You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and is turning at 90.0 rpm, and you press a steel axe against it with a radial force of 20.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstone. (b) How many turns will the stone make before coming to rest?

A grindstone in the shape of a solid disk with diameter 0.520 m and a mass of 50.0 kg is rotating at 850 rev/min. You press an ax against the rim with a normal force of 160 N, and the grindstone comes to rest in 7.50 s. Find the coefficient of friction between the ax and the grindstone. You can ignore friction in the bearings.

You are told that a basketball player spins the ball with an angular acceleration of 100 rad/s$^2$. (a) What is the ball's final angular velocity if the ball starts from rest and the acceleration lasts 2.00 s? (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?

A gyroscope slows from an initial rate of 32.0 rad/s at a rate of 0.700 rad/s$^2$ . (a) How long does it take to come to rest? (b) How many revolutions does it make before stopping?

A flywheel (I = 50 kg-m²) starting from rest acquires an angular velocity of 200.0 rad/s while subject to a constant torque from a motor for 5 s. (a) What is the angular acceleration of the flywheel? (b) What is the magnitude of the torque?

The disk of previous problem is at the angular position $\theta=0$ at time t = 0. Its angular velocity at $\omega_{0}=0.1$ rad/s, and then it experiences an angular acceleration given by $\alpha=2 \theta$, where $\theta$ is in radians and $\alpha$ is in radians per second squared. Determine the angular position of point A at time t = 2s.

Zorch, an archenemy of Superman, decides to slow Earth's rotation to once per 28.0 h by exerting an opposing force at and parallel to the equator. Superman is not immediately concerned, because he knows Zorch can only exert a force of $4.00\times10^7$ N (a little greater than a Saturn V rocket's thrust). How long must Zorch push with this force to accomplish his goal? (This period gives Superman time to devote to other villains.) Explicitly show how you follow the steps found in Problem-Solving Strategy for Rotational Dynamics.

Suppose you have a coffee mug with a circular cross-section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee has the same density as water.

Evaluate the line integral in Stokes’ Theorem to determine the value of the sursurface integral

$$\iint _ { S } ( \nabla \times \mathbf { F } ) \cdot \mathbf { n } d S$$

. Assume that n points in an upward direction. F=

$$\langle x + y , y + z , z + x \rangle$$

; S is the tilted disk enclosed by r(t)=

$$\langle \cos t , 2 \sin t , \sqrt { 3 } \cos t \rangle$$

.

Let $f(x, y)=x^2+x y+y^2$. (a) Find the maximum and minimum points and values of $f$ along the circle $x^2+y^2=1$. (b) Moving counterclockwise along the circle $x^2+y^2=1$, is the function increasing or decreasing at the points $(\pm 1,0)$ and $(0, \pm 1)$ ? (c) Find extreme points and values for $f$ in the disk $D$ consisting of all $(x, y)$ such that $x^2+y^2 \leqslant 1$.

Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk/washer and shell methods. Check that your results agree and state which method is easier to apply. y=

$$( x - 2 ) ^ { 3 }$$

-2, x=0, and y=25; revolved about the y-axis

An $8-\text{kg}$ uniform $\operatorname{rod} A B$ is hinged to a fixed support at $A$ and is attached by means of pins $B$ and $C$ to a 12-kg disk of radius $400 \mathrm{~mm}$. A spring attached at $D$ holds the rod at rest in the position shown. If point $B$ is moved down $25 \mathrm{~mm}$ and released, determine (a) the period of vibration, $(b)$ the maximum velocity of point $B$.

A disk drive manufacturer sells storage devices with capacities of one terabyte, 500 gigabytes, and 100 gigabytes with probabilities 0.5, 0.3, and 0.2, respectively. The revenues associated with the sales in that year are estimated to be $50 million,$25 million, and $10 million, respectively. Let X denote the revenue of storage devices during that year. Determine the probability mass function of X.

a. Find the vertex and intercepts.
b. Sketch the graph.
c. Confirm your sketch using PLOT CONIC from the accompanying disk, or another suitable graphing program.
Find the vertex and intercepts of the following parabolas, and sketch the graph. Find decimal approximations for any radicals you encounter.
$y=\frac{1}{5} x^2+2 x-\frac{11}{5}$