Why is the following situation impossible? A new high-speed roller coaster is claimed to be so safe that the passengers do not need to wear seat belts or any other restraining device. The coaster is designed with a vertical circular section over which the coaster travels on the inside of the circle so that the passengers are upside down for a short time interval. The radius of the circular section is 12.0 m, and the coaster enters the bottom of the circular section at a speed of 22.0 m/s. Assume the coaster moves without friction on the track and model the coaster as a particle.

The primary coil of a transformer has $N_{1}=350$ turns, and the secondary coil has $N_{2}=2000$ turns. If the input voltage across the primary coil is $\Delta v=170 \cos \omega t,$ where $\Delta v$ is in volts and t is in seconds, what rms voltage is developed across the secondary coil?

Each of the functions is given over one period. For each function, sketch several periods and decide whether it is even or odd. Then use (9.4) or (9.5) to expand it in an appropriate Fourier series.

$$f(x)=\left\{\begin{array}{rr} -1, & -l<x<0 \\ 1, & 0<x<l \end{array}\right.$$

Verify Bauer’s formula $e^{i x w}=\sum_{0}^{\infty}(2 l+1) i^{l} j_{l}(x) P_{l}(w)$ as follows. Write the integral for the coefficients $c_l$ in the Legendre series for $e^{i x w}=\sum c_{l} P_{l}(w)$. You want to show that $c_{l}(x)=(2 l+1) i^{l} j_{l}(x)$. First show that $y = c_l(x)$ satisfies the differential equation for spherical Bessel functions. Hints: Differentiate with respect to x under the integral sign to find $y^{\prime}$ and $y^{\prime \prime}$; substitute into the left side of the differential equation. Now integrate by parts with respect to w to show that the integrand is zero because $P_l(w)$ satisfies Legendre’s equation. Thus $c_l(x)$ must be a linear combination of $j_l(x)$ and $n_l(x)$. Now consider the $c_l(x)$ integral for small x; expand $e^{i w x}$ in series and evaluate the R lowest term (which is $x^l$ since $\int_{-1}^{1} w^{n} P_{l}(w) d w=0$ for n < l). Compare with the approximate formulas for $j_1(x)$ and $n_l(x)$.

You are tutoring a bright student in his last semester of introductory physics. The particular topic of the day is bonding in solids. When your session begins, the student hands you a slip of paper with the following equation printed on it:

$$U_{0}=-\alpha k_{e} \frac{e^{2}}{r_{0}}\left(1-\frac{1}{m}\right)$$

He says that he found this equation in his online studying and that it is described as an expression for the ionic cohesive energy of a crystal formed by ionic bonding. He asks you to derive this equation.

Find the solutions of the following differential equations in terms of Bessel functions by using equations. $x y^{\prime \prime}+5 y^{\prime}+x y=0$

When you step from a hot street into an air-conditioned room, you feel cooler. Why does this change occur? 1. Heat is moving from a cold area (the room) to a hotter area (the street). 2. Heat leaves your body as it moves from a warm area (your body) to a colder area (the air in the room). 3. The motion energy used to walk into the room lowers your body temperature. 4. The motion energy used to walk into the room raises your body temperature.

You are a member of an expert witness group that provides scientific services to the legal community. Your group has been asked by a defense attorney to prove at trial that a driver was not exceeding the speed limit. You are provided with the following data: The mass of the car is $1.50 \times 10^{3} \mathrm{kg}$. The mass of the driver is 95.0 kg. The coefficient of kinetic friction between the car's tires and the roadway is 0.580. The coefficient of static friction between the car's tires and the roadway is 0.820. The posted speed limit on the road is 25 mi/h. The roadway was dry and the weather was sunny at the time of the incident. You are also provided with the following description of the incident: The driver was driving up a hill that makes an angle of $17.5^{\circ}$ with the horizontal. The driver saw a dog run into the street, slammed on the brakes and left a skid mark 17.0 m long. The car came to rest at the end of the skid mark. The driver did not hit the dog, but the sound of the screeching tires drew the attention of a nearby policeman, who ticketed the driver for speeding. Should your group agree to offer testimony for the defense in this case?

Water is pumped into a tall, vertical cylinder at a volume flow rate R. The radius of the cylinder is r, and at the open top of the cylinder a tuning fork is vibrating with a frequency f. As the water rises, what time interval elapses between successive resonances?

Find $f^{\prime}(x)$ and find the equation of the line tangent to the graph of f at the indicated value of x. $f(x)=\frac{x}{(2 x-5)^{3}} ; \quad x=3$

A uniform flexible chain of given length is suspended at given points $(x_1, y_1)$ and $(x_2, y_2)$. Find the curve in which it hangs. Hint: It will hang so that its center of gravity is as low as possible.

Use power series to evaluate the function at the given point. Compare with computer results, using the computer to find the series, and also to do the problem without series. Resolve any disagreement in results. $\frac{1}{\sqrt{1+x^{4}}}-\cos x^{2}$ at $x=0.012$

Find the slope of the graph of the function at the given point.

$$\begin{matrix} \text{Equation} & \text{Point}\\ \text{}{\left(x^{2}+y^{2}\right)^{2}=8 x^{2} y} & \text{(2, 2)}\\ \end{matrix}$$

Identify each set as finite or infinite. Then determine whether 10 is an element of the set. $\{1,2,3,4,5, \dots, 75\}$