A mass $m$ is gently placed on the end of a freely hanging spring. The mass then falls $27.0 \mathrm{~cm}$ before it stops and begins to rise. What is the frequency of the oscillation?

A $950 \mathrm{~kg}$ car strikes a huge spring at a speed of $25 \mathrm{~m} / \mathrm{s}$, compressing the spring $4.0 \mathrm{~m}$. What is the spring stiffness constant of the spring?

A set of reference books consists of eight volumes numbered 1 through 8. (a) In how many ways can the eight books be arranged on a shelf? (b) In how many ways can the eight books be arranged on a shelfso that at least one book is out of order?

A computer password consists of four letters (A through Z) followed by a single digit (0 through 9). Assume that the passwords are not case sensitive (i.e., that an uppercase letter is the same as a lowercase letter). (a) How many different passwords are possible? (b) How many different passwords end in 1? (c) How many different passwords do not start with Z? (d) How many different passwords have no Z's in them?

A computer system uses passwords that are exactly seven characters, and each character is one of the 26 letters (a–z) or 10 integers (0–9). Uppercase letters are not used. a. How many passwords are possible? b. If a password consists of exactly 6 letters and 1 number, how many passwords are possible? c. If a password consists of 5 letters followed by 2 numbers, how many passwords are possible?

The exercises concerns an unprepared student who takes a 10-question true-false quiz and guesses at the answer to every question. What is the probability that the student answers every question incorrectly?

The sample spaces are too big to write down in full In these exercises, you should describe the sample space .either by describing a generic outcome or by listing some outcomes and then using the ... notation. In the latter case, you should write down enough outcomes to make the description reasonably clear. A student randomly guesses the answers to a 10-question true-or-false quiz. The observation is the student's answer (T or F) for each question. Describe the sample space.

Using set notation, write out the sample space for each of the following random experiments: (a) Three runners {A, B, and C) are running in a race (assume that there are no ties). The observation in the order in which the three runners cross the finish line. (b) Four runners (A, B, C, and D) are running in a qualifying race (assume that there are no ties). The top two finishers qualify for the finials. The observation is the pair of runners that qualify for the finals.

Using set notation, write out the sample space for each of the following random experiments. (a) A coin is tossed four times in a row. The observation is how the coin lands (H or T) on each toss. (b) A student randomly guesses the answers to a four-question true-or-false quiz.The observation is the student's answer (T or F) for each question. (c) A coin is tossed four times in a row. The observation is the percentage of tosses that are heads. (d) A student randomly guesses the answer to a four-question true-or-false quiz. The observation is the percentage of correct answers in the test.

A diving board oscillates with simple harmonic motion of frequency $2.8$ cycles per second. What is the maximum amplitude with which the end of the board can oscillate in order that a pebble placed there does not lose contact with the board during the oscillation?

A $320 \mathrm{~kg}$ wooden raft floats on a lake. When a $68 \mathrm{~kg}$ man stands on the raft, it sinks $3.5 \mathrm{~cm}$ deeper into the water. When he steps off, the raft oscillates for a while. What is the total energy of oscillation (ignoring damping)?

A $320 \mathrm{~kg}$ wooden raft floats on a lake. When a $68 \mathrm{~kg}$ man stands on the raft, it sinks $3.5 \mathrm{~cm}$ deeper into the water. When he steps off, the raft oscillates for a while. What is the frequency of oscillation?

A student randomly guesses the answers to a four-question true-or-false (T-F) quiz. Find the probability of each of the following event.s (a)

$$E_1$$

, "the student answers F on two of the four questions." (b)

$$E_2$$

: "the student answers F on at least two of the four questions." (c)

$$E_3$$

: "the student answers F on at most two of the four questions." (d)

$$E_4$$

: "the student answers F to the first two question."

A rectangular block of wood floats in a calm lake. Show that, if friction is ignored, when the block is pushed gently down into the water and then released, it will then oscillate with SHM. Also, determine an equation for the force constant.