit volume). In disks 1and 3, the denser material forms the outer half of the disk area. In disk 2, it forms the inner half of the disk area. Forces with identical magnitudes are applied tangentially to the disk, either at the outer edge or at the interface of the two materials, as shown. Rank the disks according to (a) the torque about the disk center, (b) the rotational inertia about the disk center, and (c) the angular acceleration of the disk, greatest first.

A disk, initially rotating at $\mathbf{120}\mathbf{}\mathbf{\text{ }}\frac{\mathbf{rad}}{\mathbf{s}}$, is slowed down with a constant angular acceleration of magnitude $\mathbf{120}\mathbf{}\mathbf{\text{ }}\frac{\mathbf{rad}}{{\mathbf{s}}^{\mathbf{2}}}$.

(a) How much time does the disk take to stop?

(b) Through what angle does the disk rotate during that time?

The angular speed of an automobile engine is increased at a constant rate from $\mathbf{1200}\mathbf{}\raisebox{1ex}{${\displaystyle \mathbf{\text{rev}}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{\text{min}}$}\right.$ to$\mathbf{3000}\raisebox{1ex}{${\displaystyle \text{rev}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{\text{min}}$}\right.$ in $\mathbf{12}\mathbf{\text{s}}$. (a) What is its angular acceleration in revolutions per minute-squared? (b) How many revolutions does the engine make during this$\mathbf{12}\mathbf{\text{s}}$ interval?

A flywheel turns through $\mathbf{40}\mathbf{}\mathbf{\text{ }}\mathbf{rev}\mathbf{}$ as it slows from an angular speed of $\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{\text{ }}\mathbf{rad}\mathbf{/}\mathbf{s}\mathbf{}$to a stop.

(a) Assuming a constant angular acceleration, find the time for it to come to rest.

(b) What is its angular acceleration?

(c) How much time is required for it to complete the first $\mathbf{20}\mathbf{}$ of the $\mathbf{40}$ revolutions?