The period of a simple pendulum, defined as the time necessary for one complete oscillation, is measured in time units and is given by

$$

T=2 \pi \sqrt{\frac{\ell}{g}}

$$

where $\ell$ is the length of the pendulum and $g$ is the acceleration due to gravity, in units of length divided by time squared. Show that this equation is dimensionally consistent. (You might want to check the formula using your keys at the end of a string and a stopwatch.)

Averell H.

Carnegie Mellon University

(a) Suppose the displacement of an object is related to time according to the expression $x=B t^{2}$ . What are the dimensions of $B$ ? (b) A displacement is related to time as $x=A \sin (2 \pi f t)$, where $A$ and $f$ are constants. Find the dimensions of $A .$ Hint: A trigonometric function appearing in an equation must be dimensionless.

Averell H.

Carnegie Mellon University

A shape that covers an area $A$ and has a uniform height $h$ has a volume $V=A h .$ (a) Show that $V=A h$ is dimensionally correct. (b) Show that the volumes of a cylinder and of a rectangular box can be written in the form $V=A h,$ identifying $A$ in each case. (Note that $A,$ sometimes called the "footprint" of the object, can have any shape and that the height can, in general, be replaced by the average thickness of the object.)

Averell H.

Carnegie Mellon University

Each of the following equations was given by a student during an examination: ( a ) $\frac{1}{2} m v^{2}=\frac{1}{2} m v_{0}^{2}+\sqrt{m g h}(\mathrm{b}) v=v_{0}$ $+a t^{2}(\mathrm{c}) m a=v^{2} .$ Do dimensional analysis of each equation and explain why the equation can't be correct.

Averell H.

Carnegie Mellon University

Newton's law of universal gravitation is represented by

$$

F=G \frac{M m}{r^{2}}

$$

where $F$ is the gravitational force, $M$ and $m$ are masses, and $r$ is a length. Force has the SI units $\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2}$ . What are the SI units of the proportionality constant $G$ ?

Averell H.

Carnegie Mellon University

Kinetic energy $K E$ (Topic 5) has dimensions $\mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}$ It can be written in terms of the momentum $p(\text { Topic } 6)$ and mass $m$ as

$$

K E=\frac{p^{2}}{2 m}

$$

(a) Determine the proper units for momentum using dimensional analysis. (b) Refer to Problem 5 . Given the units of force, write a simple equation relating a constant force $F$ exerted on an object, an interval of time $t$ during which the force is applied, and the resulting momentum of the object, p.

Averell H.

Carnegie Mellon University

A rectangular airstrip measures 32.30 $\mathrm{m}$ by $210 \mathrm{m},$ with the width measured more accurately than the length. Find the area, taking into account significant figures.

Averell H.

Carnegie Mellon University

Use the rules for significant figures to find the answer to the addition problem $21.4+15+17.17+4.003 .$

Averell H.

Carnegie Mellon University

A carpet is to be installed in a room of length 9.72 $\mathrm{m}$ and width 5.9 $\mathrm{m}$ . Find the area of the room retaining the proper number of significant figures.

Averell H.

Carnegie Mellon University

Use your calculator to determine $(\sqrt{8})^{3}$ to three significant figures in two ways: (a) Find $\sqrt{8}$ to four significant figures; then cube this number and round to three significant figures. (b) Find $\sqrt{8}$ to three significant figures; then cube this number and round to three significant figures. (c) Which answer is more accurate? Explain.

Averell H.

Carnegie Mellon University

How many significant figures are there in (a) $78.9 \pm 0.2,(\mathrm{b})$ $3.788 \times 10^{9},(\mathrm{c}) 2.46 \times 10^{26},(\mathrm{d}) 0.0032$

Averell H.

Carnegie Mellon University

The speed of light is now defined to be $2.99792458 \times 10^{8} \mathrm{m} / \mathrm{s}$ . Express the speed of light to (a) three significant figures, (b) five significant figures, and (c) seven significant figures.

Averell H.

Carnegie Mellon University

A rectangle has a length of $(2.0 \pm 0.2) \mathrm{m}$ and a width of $(1.5$ $\pm 0.1 ) \mathrm{m}$ . Calculate (a) the area and (b) the perimeter of the rectangle, and give the uncertainty in each value.

Averell H.

Carnegie Mellon University

The radius of a circle is measured to be $(10.5 \pm 0.2) \mathrm{m}$ . Calculate (a) the area and (b) the circumference of the circle, and give the uncertainty in each value.

Averell H.

Carnegie Mellon University

The edges of a shoebox are measured to be $11.4 \mathrm{cm}, 17.8 \mathrm{cm},$ and 29 $\mathrm{cm} .$ Determine the volume of the box retaining the proper number of significant figures in your answer.

Averell H.

Carnegie Mellon University

Carry out the following arithmetic operations: (a) the sum of the measured values $756,37.2,0.83,$ and $2.5 ;(\mathrm{b})$ the product $0.0032 \times 356.3 ;(\mathrm{c})$ the product $5.620 \times \pi .$

Averell H.

Carnegie Mellon University

The Roman cubitus is an ancient unit of measure equivalent to about 0.445 $\mathrm{m}$ . Convert the $2.00-\mathrm{m}$ height of a basketball forward to cubiti.

Averell H.

Carnegie Mellon University

A house is advertised as having 1420 square feet under roof. What is the area of this house in square meters?

Averell H.

Carnegie Mellon University

A fathom is a unit of length, usually reserved for measuring the depth of water. A fathom is approximately 6 $\mathrm{ft}$ in length. Take the distance from Earth to the Moon to be 250000 miles, and use the given approximation to find the distance in fathoms.

Averell H.

Carnegie Mellon University

A small turtle moves at a speed of 186 furlongs per fortnight. Find the speed of the turtle in centimeters per second. Note that 1 furlong $=220$ yards and 1 fortnight $=14$ days.

Averell H.

Carnegie Mellon University

A firkin is an old British unit of volume equal to 9 gallons. How many cubic meters are there in 6.00 firkins?

Averell H.

Carnegie Mellon University

Find the height or length of these natural wonders in kilometers, meters, and centimeters: (a) The longest cave system in the world is the Mammoth Cave system in Central Kentucky, with a mapped length of 348 miles. (b) In the United States, the waterfall with the greatest single drop is Ribbon Falls in California, which drops 1612 $\mathrm{ft}$ (c) At 20320 feet, Mount Mckinley in Alaska is America's highest mountain. (d) The deepest canyon in the United States is King's Canyon in California, with a depth of 8200 $\mathrm{ft}$ .

Averell H.

Carnegie Mellon University

A car is traveling at a speed of 38.0 $\mathrm{m} / \mathrm{s}$ on an interstate high-way where the speed limit is 75.0 $\mathrm{mi} / \mathrm{h}$ . Is the driver exceeding the speed limit? Justify your answer.

Averell H.

Carnegie Mellon University

A certain car has a fuel efficiency of 25.0 miles per gallon $(\mathrm{mi} /$ gal). Express this efficiency in kilometers per liter (km/L).

Averell H.

Carnegie Mellon University

The diameter of a sphere is measured to be 5.36 in. Find (a) the radius of the sphere in centimeters, (b) the surface area of the sphere in square centimeters, and (c) the volume of the sphere in cubic centimeters.

Averell H.

Carnegie Mellon University

Suppose your hair grows at the rate of 1$/ 32$ inch per day. Find the rate at which it grows in nanometers per second. Because the distance between atoms in a molecule is on the order of $0.1 \mathrm{nm},$ your answer suggests how rapidly atoms are assembled in this protein synthesis.

Averell H.

Carnegie Mellon University

The speed of light is about $3.00 \times 10^{8} \mathrm{m} / \mathrm{s}$ . Convert this figure to miles per hour.

Averell H.

Carnegie Mellon University

A house is 50.0 $\mathrm{ft}$ long and 26 $\mathrm{ft}$ wide and has 8.0 $\mathrm{ft}$ -high ceilings. What is the volume of the interior of the house in cubic meters and in cubic centimeters?

Averell H.

Carnegie Mellon University

The amount of water in reservoirs is often measured in acre-ft. One acre-ft is a volume that covers an area of one acre to a depth of one foot. An acre is 43560 $\mathrm{ft}^{2}$ . Find the volume in SI units of a reservoir containing 25.0 acre-ft of water.

Averell H.

Carnegie Mellon University

The base of a pyramid covers an area of 13.0 accres $(1 \text { acre }=$ 43560 $\mathrm{ft}^{2}$ ) and has a height of 481 $\mathrm{ft}$ (Fig. $\mathrm{P} 1.30 )$ . If the volume of a pyramid is given by the expression $V=b h / 3,$ where $b$ is the area of the base and $h$ is the height, find the volume of this

pyramid in cubic meters.

Averell H.

Carnegie Mellon University

A quart container of ice cream is to be made in the form of a cube. What should be the length of a side, in centimeters? (Use the conversion 1 gallon $=3.786$ liter.)

Averell H.

Carnegie Mellon University

Estimate the number of steps you would have to take to walk a distance equal to the circumference of the Earth.

Averell H.

Carnegie Mellon University

Estimate the number of breaths taken by a human being during an average lifetime.

Averell H.

Carnegie Mellon University

Estimate the number of people in the world who are suffering from the common cold on any given day. (Answers may vary. Remember that a person suffers from a cold for about a week.)

Averell H.

Carnegie Mellon University

The habitable part of Earthâ€™s surface has been estimated to cover 60 trillion square meters. Estimate the percent of this area occupied by humans if Earthâ€™s current population stood packed together as people do in a crowded elevator.

Averell H.

Carnegie Mellon University

Treat a cell in a human as a sphere of radius 1.0 mm. (a) Determine the volume of a cell. (b) Estimate the volume of your body. (c) Estimate the number of cells in your body.

Averell H.

Carnegie Mellon University

An automobile tire is rated to last for 50 000 miles. Estimate the number of revolutions the tire will make in its lifetime.

Averell H.

Carnegie Mellon University

A study from the National Institutes of Health states that the human body contains trillions of microorganisms that make up 1% to 3% of the bodyâ€™s mass. Use this information to estimate the average mass of the bodyâ€™s approximately 100 trillion microorganisms.

Averell H.

Carnegie Mellon University

A point is located in a polar coordinate system by the coordinates $r=2.5 \mathrm{m}$ and $\theta=35^{\circ} .$ Find the $x$ - and $y$ -coordinates of this point, assuming that the two coordinate systems have the same origin.

Averell H.

Carnegie Mellon University

A certain corner of a room is selected as the origin of a rectangular coordinate system. If a fly is crawling on an adjacent wall at a point having coordinates $(2.0,1.0),$ where the units are meters, what is the distance of the fly from the corner of the room?

Averell H.

Carnegie Mellon University

Express the location of the fly in Problem 40 in polar coordinates.

Averell H.

Carnegie Mellon University

Two points in a rectangular coordinate system have the coordinates $(5.0,3.0)$ and $(-3.0,4.0),$ where the units are centimeters. Determine the distance between these points.

Averell H.

Carnegie Mellon University

Two points are given in polar coordinates by $(r, \theta)=(2.00 \mathrm{m},$ $50.0^{\circ} )$ and $(r, \theta)=\left(5.00 \mathrm{m},-50.0^{\circ}\right),$ respectively. What is the distance between them?

Averell H.

Carnegie Mellon University

Given points $\left(r_{1}, \theta_{1}\right)$ and $\left(r_{2}, \theta_{2}\right)$ in polar coordinates, obtain a general formula for the distance between them. Simplify it as much as possible using the identity $\cos ^{2} \theta+\sin ^{2} \theta=1$ Hint: Write the expressions for the two points in Cartesian coordinates and substitute into the usual distance formula.

Averell H.

Carnegie Mellon University

For the triangle shown in Figure $P 1.45,$ what are $(a)$ the length of the unknown side, (b) the tangent of $\theta,$ and $(c)$ the sine of $\phi ?$

Averell H.

Carnegie Mellon University

A ladder 9.00 $\mathrm{m}$ long leans against the side of a building. If the ladder is inclined at an angle of $75.0^{\circ}$ to the horizontal, what is the horizontal distance from the bottom of the ladder to the building?

Averell H.

Carnegie Mellon University

A high fountain of water is located at the center of a circular pool as shown in Figure P1.47. Not wishing to get his feet wet, a student walks around the pool and measures its circumference to be 15.0 $\mathrm{m}$ . Next, the student stands at the edge of the pool and uses a protractor to gauge the angle of elevation at the bottom of the fountain to be $55.0^{\circ} .$ How high is the fountain?

Averell H.

Carnegie Mellon University

A right triangle has a hypotenuse of length $3.00 \mathrm{m},$ and one of its angles is $30.0^{\circ} .$ What are the lengths of (a) the side opposite the $30.0^{\circ}$ angle and (b) the side adjacent to the $30.0^{\circ}$ angle?

Averell H.

Carnegie Mellon University

In Figure P1.49, find (a) the side opposite $\theta,(b)$ the side adjacent to $\phi,(c) \cos \theta,(\mathrm{d}) \sin \phi,$ and (e) $\tan \phi .$

Averell H.

Carnegie Mellon University

In a certain right triangle, the two sides that are perpendicular to each other are 5.00 $\mathrm{m}$ and 7.00 $\mathrm{m}$ long. What is the length of the third side of the triangle?

Averell H.

Carnegie Mellon University

In Problem 50 , what is the tangent of the angle for which 5.00m is the opposite side?

Averell H.

Carnegie Mellon University

A woman measures the angle of elevation of a mountaintop as $12.0^{\circ}$ . After walking 1.00 $\mathrm{km}$ closer to the mountain on level ground, she finds the angle to be $14.0^{\circ} .$ Find the mountain's height, neglecting the height of the woman's eyes above the ground. Hint: Distances from the mountain x and $x-1 \mathrm{km}$ ) and the mountain's height are unknown. Draw two triangles, one for each of the woman's locations, and equate expressions for the mountain's height. Use that expression to find the first distance $x$ from the mountain and substitute to find the mountain's height.

Averell H.

Carnegie Mellon University

A surveyor measures the distance across a straight river by the following method: starting directly across from a tree on the opposite bank, he walks $x=1.00 \times 10^{2} \mathrm{m}$ along the riverbank to establish a baseline. Then he sights across to the tree. The angle from his baseline to the tree is $\theta=35.0^{\circ}(\mathrm{Fig.} \text { P1.53 })$ . How wide is the river?

Averell H.

Carnegie Mellon University

Vector $\vec{A}$ has a magnitude of 8.00 units and makes an angle of $45.0^{\circ}$ with the positive $x$ -axis. Vector $\overrightarrow{\mathbf{B}}$ also has a magnitude of 8.00 units and is directed along the negative $x$ -axis. Using graphical methods, find (a) the vector sum $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ and (b) the vector difference $\overline{\mathbf{A}}-\overrightarrow{\mathbf{B}}$ .

Averell H.

Carnegie Mellon University

Vector $\vec{A}$ has a magnitude of 29 units and points in the positive $y$ -direction. When vector $\overrightarrow{\mathbf{B}}$ is added to $\overrightarrow{\mathbf{A}},$ the resultant vector $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$ points in the negative $y$ -direction with a magnitude of 14 units. Find the magnitude and direction of $\overrightarrow{\mathrm{B}}$ .

Averell H.

Carnegie Mellon University

An airplane flies $2.00 \times 10^{2} \mathrm{km}$ due west from city A to city $\mathrm{B}$ and then $3.00 \times 10^{2} \mathrm{km}$ in the direction of $30.0^{\circ}$ north of west from city $\mathrm{B}$ to city $\mathrm{C}$ (a) In straight-line distance, how far is city C from city A? (b) Relative to city $A,$ in what direction is city $\mathrm{C} ?$ (c) Why is the answer only approximately correct?

Averell H.

Carnegie Mellon University

Vector $\overrightarrow{\mathrm{A}}$ is 3.00 units in length and points along the positive $x$ -axis. Vector $\overrightarrow{\mathrm{B}}$ is 4.00 units in length and points along the negative $y$ -axis. Use graphical methods to find the magnitude and direction of the vectors (a) $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$ and $(\mathrm{b}) \overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}$

Averell H.

Carnegie Mellon University

A force $\overrightarrow{\mathbf{F}}_{1}$ of magnitude 6.00 units acts on an object at the origin in a direction $\theta=30.0^{\circ}$ above the positive $x$ -axis $(\text { Fig. } \mathrm{P} 1.58) .$ A second force $\overrightarrow{\mathrm{F}}_{2}$ of magnitude 5.00 units acts on the object in the direction of the positive $y$ -axis. Find graphically the magnitude and direction of the resultant force $\overrightarrow{\mathbf{F}}_{1}+\overrightarrow{\mathbf{F}}_{2} .$

Averell H.

Carnegie Mellon University

A roller coaster moves $2.00 \times 10^{2} \mathrm{ft}$ horizontally and then rises 135 $\mathrm{ft}$ at an angle of $30.0^{\circ}$ above the horizontal. Next, it travels 135 $\mathrm{ft}$ at an angle of $40.0^{\circ}$ below the horizontal. Use graphical techniques to find the roller coaster's displacement

from its starting point to the end of this movement.

Averell H.

Carnegie Mellon University

Calculate (a) the $x$ -component and (b) the $y$ -component of the vector with magnitude 24.0 $\mathrm{m}$ and direction $56.0^{\circ} .$

Averell H.

Carnegie Mellon University

A vector $\vec{A}$ has components $A_{x}=-5.00 \mathrm{m}$ and $A_{y}=9.00 \mathrm{m} .$ Find (a) the magnitude and (b) the direction of the vector.

Averell H.

Carnegie Mellon University

A person walks $25.0^{\circ}$ north of east for 3.10 $\mathrm{km} .$ How far due north and how far due east would she have to walk to arrive at the same location?

Averell H.

Carnegie Mellon University

The magnitude of vector $\vec{A}$ is 35.0 units and points in the direction $325^{\circ}$ counterclockwise from the positive $x$ -axis. Calculate the $x$ - and $y$ -components of this vector.

Averell H.

Carnegie Mellon University

A figure skater glides along a circular path of radius 5.00 $\mathrm{m}$ . If she coasts around one half of the circle, find (a) her distance from the starting location and (b) the length of the path she skated.

Averell H.

Carnegie Mellon University

A girl delivering newspapers covers her route by traveling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east.

(a) What is her final position relative to her starting location?

(b) What is the length of the path she walked?

Averell H.

Carnegie Mellon University

A quarterback takes the ball from the line of scrimmage, runs backwards for 10.0 yards, and then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 50.0 -yard forward pass straight downfield, perpendicular to the line of scrimmage. How far is the football from its original

location?

Averell H.

Carnegie Mellon University

A vector has an $x$ -component of $-25.0$ units and a $y$ -component of 40.0 units. Find the magnitude and direction of the vector.

Averell H.

Carnegie Mellon University

A map suggests that Atlanta is $730 .$ miles in a direction $5.00^{\circ}$ north of east from Dallas. The same map shows that Chicago is 560 . miles in a direction $21.0^{\circ}$ west of north from Atlanta. Figure $\mathrm{P} 1.68$ shows the location of these three cities. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.

Averell H.

Carnegie Mellon University

The eye of a hurricane passes over Grand Bahama Island in a direction $60.0^{\circ}$ north of west with a speed of 41.0 $\mathrm{km} / \mathrm{h}$ . Three hours later the course of the hurricane suddenly shifts due north, and its speed slows to 25.0 $\mathrm{km} / \mathrm{h}$ . How far from Grand Bahama is the hurricane 4.50 $\mathrm{h}$ after it passes over the island?

Averell H.

Carnegie Mellon University

The helicopter view in Figure $P 1.70$ shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown and (b) the force a third person would have to exert on the mule to make the net force equal to zero. The forces are measured in units of newtons (N).

Averell H.

Carnegie Mellon University

A commuter airplane starts from an airport and takes the route shown in Figure $P 1.71 .$ The plane first flies to city $A$ located 175 $\mathrm{km}$ away in a direction $30.0^{\circ}$ north of east. Next, it flies for $150 . \mathrm{km} 20.0^{\circ}$ west of north, to city $B$ . Finally, the plane flies $190 . \mathrm{km}$ due west, to city $C$ . Find the location of city $C$ relative to the location of the starting point.

Averell H.

Carnegie Mellon University

(a) Find a conversion factor to convert from miles per hour to kilometers per hour. (b) For a while, federal law mandated that the maximum highway speed would be 55 mi/h. Use the conversion factor from part (a) to find the speed in kilometers per hour. (c) The maximum highway speed has been raised to 65 mi/h in some places. In kilometers per hour, how much of an increase is this over the 55-mi/h limit?

Averell H.

Carnegie Mellon University

The displacement of an object moving under uniform acceleration is some function of time and the acceleration. Suppose we write this displacement as $s=k a^{m} t^{n},$ where $k$ is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if $m=1$ and $n=2$ . Can the analysis give the value of $k ?$

Averell H.

Carnegie Mellon University

Assume it takes 7.00 minutes to fill a 30.0 -gal gasoline tank, (a) Calculate the rate at which the tank is filled in gallons per second. (b) Calculate the rate at which the tank is filled in cubic meters per second. (c) Determine the time interval, in hours, required to fill a $1.00-\mathrm{m}^{3}$ volume at the same rate. (1 U.S. gal = 231 $\mathrm{in} .^{3}$ )

Averell H.

Carnegie Mellon University

One gallon of paint (volume $=3.79 \times 10^{-3} \mathrm{m}^{3} )$ covers an area of 25.0 $\mathrm{m}^{2}$ . What is the thickness of the fresh paint on the wall?

Averell H.

Carnegie Mellon University

A sphere of radius $r$ has surface area $A=4 \pi r^{2}$ and volume $V=(4 / 3) \pi r^{3} .$ If the radius of sphere 2 is double the radius of sphere $1,$ what is the ratio of (a) the areas, $A_{2} / A_{1}$ and (b) the volumes, $V_{2} / V_{1} ?$

Averell H.

Carnegie Mellon University

Assume there are 100 million passenger cars in the United States and that the average fuel consumption is 20 $\mathrm{mi} / \mathrm{gal}$ of gasoline. If the average distance traveled by each car is 10000

$\mathrm{mi} / \mathrm{yr}$ , how much gasoline would be saved per year if average fuel consumption could be increased to 25 $\mathrm{mi} / \mathrm{gal} ?$

Averell H.

Carnegie Mellon University

In 2015 , the U.S. national debt was about $\$ 18$ tillion. (a) If payments were made at the rate of $\$ 1000$ per second, how many years would it take to pay off the debt, assuming that no interest were charged? (b) A dollar bill is about 15.5 cm long. If 18 trillion dollar bills were laid end to end around the Earth's equator, how many times would they encircle the planet? Take the radius of the Earth at the equator to be 6378 $\mathrm{km}$ . (Note Before doing any of these calculations, try to guess at the answers. You may be very surprised.)

Averell H.

Carnegie Mellon University

(a) How many Earths could fit inside the Sun? (b) How many of Earth's Moons could fit inside the Earth?

Averell H.

Carnegie Mellon University

An average person sneezes about three times per day. Estimate the worldwide number of sneezes happening in a time interval approximately equal to one sneeze.

Averell H.

Carnegie Mellon University

The nearest neutron star (a collapsed star made primarily of neutrons) is about $3.00 \times 10^{18} \mathrm{m}$ away from Earth. Given that the Milky Way galaxy (Fig. P1.81) is roughly a disk of diameter $\sim 10^{21} \mathrm{m}$ and thickness $\sim 10^{19} \mathrm{m},$ estimate the number of neutron stars in the Milky Way to the nearest order of magnitude.

Averell H.

Carnegie Mellon University