Mechanics

BIO “Seeing” Surfaces at the nanoscale. One technique for making images of surfaces at the nanometer scale, including membranes and biomolecules, is dynamic atomic force microscopy. In this technique, a small tip is attached to a cantilever, which is a flexible, rectangular slab supported at one end, like a diving board. The cantilever vibrates, so the tip moves up and down in simple harmonic motion. In one operating mode, the resonant frequency for a cantilever with force constant k = 1000 N/m is 100 kHz. As the oscillating tip is brought within a few nanometers of the surface of a sample (as shown in the figure), it experiences an attractive force from the surface. For an oscillation with a small amplitude (typically, 0.050 nm), the force F that the sample surface exerts on the tip varies linearly with the displacement x of the tip, \(\left| F \right| = {k_{{\rm{surf}}}}x\), where \({k_{{\rm{surf}}}}\) is the effective force constant for this force. The net force on the tip is therefore\(\left( {k + {k_{{\rm{surf}}}}} \right)x\), and the frequency of the oscillation changes slightly due to the interaction with the surface. Measurements of the frequency as the tip moves over different parts of the sample’s surface can provide information about the sample.

In the model of Problem 14.94, what is the mechanical energy of the vibration when the tip is not interacting with the surface? (a) \({\bf{1}}.{\bf{2}} \times {\bf{1}}{{\bf{0}}^{ - {\bf{18}}}}{\rm{ }}{\bf{J}}\); (b) \({\bf{1}}.{\bf{2}} \times {\bf{1}}{{\bf{0}}^{ - {\bf{1}}6}}{\rm{ }}{\bf{J}}\);(c) \({\bf{1}}.{\bf{2}} \times {\bf{1}}{{\bf{0}}^{ - 9}}{\rm{ }}{\bf{J}}\); (d) \(5.0 \times {\bf{1}}{{\bf{0}}^{ - {\bf{8}}}}{\rm{ }}{\bf{J}}\).

Two blocks, with masses 4.00 kg and 8.00 kg, are connected by a string and slide down a 30.0° inclined plane (Fig. P5.96). The coefficient of kinetic friction between the 4.00-kg block and the plane is 0.25; that between the 8.00-kg block and the plane is 0.35. Calculate 

(a) the acceleration of each block and 

(b) the tension in the string. 

(c) What happens if the positions of the blocks are reversed, so that the 4.00-kg block is uphill from the 8.00-kg block?

Energy of locomotion. On flat ground, a 70 - kg person requires about 300 W of metabolic power to walk at a steady pace of 5 -km/hr (1.4 m/s). Using the same metabolic power output, that person can bicycle over the same ground at 15 - km/h.

How many times greater is the kinetic energy of the person when biking than when walking? Ignore the mass of the bike. (a) 1.7; (b) 3; (c) 8; (d) 9

For the Texas Department of Public Safety, you are investigating an accident that occurred early on a foggy morning in a remote section of the Texas Panhandle. A 2012 Prius traveling due north collided in a highway intersection with a 2013 Dodge Durango that was traveling due east. After the collision, the wreckage of the two vehicles was locked together and skidded across the level ground until it struck a tree. You measure that the tree is $\mathbf{35}\mathbf{\text{ ft}}$ from the point of impact. The line from the point of impact to the tree is in a direction $\mathbf{39}\mathbf{°}$ north of east. From experience, you estimate that the coefficient of kinetic friction between the ground and the wreckage is $\mathbf{0}\mathbf{.}\mathbf{45}$. Shortly before the collision, a highway patrolman with a radar gun measured the speed of the Prius to be $\mathbf{50}\mathbf{\text{ mph}}$ and, according to a witness, the Prius driver made no attempt to slow down. Four people with a total weight of $\mathbf{460}\mathbf{\text{ lb}}$  were in the Durango. The only person in the Prius was the $\mathbf{\text{150 lb}}$ driver. The Durango with its passengers had a weight of $\mathbf{6500}\mathbf{\text{ lb}}$ , and the Prius with its driver had a weight of $\mathbf{\text{3042 lb}}$. (a) What was Durango’s speed just before the collision? (b) How fast was the wreckage traveling just before it struck the tree?

How many correct experiments do we need to disprove a theory? How many do we need to prove a theory? Explain.

Starting with the definition 1 in. = 2.54 cm, find the number of (a) kilometers in 1.00 mile and (b) feet in 1.00 km

Question: According to the label on a bottle of salad dressing, the volume of the contents is 0.473 liter (L). Using only the conversions 1 L = 1000 cm³ and 1 in. = 2.54 cm, express this volume in cubic inches.

Question:The density of gold is 19.3 g/cm3 . What is this value in kilograms per cubic meter?

Starting with the definition 1 in. = 2.54 cm, find the number of (a) kilometers in 1.00 mile and (b) feet in 1.00 km.

Question: Starting with the definition 1 in. = 2.54 cm, find the number of (a) kilometers in 1.00 mile and (b) feet in 1.00 km.

According to the label on a bottle of salad dressing, the volume of the contents is 0.473 liter (L). Using only the conversions 1 L = 1000 cm³ and 1 in. = 2.54 cm, express this volume in cubic inches.

What is your height in centimeters? What is your weight in Newton?

How many nanoseconds does it take light to travel 1.00 ft in vacuum? (This result is a useful quantity to remember.)?

Question: How many nanoseconds does it take light to travel 1.00 ft in vacuum? (This result is a useful quantity to remember.)?

The U.S. National Institute of Standards and Technology (NIST) maintains several accurate copies of the international standard kilogram. Even after careful cleaning, these national standard kilograms are gaining mass at an average rate of about $\mathbf{1}\mathbf{ }\mathit{\mu }\mathit{g}\mathbf{/}\mathit{y}$ (y = year) when compared every  years or so to the standard international kilogram. Does this apparent increase have any importance? Explain.

The density of gold is 19.3 g/cm³ . What is this value in kilograms per cubic meter?

What physical phenomena (other than a pendulum or cesium clock) could you use to define a time standard?

The most powerful engine available for the classic 1963 Chevrolet Corvette Sting Ray developed 360 horsepower and had a displacement of 327 cubic inches. Express this displacement in liters (L) by using only the conversions 1 L = 1000 cm³ and 1 in. = 2.54 cm.

The most powerful engine available for the classic 1963 Chevrolet Corvette Sting Ray developed 360 horsepower and had a displacement of 327 cubic inches. Express this displacement in liters (L) by using only the conversions 1 L = 1000 cm³ and 1 in. = 2.54 cm.

Describe how you could measure the thickness of a sheet of paper with an ordinary ruler.

The quantity $\mathbf{\pi }\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{14159}$ is a number with no dimensions since it is a ratio of two lengths. Describe two or three other geometrical or physical quantities that are dimensionless.

What are the units of volume? Suppose another student tells you that a cylinder of radius r and height h has volume given by ${\mathbf{\pi r}}^{\mathbf{3}}\mathbf{h}$. Explain why this cannot be right.

Three archers each fire four arrows at a target. Joe’s four arrows hit at points 10 cm above,  10 cm below, 10 cm to the left, and 10 cm to the right of the centre of the target. All four of Moe’s arrows hit within 1 cm of a point 20 cm from the centre, and Flo’s four arrows hit within 1 cm of the centre. The contest judge says that one of the archers is precise but not accurate, another archer is accurate but not precise, and the third archer is both accurate and precise. Which description applies to which archer? Explain.

Is the vector $\left(\overbrace{i}+\overbrace{j}+\overbrace{k}\right)$ a unit vector? Is the vector  $\left(3.0\overbrace{i}-2.0\overbrace{j}\right)$ a unit vector? Justify your answers.

A circular racetrack has a radius of 500 m. What is the displacement of a bicyclist when she travels around the track from the north side to the south side? When she makes one complete circle around the track? Explain.

Can you find two vectors with different lengths that have a vector sum of zero? What length restrictions are required for three vectors to have a vector sum of zero? Explain.

The “direction of time” is said to proceed from past to future. Does this mean that time is a vector quantity? Explain.

Air traffic controllers give instructions called “vectors” to tell airline pilots in which direction they are to fly. If these are the only instructions given, is the name “vector” used correctly? Why or why not?

Can you find a vector quantity that has a magnitude of zero but components that are not zero? Explain. Can the magnitude of a vector be less than the magnitude of any of its components? Explain.

Question-. (a) Does it make sense to say that a vector is negative? Why? (b) Does it make sense to say that one vector is the negative of another? Why? Does your answer here contradict what you said in part (a)?

If $\overrightarrow{\mathbf{C}}\mathbf{=}\overrightarrow{\mathbf{A}}\mathbf{+}\overrightarrow{\mathbf{B}}$ , what must be true about the directions and magnitudes of $\overrightarrow{\mathbf{A}}$  and $\overrightarrow{\mathbf{B}}$ if $\mathit{C}\mathbf{=}\mathit{A}\mathbf{+}\mathit{B}$ ? What must be true about the directions and magnitudes of $\overrightarrow{\mathbf{A}}$and $\overrightarrow{\mathbf{B}}$  if $\mathit{C}\mathbf{=}\mathbf{0}$ ?

If $\overrightarrow{\mathit{A}}$ and $\overrightarrow{\mathbf{B}}$ are nonzero vectors, is it possible for both $\overrightarrow{\mathbf{A}}\mathbf{·}\overrightarrow{\mathbf{B}}$ and $\overrightarrow{\mathbf{A}}\times \overrightarrow{\mathbf{B}}$ to be zero? Explain.

What does $\overrightarrow{A} . \overrightarrow{A}$ , the scalar product of a vector with itself, give? What about  $\overrightarrow{A}\times \overrightarrow{A}$, the vector product of a vector with itself?

Let A represent any nonzero vector. Why is  $\raisebox{1ex}{$\overrightarrow{A}$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$     a unit vector, and what is its direction? If $\theta$  is the angle that A makes with the +x-axis, explain why  $\raisebox{1ex}{$\overrightarrow{A}$}\!\left/ \!\raisebox{-1ex}{$A$}\right.$  is called the direction cosine for that axis.

Figure 1.7 shows the result of an unacceptable error in the stopping position of a train. If a train travels 890 km from Berlin to Paris and then overshoots the end of the track by 10.0 m, what is the percent error in the total distance covered? Is it correct to write the total distance covered by the train as 890,010 m? Explain.

Let A represent any nonzero Which of the following are legitimate mathematical operations: (a)  $\overrightarrow{A}⦁\left(\overrightarrow{B}-\overrightarrow{C}\right)$ 2; (b)   $\left(\overrightarrow{B}-\overrightarrow{C}\right)\times \overrightarrow{A}$; (c)  $\overrightarrow{A}⦁\left(\overrightarrow{B}\times \overrightarrow{C}\right)$ ; (d)  $\overrightarrow{A}\times \left(\overrightarrow{B}\times \overrightarrow{C}\right)$ ; (e)  $\overrightarrow{A}\times \left(\overrightarrow{B}⦁\overrightarrow{C}\right)$ ? In each case, give the reason for your answer.

Show that, no matter what $\overrightarrow{A}$ and $\overrightarrow{B}$are, $\overrightarrow{A}.\left(\overrightarrow{A}\times \overrightarrow{B}\right)$ =0

If $\overrightarrow{A}$.$\overrightarrow{B}$=0 does it necessarily follow that A = 0 or B= 0? Explain. (b) If $\overrightarrow{A}\times \overrightarrow{B}=0$ $\overrightarrow{A}$ = 0, does it necessarily follow that A = 0 or B = 0? Explain.

If$\overrightarrow{A}$ = 0 for a vector in the xy-plane, does it follow that A_x = -A_y? What can you say about A_x and A_y?

If $\overrightarrow{A}$ = 0 for a vector in the xy-plane, does it follow that A_x = -A_y? What can you say about A_x and A_y?

How many years older will you be 1.00 gigasecond from now? (Assume a 365-day year.)

While driving in an exotic foreign land, you see a speed limit sign that reads 180,000 furlongs per fortnight. How many miles per hour is this? (One furlong is 1/8 mile, and a fortnight is 14 days. A furlong originally referred to the length of a plowed furrow.)

While driving in an exotic foreign land, you see a speed limit sign that reads 180,000 furlongs per fortnight. How many miles per hour is this? (One furlong is 1/8 mile, and a fortnight is 14 days. A furlong originally referred to the length of a plowed furrow.)

A certain fuel-efficient hybrid car gets gasoline mileage of 55.0 mpg (miles per gallon). (a) If you are driving this car in Europe and want to compare its mileage with that of other European cars, express this mileage in km/L (1L = liter). Use the conversion factors in Appendix E. (b) If this car’s gas tank holds 45 L, how many tanks of gas will you use to drive 1500 km?

The following conversions occur frequently in physics and are very useful. (a) Use 1 mi = 5280 ft and 1 h = 3600 s to convert 60 mph to units of ft/s. (b) The acceleration of a freely falling object is 32 ft/s². Use 1 ft = 30.48 cm to express this acceleration in units of m/s². (c) The density of water is 1.0 g/cm³. Convert this density to units of kg/m³.

Question- Neptunium. In the fall of 2002, scientists at Los Alamos National Laboratory determined that the critical mass of neptunium-237 is about 60 kg. The critical mass of a fissionable material is the minimum amount that must be brought together to start a nuclear chain reaction. Neptunium-237 has a density of 19.5 g/cm³. What would be the radius of a sphere of this material that has a critical mass?

(a) The recommended daily allowance (RDA) of the trace metal magnesium is 410 mg/day for males. Express this quantity in µg/day. (b) For adults, the RDA of the amino acid lysine is 12 mg per kg of body weight. How many grams per day should a 75-kg adult receive? (c) A typical multivitamin tablet can contain 2.0 mg of vitamin B2 (riboflavin), and the RDA is 0.0030 g/day. How many such tablets should a person take each day to get the proper amount of this vitamin, if he gets none from other sources? (d) The RDA for the trace element selenium is 0.000070 g/day. Express this dose in mg/day.

Bacteria vary in size, but a diameter of$2.0\mu m$is not unusual. What are the volume (in cubic centimeters) and surface area (in square millimeters) of a spherical bacterium of that size? (Consult Appendix B for relevant formulas.)

With a wooden ruler, you measure the length of a rectangular piece of sheet metal to be 12 mm. With micrometer calipers, you measure the width of the rectangle to be 5.98 mm. Use the correct number of significant figures: What is (a) the area of the rectangle; (b) the ratio of the rectangle’s width to its length; (c) the perimeter of the rectangle; (d) the difference between the length and the width; and (e) the ratio of the length to the width?

A useful and easy-to-remember approximate value for the number of seconds in a year is π x ${10}^7$. Determine the percent error in this approximate value. (There are $365.24$ days in one year.)