Consider a mass m moving in two dimensions with potential energy U (x, y) = 1/2 kr², where r² = x² + y². Write down the Lagrangian, using coordinates x and y, and find the two Lagrange equations of motion. Describe their solutions.

Verbally describe the subset of real numbers represented by the inequality. Sketch the subset on the real number line, and state whether the interval is bounded or unbounded.

$$-4<x<2$$

The velocity of an object plotted as a function of time. (a) Calculate the instantaneous acceleration at $t=3 \mathrm{~s}, t=10 \mathrm{~s}$, and $t=13 \mathrm{~s}$. (b) What is the average acceleration for the complete motion?

Fill in the blank with always, sometimes, or never to make a true statement. $\overrightarrow{J K}$ and $\overrightarrow{J L}$ are _____ the same ray.

A particle of mass m moving in three dimensions under the potential energy function $V(x, y, z)=\alpha x+\beta y^{2}+\gamma z^{3}$ has speed $v_0$ when it passes through the origin. (a) What will its speed be if and when it passes through the point (1,1,1)? (b) If the point (1, 1, 1) is a turning point in the motion (v = 0), what is $v_{0}$? (c) What are the component differential equations of motion of the particle?

Goldstein observed positively charged particles moving in the opposite direction to electrons in a cathode ray tube. From their mass, he concluded that these particles were formed from residual gas in the tube. For example, if the cathode ray tube contained helium, the canal rays consisted of $\mathrm{He^+}$ ions. Describe the process that forms these ions.

The following measurements from yaw marks left at the scene of an accident were taken by police. Using a 40-foot length chord, the middle ordinate measured approximately 4 feet. The drag factor for the road surface was determined to be 0.95. a. Determine the radius of the curved yaw mark to the nearest tenth of a foot. b. Determine the minimum speed that the car was going when the skid occurred to the nearest tenth.

Three nested hollow spheres have the same center. The innermost sphere has a radius of 2 cm and carries a uniformly distributed charge of $6 \mathrm{nC}\left(1 \mathrm{nC}=1 \times 10^{-9} \mathrm{C}\right).$ The middle sphere has a radius of 5 cm and carries a uniformly distributed charge of -4 nC. The outermost sphere has a radius of 10 cm and carries a uniformly distributed charge of 8 nC. What is the magnitude of the electric field at a distance of 4 cm from the center?

Suppose that you roll a tetrahedral die and a six-sided die at the same time. a. Make a chart that shows the sample space of all possible outcomes. b. How many possible outcomes are there? Are they equally likely? c. Make a table for the probability distribution of the sum of the two dice. d. What is the probability that the sum is at most 3?

In the $p-V$ diagram of Fig., the gas does $5 \mathrm{~J}$ of work when taken along isotherm $a b$ and $4 \mathrm{~J}$ when taken along adiabatic $b c$. What is the change in the internal energy of the gas when it is taken along the straight path from $a$ to $c$ ?

Tell whether the statement is sometimes, always, or never true. The range of

$$f ( x ) = a \sqrt [ 3 ] { x - h },$$

where a and h are nonzero real numbers, is all real numbers.

Carl had a piece of cake in the shape of an isosceles triangle with angles measuring $26^{\circ}, 77^{\circ}$, and $77^{\circ}$. He wanted to divide it into two equal parts, so he cut it through the middle of the $26^{\circ}$ angle to the midpoint of the opposite side. He says that because he is dividing it at the midpoint of a side, the two pieces are congruent. Is this enough information?

A soccer ball rests on the field at the location x = 5.0 m. Two players run along the same straight line toward the ball. Their equations of motion are as follows:

$$\begin{array}{l} x_{1}=-8.2 \mathrm{m}+(4.2 \mathrm{m} / \mathrm{s}) t \\ x_{2}=-7.3 \mathrm{m}+(3.9 \mathrm{m} / \mathrm{s}) t \end{array}$$

(a) Which player is closer to the ball at t = 0? (b) At what time does one player pass the other player? (c) What is the location of the players when one passes the other?

Determine whether the statement is true or false. Explain your reasoning.

$$\vec { L M } \text { and } \vec { M L }$$

are the same ray.