Conceptual Questions

ssm Solution is in the Student Solutions Manual

| 1 | Is it possible for x and v to be related to an angle according to , where x has units of meters (m) and v has units of meters per second (m/s)? Account for your answer.

| 2 | ssm You can always add two numbers that have the same units. However, you cannot always add two numbers that have the same dimensions. Explain why not; include an example in your explanation.

| 3 | The following table lists four variables along with their units:

Variable | Units
x | Meters (m)
v | Meters per second (m/s)
t | Seconds (s)
a | Meters per second squared (m/s2)

These variables appear in the following equations, along with a few numbers that have no units. In which of the equations are the units on the left side of the equals sign consistent with the units on the right side?

a. |
b. |
c. |
d. |
e. |
f. |

| 4 | (a) Is it possible for two quantities to have the same dimensions but different units? (b) Is it possible for two quantities to have the same units but different dimensions? In each case, support your answer with an example and an explanation.

| 5 | In the equation you wish to determine the integer value (1, 2, etc.) of the exponent n. The dimensions of y, a, and t are known. It is also known that c has no dimensions. Can dimensional analysis be used to determine n? Account for your answer.

| 6 | Using your calculator, verify that divided by is equal to , for an angle . Try 30°, for example. Prove that this result is true in general by using the definitions for , , and given in Equations 1.1, 1.2 and 1.3.

| 7 | and are called sinusoidal functions of the angle . The way in which these functions change as changes leads to a characteristic pattern when they are graphed. This pattern arises many times in physics. (a) To familiarize yourself with the sinusoidal pattern, use a calculator and construct a graph, with plotted on the vertical axis and on the horizontal axis. Use 15° increments for between 0° and 720°. (b) Repeat for .

| 8 | Which of the following quantities (if any) can be considered a vector: (a) the number of people attending a football game, (b) the number of days in a month, and (c) the number of pages in a book? Explain your reasoning.

| 9 | Which of the following displacement vectors (if any) are equal? Explain your reasoning.

Vector | Magnitude | Direction
| 100 m | 30° north of east
| 100 m | 30° south of west
| 50 m | 30° south of west
| 100 m | 60° east of north

| 10 | Are two vectors with the same magnitude necessarily equal? Give your reasoning.

| 11 | ssm A cube has six faces and twelve edges. You start at one corner, are allowed to move only along the edges, and may not retrace your path along any edge. Consistent with these rules, there are a number of ways to arrive back at your starting point. For instance, you could move around the four edges that make up one of the square faces. The four corresponding displacement vectors would add to zero. How many ways are there to arrive back at your starting point that involve eight displacement vectors that add to zero? Describe each possibility, using drawings for clarity.

| 12 | (a) Is it possible for one component of a vector to be zero, while the vector itself is not zero? (b) Is it possible for a vector to be zero, while one component of the vector is not zero? Explain.

| 13 | Can two nonzero perpendicular vectors be added together so their sum is zero? Explain.

| 14 | Can three or more vectors with unequal magnitudes be added together so their sum is zero? If so, show by means of a tail-to-head arrangement of the vectors how this could occur.

| 15 | In preparation for this question, review Conceptual Example 7. Vectors and satisfy the vector equation . (a) How does the magnitude of compare with the magnitude of ? (b) How does the direction of compare with the direction of ? Give your reasoning.

| 16 | Vectors , , and satisfy the vector equation , and their magnitudes are related by the scalar equation . How is vector oriented with respect to vector ? Account for your answer.

| 17 | Vectors , , and satisfy the vector equation , and their magnitudes are related by the scalar equation . How is vector oriented with respect to vector ? Explain your reasoning.

| 18 | The magnitude of a vector has doubled, its direction remaining the same. Can you conclude that the magnitude of each component of the vector has doubled? Explain your answer.

| 19 | ssm The tail of a vector is fixed to the origin of an x, y axis system. Originally the vector points along the axis. As time passes, the vector rotates counterclockwise. Describe how the sizes of the x and y components of the vector compare to the size of the original vector for rotational angles of (a) 90°, (b) 180°, (c) 270°, and (d) 360°.

| 20 | A vector has a component of zero along the x axis of a certain axis system. Does this vector necessarily have a component of zero along the x axis of another (rotated) axis system? Use a drawing to justify your answer.

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