A flywheel rotates such that it sweeps out an angle at the rate of $\theta = \omega t = 45.0 t$ radians. The wheel rotates counterclockwise when viewed in the plane of the page.

(A) What is the angular velocity of the flywheel? What direction is the angular velocity?

(B) How many radians does the flywheel rotate through in 30 s?

(C) What is the tangential speed of a point on the flywheel 10 cm from the axis of rotation?

Follow Example 10.2.1.

A bicycle mechanic mounts a bicycle on the repair stand and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s.

(A) Calculate the average angular acceleration in rad/s$^2$.

(B) If she now hits the brakes, causing an angular acceleration of $−87.3$ rad/s$^2$, how long does it take the wheel to stop?

Follow Example 10.2.2.

A wind turbine (figure below) in a wind farm is being shut down for maintenance. It takes $30$ s for the turbine to go from its operating angular velocity to a complete stop in which the angular velocity function is

$$ \omega(t) = \frac{(t-30)^2}{100} $$

rad/s. If the turbine is rotating counterclockwise looking into the page,

(A) what are the directions of the angular velocity and acceleration vectors?

(B) What is the average angular acceleration?

(C) What is the instantaneous angular acceleration at t = 0.0, 15.0, 30.0 s?

Follow Example 10.2.3.

A deep-sea fisherman hooks a big fish that swims away from the boat, pulling the fishing line from his fishing reel. The whole system is initially at rest, and the fishing line unwinds from the reel at a radius of $4.50$ cm from its axis of rotation. The reel is given an angular acceleration of $110$ rad/s$^2$ for $2.00$ s (figure below).

(A) What is the final angular velocity of the reel after $2$ s?

(B) How many revolutions does the reel make?

(C) Now the fisherman applies a brake to the spinning reel, achieving an angular acceleration of $−300$ rad/s$^2$. How long does it take the reel to come to a stop?

Follow Examples 10.4 and 10.5.

A centrifuge has a radius of $20$ cm and accelerates from a maximum rotation rate of 10,000 rpm to rest in $30$ seconds under a constant angular acceleration. It is rotating counterclockwise.

(A) What is the magnitude of the total acceleration of a point at the tip of the centrifuge at $t = 29.0$ s?

(B) What is the direction of the total acceleration vector?

Follow Example 10.7.

A typical small rescue helicopter has four blades: Each is 4.00 m long and has a mass of 50.0 kg (figure below). The blades can be approximated as thin rods that rotate about one end of an axis perpendicular to their length. The helicopter has a total loaded mass of 1000 kg.

(A) Calculate the rotational kinetic energy in the blades when they rotate at 300 rpm.

(B) Calculate the translational kinetic energy of the helicopter when it flies at $20.0$ m/s, and compare it with the rotational energy in the blades.

Follow Example 10.5.2.

(A) Show the relationship between torque and force using equations and diagrams.

(B) Four forces are shown in figure below at particular locations and orientations with respect to a given xy-coordinate system. Find the torque due to each force about the origin, then use your results to find the net torque about the origin.

Follow Example 10.14.

The figure shows several forces acting at different locations and angles on a flywheel. We have $F_1= 20$ N, $F_2=30$ N, $F_3= 30$ N, and $r = 0.5$ m. Find the net torque on the flywheel about an axis through the center.

Follow Example 10.15.

Consider the father pushing a playground merry-go-round in the figure below. He exerts a force of $250$ N at the edge of the $200.0$-kg merry-go-round, which has a $1.50$-m radius. Calculate the angular acceleration produced

(A) when no one is on the merry-go-round and

(B) when an $18.0$-kg child sits $1.25$ m away from the center. Consider the merry-go-round itself to be a uniform disk with negligible friction.

Follow Example 10.16.

A string wrapped around the pulley in the figure below is pulled with a constant downward force $\overrightarrow{F}$ of magnitude $50$ N. The radius $R$ and moment of inertia $I$ of the pulley are $0.10$ m and $2.5 \times 10^{−3}$ kg m$^2$, respectively. If the string does not slip, what is the angular velocity of the pulley after $1.0$ m of string has unwound? Assume the pulley starts from rest.

Follow Example 10.18.

A robot arm on a Mars rover like Curiosity shown in the figure below is 1.0 m long and has forceps at the free end to pick up rocks. The mass of the arm is $2.0$ kg and the mass of the forceps is $1.0$ kg. The robot arm and forceps move from rest to $\omega = 0.1 \pi$ rad/s in $0.1$ s. It rotates down and picks up a Mars rock that has mass $1.5$ kg. The axis of rotation is the point where the robot arm connects to the rover.

(A) What is the angular momentum of the robot arm by itself about the axis of rotation after 0.1 s when the arm has stopped accelerating?

(B) What is the angular momentum of the robot arm when it has the Mars rock in its forceps and is rotating upwards?

(C) When the arm does not have a rock in the forceps, what is the torque about the point where the arm connects to the rover when it is accelerating from rest to its final angular velocity?

Follow Example 11.3.3.

An $80.0$-kg gymnast dismounts from a high bar. He starts the dismount at full extension, then tucks to complete a number of revolutions before landing. His moment of inertia when fully extended can be approximated as a rod of length $1.8$ m and when in the tuck a rod of half that length. If his rotation rate at full extension is $1.0$ rev/s and he enters the tuck when his center of mass is at $3.0$ m height moving horizontally to the floor, how many revolutions can he execute if he comes out of the tuck at $1.8$ m height?

Follow Example 11.4.2A.