A 50.0-kg grindstone is a solid disk 0.520 m in diameter. You press an ax down on the rim with a normal force of 160 N. The coefficient of kinetic friction between the blade and the stone is 0.60, and there is a constant friction torque of $6.50 \mathrm { N } \cdot \mathrm { m }$ between the axle of the stone and its bearings. How much force must be applied tangentially at the end of a crank handle 0.500 m long to bring the stone from rest to 120 rev/min in 9.00 s?

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?

An object has an angular velocity. It also has an angular acceleration due to torques that are present. Therefore, the angular velocity is changing. What happens to the angular velocity if additional torques are applied so as to make the net torque suddenly equal to zero?

Let $F: R^{3} \rightarrow R^{3}$ be the map (a similarity) defined by $F(p)=c p, p \in R^{3}, c$ a positive constant. Let $S \subset R^{3}$ be a reguar surface and set $F(S)=\overline{S}$. Show that $\bar{S}$ is a regular surface, and find formulas relating the Gaussian and mean curvatures, K and H, of S with the Gaussian and mean curvatures, $\overline{K}$ and $\overline{H},$ of $\overline{S}$.

A block with mass m = 5.00 kg slides down a surface inclined $36.9 ^ { \circ }$ to the horizontal. The coefficient of kinetic friction is 0.25. A string attached to the block is wrapped around a flywheel on a fixed axis at O. The flywheel has mass 25.0 kg and moment of inertia $0.500 \mathrm { kg } \cdot \mathrm { m } ^ { 2 }$ with respect to the axis of rotation. The string pulls without slipping at a perpendicular distance of 0.200 m from that axis. What is the acceleration of the block down the plane?

Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. (a) Claim. Let $\circ$ be the operation on $\mathbb{R} \times \mathbb{R}$ defined by setting $(a, b) \circ(c, d)=(a+c, b+d),$ and let - be the usual subtraction on $\mathbb{R}.$ Then the function $f$ given by $f(a, b) =a - 3b$ is an OP map from $(\mathbb{R} \times \mathbb{R}, \circ) \quad \text{to} \quad (\mathbb{R},-).$ "Proof." (4, 2) and (3, 1) are in $\mathbb{R} \times \mathbb{R}.$ Then $f((4,2) \circ(3,1))= f(7,3)=7-3 \cdot 3=-2,$ whereas $f(4,2)-f(3,1)=-2-0=-2,$ so $f$ is operation preserving. (b) Claim. Let $f:(G, *) \rightarrow(H, \cdot) \text { and } g:(H, \cdot) \rightarrow(K, \otimes)$ be OP maps. Then the composite $g \circ f:(G, *) \rightarrow(K, \otimes)$ is an OP map. "Proof." $g \circ f(a b)=g(f(a b))=g(f(a) f(b))=g(f(a)) g(f(b))= (g \circ f(a))(g \circ f(b)).$

A lawn roller in the form of a thin-walled, hollow cylinder with mass M is pulled horizontally with a constant horizontal force F applied by a handle attached to the axle. If it rolls without slipping, find the acceleration and the friction force.

Underline the verbal phrase in each of the following sentences. Then, on the line provided, identify the phrase by writing part for participial phrase or inf for infinitive phrase.

$\underline{\hspace{1cm}}$ Splashing in the shallow water, the children enjoyed their day at the beach.

Match the check-writing policies for a small business to the control activities that follow.

$$\begin{array}{ll} \text{a. Authorization} & \text{1. The person who writes the checks to pay bills is different}\\ \text{b. Recording transactions} & \text{$\quad$from the people who authorize the payments and}\\ \text{c. Documents and} & \text{$\quad$keep records of the payments.}\\ \text{$\quad$records} & \text{2. The checks are kept in a locked drawer. The only person}\\ \text{d. Physical controls} & \text{$\quad$who has the key is the person who writes the checks.}\\ \text{e. Periodic independent} & \text{3. The person who writes the checks is bonded.}\\ \text{$\quad$verification} & \text{4. Once each month the owner compares and reconciles}\\ \text{f. Separation of duties} & \text{$\quad$the amount of money shown in the accounting records}\\ \text{g. Sound personnel} & \text{$\quad$with the amount in the bank account.}\\ \text{$\quad$practices} & \text{5. The owner of the business approves each check before}\\ & \text{$\quad$it is mailed.}\\ & \text{6. Information pertaining to each check is recorded on}\\ & \text{$\quad$the check stub.}\\ & \text{7. Every day, all checks are recorded in the accounting}\\ & \text{$\quad$records, using the information on the check stubs.}\\ \end{array}$$

Prove that if $\lim _{x \rightarrow c} f(x)$ exists and $\lim _{x \rightarrow c}[f(x)+g(x)]$ does not exist, then $\lim _{x \rightarrow c} g(x)$ does not exist.

While following a treasure map, you start at an old oak tree. You first walk 825 m directly south, then turn and walk 1.25 km at $30.0^{\circ}$ west of north, and finally walk 1.00 km at $32.0^{\circ}$ north of east, where you find the treasure: a biography of Isaac Newton! (a) To return to the old oak tree, in what direction should you head and how far will you walk? Use components to solve this problem. (b) To see whether your calculation in part (a) is reason-able, compare it with a graphical solution drawn roughly to scale.

One of the characteristics of a haiku is that it presents a moment of discovery or revelation. In your own words, describe the moment frozen in each of the haiku on page 500.

Energy may be produced from solid waste in two ways: (1) generate methane from anaerobic decomposition of the waste and burn it (landfill-gas-to-energy, or LFGTE) or (2) burn the waste directly (waste-to-energy, or WTE). The heat generated by either method can be used to produce steam, which impinges on a turbine rotor connected to a generator to produce electricity. LFGTE produces about 215 kWh electricity/ton of waste, and WTE produces roughly 600 kWh/ton of waste. The average output of a large power plant is 1 GW, which is enough to supply the annual residential energy consumption of a city of roughly 800,000 people. a) The current rate of municipal solid-waste generation in the United States is approximately 413 million tons per year. If all of it were used for energy recovery, how many 1 GW power plants could LFGTE supply? How many if WTE is used? b) A municipality trying to decide between LFGTE, WTE, and a natural gas-fired combustion turbine has called you in as a consultant. Use information in the sources cited below to summarize the pros and cons of each choice.

The rotor blades of a helicopter are $25 \text{ft}$ long and are rotating at $200 \text{rpm}$.

(a) Find the angular speed of the rotor.

(b) Find the linear speed of a point on the tip of a blade.