From Appendix A, the Volume Vof a Sphere Is Given in Terms of Its Radius As

Solutions

Chapter 1

Problems:

1-12

Bacteria vary somewhat in size, but a diameter of 2.0μm is not unusual.What would be the volume (in cubic centimeters) and surface area (in square millimeters) of such a bacterium, assuming that it is spherical?

Solution:

From Appendix A, the volume Vof a sphere is given in terms of its radius as

while its surface area Ais given as

The necessary equalities for this problem are:

1-23:

A brass washer has an outside diameter of 4.50cm with a hole of diameter 1.25cm and is 1.50mm thick. The density of brass is 8600kg/m3. If you put this washer on a laboratory balance, what it will weight in grams?

The density of the washer material is:

ρ=

The mass of the washer is:

1-35:

A runner jogs around a circular track 150ft in diameter. (a) Clearly sketch this runner’s displacement vector one he has completed one-forth of a lap; one-half of a lap; one lap. (b) Find the magnitude and direction of the runner’s displacement vector for each case in part (a).

Solution:

The displacement vector is directed from the initial position of the object to the final position. In each case, its magnitude dis the length of the line that connects points 1 and 2.

(a) The initial position (point 1), the final position (point 2), the displacement vector and its direction φ is shown in Figure. for each case.

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Quarter lap Half lap Full lap

(b) Quarter lap: The magnitude is

From Figure

Half lap: The magnitude is The direction isφ =1800

Full lap: The final position equals the initial position. The displacement is therefore equal to zero and the direction is undefined.

Reflect: In each case, the magnitude of the displacement is less than the distance traveled.

1-47:

Two vectors of equal magnitude act in a vertical plane perpendicular to each other. If their resultant is 75 N directly downward, (a) sketch these two vectors and the resultant, and (b) use components to find the magnitude of each of the two vectors and the angle each makes with the vertical.

Solution:

Use coordinates for which the axis is downward. Let the two vectors be and and let the angles they make on either side of the axis be φA and φB . Since the resultant, is in the direction, The two vectors and their components are shown in Figure(a).

(a) (b)

(a) Since and we can conclude that and . Similarly, since The vector addition diagram for is given in Figure b.

(b) and and so and

1-63

A patient with a dislocated shoulder is put into a traction apparatus as shown in Figure. The pulls and have equal magnitudes and must combine to produce an outward traction force of 5.60 N on the patient’s arm. How large should these pulls be?

Solution:

Use coordinates having a horizontal axis and an upward axis. Then

and

Since

and