Why is the following situation impossible? An air rifle is used to shoot 1.00-g particles at a speed of $v_{x}=100 \mathrm{m} / \mathrm{s} .$ The rifle's barrel has a diameter of 2.00 mm. The rifle is mounted on a perfectly rigid support so that it is fired in exactly the same way each time. Because of the uncertainty principle, however, after many firings, the diameter of the spray of pellets on a paper target is 1.00 cm.

Suppose that an honest die is rolled n=180 times. Let the random variable X represent the number of times the number 6 is rolled. (a) Find the mean and standard deviation for the distribution of X. (Hint: Use the dishonest-coin principle with p=1/6 to find

$$\mu$$

and

$$\sigma$$

.) (b) Find the probability that a 6 will be rolled more than 40 times. (c) Find the probability that a 6 will be rolled somewhere between 30 and 35 times.

Suppose you attempt to pick up a very heavy object. Before you tried to pick it up, the object was sitting still—its momentum was not changing. You pull very hard, but do not succeed in moving the object. Is this a violation of the Momentum Principle? How can you be exerting a large force on the object without causing a change in its momentum? What does change when you apply this force?

Rayleigh’s criterion is used to determine when two objects are barely resolved by a lens of diameter d. The angular separation must be greater than $\theta_{R}$ where $\theta_{R} =1.22 \lambda /d$ . In order to resolve two objects 4000 nm apart at a distance of 20 cm with a lens of diameter 5 cm, what energy (a) photons or (b) electrons should be used? Is this consistent with the uncertainty principle?

Show how a single block of glass can be used to turn a p-polarized beam of light through $180^{\circ},$ with the light suffering (in principle) zero reflective loss. The light is incident from air, and the returning beam (also in air) may be displaced sideways from the incident beam. Specify all pertinent angles and use n = 1.45 for glass. More than one design is possible here.

Use the uncertainty principle to estimate the binding energy of the $\mathrm{H}_2$ molecule by calculating the difference in kinetic energy of the electrons between when they are in separate atoms and when they are in the molecule. Take $\Delta x$ for the electrons in the separated atoms to be the radius of the first Bohr orbit, $0.053 \mathrm{~nm}$, and for the molecule take $\Delta x$ to be the separation of the nuclei, $0.074 \mathrm{~nm}$. [Hint: Let $\left.\Delta p \approx \Delta p_x.\right]$

Refer to the figure and use the principle of superposition to Find the component of the current $i$ through $R_{3}$ that is due to $V_{S 2}$.

$$\begin{aligned} & V_{S 1}=V_{S 2}=450 \mathrm{~V} \\ & R_{1}=7 \Omega \quad R_{2}=5 \Omega \\ & R_{3}=10 \Omega \quad R_{4}=R_{5}=1 \Omega \end{aligned}$$

Object A has mass m$_A$ = 10 kg and initial momentum p$_{A,i}$ = (21, -7, 0) kg·m/s, just before it strikes object B, which has mass m$_B$= 14 kg. Just before the collision object B has initial momentum p$_{B,i}$= (3, 6, 0) kg·m/s. Therefore, what does the Momentum Principle predict that the total final momentum of the system will be, just after the collision?

Suppose $f$ is differentiable at points on a closed path $\gamma$ and at all points in the region G enclosed by $\gamma,$ except possibly at a finite number of poles of $f$ in G. Let Z be the number of zeros of $f$ in G, and P the number of poles of $f$ in G, with each zero and pole counted as many times as its multiplicity. Show that $\frac{1}{2 \pi i} \oint_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z=Z-P.$ This formula is known as the argument principle.

The following exercise is a problem that may require permutations, combinations, or the multiplication principle.

According to the Baskin-Robbins Web site, there are 21 "classic flavors" of ice cream.

(a) How many different double-scoop cones can be made if order does not matter (for example, putting chocolate on top of vanilla is equivalent to putting vanilla on top of chocolate)?

(b) How many different triple-scoop cones can be made if order does matter?

Show that, in principle, $\mathrm{Na}_2 \mathrm{CO}_3(\mathrm{aq})$ can be converted almost completely to $\mathrm{NaOH}(\mathrm{aq})$ by the reaction

$$\begin{aligned} \mathrm{Ca}(\mathrm{OH})_2(\mathrm{~s})+\mathrm{Na}_2 \mathrm{CO}_3(\mathrm{aq}) & \longrightarrow \\ & \mathrm{CaCO}_3(\mathrm{~s})+2 \mathrm{NaOH}(\mathrm{aq}) \end{aligned}$$

Underline any verbs that are used incorrectly. In the space provided, write the correct form (or $C$ if correct) and the number of the $\mathrm{G} / \mathrm{M}$ principle illustrated. When you finish, compare your responses with those provided at the bottom of this page. If your responses differ, study carefully the principles in parentheses.

Everyone except the temporary workers employed during the past year has become eligible for health benefits.

A soap bubble will enclose the maximum space with a minimum amount of surface material. Architects have used this principle to create buildings that enclose a great amount of space with a small amount of building material. If a soap bubble has a surface area of A, then its volume V is given by the equation $V=0.094 \sqrt{A^{3}}$. Find the surface area of a bubble with a volume of $1.75 \times 10^{2}$ cubic millimetres.

Hoot Enterprises buys a warehouse for $590,000 to use for its East Coast distribution operations. On the date of the purchase, a professional appraisal shows a value of$650,000 for the warehouse. The seller had originally purchased the building for $480,000. Hoot has a similar warehouse on the West Coast that has a book value of$603,000. Under the historical cost principle, Hoot should record the building for

a. $650,000.

b.$480,000.

c. $590,000.

d.$603,000.