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 hand-driven grinding wheel is a uniform 60-kg disk with a $45-\mathrm{cm}$ radius. It has a handle of negligible mass $65 \mathrm{~cm}$ from the rotation axis. A compact $25-\mathrm{kg}$ load is attached to the handle when it is at the same height as the horizontal rotation axis. Ignoring the effects of friction, find

$(a)$ the initial angular acceleration of the wheel.

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.

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.

A 3-kg slender bar is rigidly attached to a 2- $\mathrm{kg}$ thin circular disk. In terms of the body-fixed coordinate system shown, the angular velocity of the composite object is $\omega=1001-4 J+6 k(\mathrm{rad} / \mathrm{s})$. What is the object's angular momentum about its cehter of mass?

A $1.5-\mathrm{kg}$ slender rod is welded to a $5-\mathrm{kg}$ uniform disk as shown. The assembly swings freely about $C$ in a vertical plane. Knowing that in the position shown the assembly has an angular velocity of $10 \mathrm{rad} / \mathrm{s}$ clockwise, find $(a)$ the angular acceleration of the assembly, $(b)$ the components of the reaction at $C$.

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) A lamina has constant density $\rho$ and takes the shape of a disk with center the origin and radius $R$. Use Newton’s Law of Gravitation to show that the magnitude of the force of attraction that the lamina exerts on a body with mass $m$ located at the point $(0, 0, d)$ on the positive z-axis is

$$F = 2\pi Gm\rho d\left(\dfrac{1}{d} - \dfrac{1}{\sqrt{R^2 + d^2}}\right)$$

[Hint: Divide the disk as in the figure and first compute the vertical component of the force exerted by the polar subrectangle $R_{ij}$.]

(b) Show that the magnitude of the force of attraction of a lamina with density $\rho$ that occupies an entire plane on an object with mass $m$ located at a distance $d$ from the plane is

$$F = 2\pi Gm\rho$$

Notice that this expression does not depend on $d$.

Volume of a sphere Let $R$ be the region bounded by the upper half of the circle $x^{2}+y^{2}=r^{2}$ and the $x$ -axis. A sphere of radius $r$ is obtained by revolving $R$ about the $x$ -axis.
a. Use the shell method to verify that the volume of a sphere of radius $r$ is $\frac{4}{3} \pi r^{3}$ . b. Repeat part (a) using the disk method.

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)=

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

.

A conical domain with vertex (0, 0, b) and axis along the z-axis has as base a disk of radius a in the x y-plane. Find the flux of $\mathbf{F}=\left(x+y^{2}\right) \mathbf{i}+\left(3 x^{2} y+y^{3}-x^{3}\right) \mathbf{j}+(z+1) \mathbf{k}$ upward through the conical part of the surface of the domain.

(II) A thin $7.0\text{-kg}$ wheel of radius $34 \text{~cm}$ is a uniform disk and is weighted to one side by a $1.50\text{-kg}$ weight, small in size, placed $24 \text{~cm}$ from the center of the wheel. Calculate $(b)$ the moment of inertia about an axis through its $\text{\scriptsize{CM}}$, perpendicular to its face.

a. Complete the square (if necessary), find the center, draw the asymptotes, plot the vertices, then sketch the graph.
b. Calculate the focal radius, and plot the foci.
c. Confirm your sketch by plotting the original equation, using PLOT CONIC on the accompanying disk, or a similar program.
$9 x^2-4 y^2-54 x-16 y-79=0$

Find the potential Φ(r, θ) in the unit disk r<1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch a figure of the equipotential lines. Φ(1, θ)=1 if -½π<θ<½π and 0 otherwise