<< Chapter < Page Chapter >> Page >

Example (Gravitational Force)

The mass of the sun is 2 × 10 30 kg size 12{2 times "10" rSup { size 8{"30"} } ` ital "kg"} {} . The mass of the earth is 5 . 97 × 10 24 kg size 12{5 "." "97" times "10" rSup { size 8{"24"} } ital "kg"} {} . The approximate distance between the sun and the earth is 149 . 6 × 10 6 km . size 12{"149" "." 6 times "10" rSup { size 8{6} } ital "km" "." } {}

Question: What is the magnitude of the gravitational force that the sun exerts on the earth? (Express the result in scientific notation.)

Solution:

F = ( 6 . 67 × 10 11 N m 2 / kg 2 ) ( 2 × 10 30 kg ) ( 5 . 97 × 10 24 kg ) / ( 149 . 6 × 10 6 km ) 2 size 12{F= \( 6 "." "67"` times `"10" rSup { size 8{ - "11"} } `N - m rSup { size 8{2} } / ital "kg" rSup { size 8{2} } \) ` \( 2 times "10" rSup { size 8{"30"} } ital "kg" \) \( 5 "." "97" times "10" rSup { size 8{"24"} } ital "kg" \) /` \( "149" "." 6 times "10" rSup { size 8{6} } ital "km" \) rSup { size 8{2} } } {}
F = ( 6 . 67 × 10 11 N m 2 / kg 2 ) ( 2 × 10 30 kg ) ( 5 . 97 × 10 24 kg ) / ( 149 . 6 × 10 9 m ) 2 size 12{F= \( 6 "." "67"` times `"10" rSup { size 8{ - "11"} } `N - m rSup { size 8{2} } / ital "kg" rSup { size 8{2} } \) ` \( 2 times "10" rSup { size 8{"30"} } ital "kg" \) \( 5 "." "97" times "10" rSup { size 8{"24"} } ital "kg" \) /` \( "149" "." 6 times "10" rSup { size 8{9} } m \) rSup { size 8{2} } } {}
F = ( 6 . 67 × 10 11 N m 2 ) ( 2 × 10 30 ) ( 5 . 97 × 10 24 ) / ( 149 . 6 × 10 9 m ) 2 size 12{F= \( 6 "." "67"` times `"10" rSup { size 8{ - "11"} } `N - m rSup { size 8{2} } \) ` \( 2 times "10" rSup { size 8{"30"} } \) \( 5 "." "97" times "10" rSup { size 8{"24"} } \) /` \( "149" "." 6 times "10" rSup { size 8{9} } m \) rSup { size 8{2} } } {}
F = ( 6 . 67 × 10 11 N m 2 ) ( 2 × 10 30 ) ( 5 . 97 × 10 24 ) / ( 22 , 380 × 10 18 m 2 ) size 12{F= \( 6 "." "67"` times `"10" rSup { size 8{ - "11"} } `N - m rSup { size 8{2} } \) ` \( 2 times "10" rSup { size 8{"30"} } \) \( 5 "." "97" times "10" rSup { size 8{"24"} } \) /` \( "22","380" times "10" rSup { size 8{"18"} } m rSup { size 8{2} } \) } {}
F = ( 6 . 67 × 10 11 N ) ( 2 × 10 30 ) ( 5 . 97 × 10 24 ) / ( 22 , 380 × 10 18 ) size 12{F= \( 6 "." "67"` times `"10" rSup { size 8{ - "11"} } `N \) ` \( 2 times "10" rSup { size 8{"30"} } \) \( 5 "." "97" times "10" rSup { size 8{"24"} } \) /` \( "22","380" times "10" rSup { size 8{"18"} } \) } {}
F = ( 6 . 67 × 2 × 5 . 97 ) × ( 10 11 × 10 30 × 10 24 ) N / ( 2 . 238 × 10 22 ) size 12{F= \( 6 "." "67"` times `2 times 5 "." "97" \) times \( "10" rSup { size 8{ - "11"} } times "10" rSup { size 8{"30"} } times "10" rSup { size 8{"24"} } \) `N/` \( 2 "." "238" times "10" rSup { size 8{"22"} } \) } {}
F = ( 79 . 64 × 10 43 ) N / ( 2 . 238 × 10 22 ) size 12{F= \( "79" "." "64" times "10" rSup { size 8{"43"} } \) `N/` \( 2 "." "238" times "10" rSup { size 8{"22"} } \) } {}
F = ( 79 . 64 / 2 . 238 ) × 10 43 22 N size 12{F= \( "79" "." "64"/2 "." "238" \) times "10" rSup { size 8{"43" - "22"} } `N} {}
F = ( 35 . 6 ) × 10 21 N size 12{F= \( "35" "." 6 \) times "10" rSup { size 8{"21"} } `N} {}
F = 3 . 56 × 10 22 N size 12{F=3 "." "56" times "10" rSup { size 8{"22"} } `N} {}

Coulomb’s law

Electrons and protons are examples of charged particles. In the case of the proton, charge is positive. The charge of a proton is thus said to have positive polarity. On the other hand, an electron has a negative charge. Thus the charge of an electron is said to have negative polarity.

An electrostatic force exists between two charged particles. If the charges associated with the two particles are of the same polarity, the electrostatic force will be repulsive . On the other hand if the charges for two particles have opposite polarity, the electrostatic force will be attractive .

Through extensive laboratory work, the physicist Charles Coulomb first established a mathematical expression that calculates the magnitude of the electrostatic force that results from the interaction of two charged particles. This expression which is known as Coulomb’s law for electrostatic forces is given by

F = k q Q r 2 size 12{F=k` left ( { {q`Q} over {r rSup { size 8{2} } } } right )} {}

where q and Q are the values of the charges measured in the units Coulombs, r is the distance measured in meters that separates the charged particles and k is a constant

k = 9 × 10 9 Newton meters 2 / Coulomb 2 size 12{k=9` times `"10" rSup { size 8{9} } ` ital "Newton" - ital "meters" rSup { size 8{2} } / ital "Coulomb" rSup { size 8{2} } } {}

It is interesting to note the similarity of the general form of Coulomb’s law for electrostatic forces with Newton’s law of gravitational force. Just as was the case with the gravitational force, the magnitude of the electrostatic force decreases with the square of the distance separating the two particles.

Example (Electrostatic Force)

The charge on an electron is given by the equation

q e = 1 . 60 × 10 19 Coulombs ( C ) size 12{q rSub { size 8{e} } = - 1 "." "60" times "10" rSup { size 8{ - "19"} } ital "Coulombs"` \( C \) } {}

The charge associated with a proton is

q p =+ 1 . 60 × 10 19 Coulombs ( C ) size 12{q rSub { size 8{p} } "=+"1 "." "60" times "10" rSup { size 8{ - "19"} } ital "Coulombs"` \( C \) } {}

Suppose that an electron and a proton are separated by a distance of 10 6 nanometers . size 12{"10" rSup { size 8{ - 6} } ital "nanometers" "." } {}

Questions: What is the magnitude of the electrostatic force that the protonexerts on the electron? Is the force attractive or repulsive?

Solution:

F = k q e q p r 2 size 12{F=k` { {q rSub { size 8{e} } `q rSub { size 8{p} } } over {r rSup { size 8{2} } } } } {}
F = ( 9 × 10 9 N m 2 / C 2 ) ( 1 . 60 × 10 19 C ) ( 1 . 60 × 10 19 C ) ( 10 6 × nanometers ) 2 size 12{F= { { \( 9` times `"10" rSup { size 8{9} } `N - m rSup { size 8{2} } /C rSup { size 8{2} } \) ` \( - 1 "." "60" times "10" rSup { size 8{ - "19"} } C \) \( 1 "." "60" times "10" rSup { size 8{ - "19"} } C \) } over { \( "10" rSup { size 8{ - 6} } times ital "nanometers" \) rSup { size 8{2} } } } } {}
F = ( 9 × 10 9 ) ( 1 . 60 × 10 19 ) ( 1 . 60 × 10 19 ) N m 2 ( 10 6 × 10 9 m ) 2 size 12{F= { { \( 9` times `"10" rSup { size 8{9} } \) ` \( - 1 "." "60" times "10" rSup { size 8{ - "19"} } \) \( 1 "." "60" times "10" rSup { size 8{ - "19"} } \) `N - m rSup { size 8{2} } } over { \( "10" rSup { size 8{ - 6} } times "10" rSup { size 8{ - 9} } `m \) rSup { size 8{2} } } } } {}
F = ( 9 ) ( 1 . 60 ) ( 1 . 60 ) ( 10 9 × 10 19 × 10 19 ) N m ( 10 15 m ) 2 size 12{F= { { \( 9` \) ` \( - 1 "." "60" \) \( 1 "." "60" \) ` \( "10" rSup { size 8{9} } times "10" rSup { size 8{ - "19"} } times "10" rSup { size 8{ - "19"} } \) N - m} over { \( "10" rSup { size 8{ - "15"} } `m \) rSup { size 8{2} } } } } {}
F = ( 23 . 0 × 10 29 ) N ( 10 15 ) 2 size 12{F= { { \( "23" "." 0` times "10" rSup { size 8{ - "29"} } \) N} over { \( "10" rSup { size 8{ - "15"} } \) rSup { size 8{2} } } } } {}
F = 23 . 0 × 10 29 × 10 30 N size 12{F="23" "." 0 times "10" rSup { size 8{ - "29"} } times "10" rSup { size 8{"30"} } `N} {}
F = 23 . 0 × 10 1 N size 12{F="23" "." 0 times "10" rSup { size 8{1} } `N} {}
F = 230 N size 12{F="230"`N} {}

Because the charges on the electron and the proton differ in polarity, the electrostatic force is attractive .

Exercises

  1. The weight of the Space Shuttle is 4,470,000 lb. Express this weight in scientific notation.
  2. The weight of a honey bee is 0.000 385 05 lb. Express this weight in scientific notation.
  3. Perform the following multiplications. (a) (37.5 x 10 7 ) x (2.87 x 10 5 ), (b) (37.5 x 10 7 ) x (2.87 x 10 -5 ).
  4. Perform the following divisions. (a) (37.5 x 10 7 ) / (2.87 x 10 5 ), (b) (37.5 x 10 7 ) / (2.87 x 10 -5 ).
  5. The area of the United States is 9.83 x 10 6 km 2 . Its population is 3.10 x 10 6 . What is the population density (people/km 2 ) of the United States?
  6. The distance between the Sun and the Earth is 1.47 x 10 11 m. The speed of light is 299,792,458 m/s. How long does it take for light to travel between the Sun and the Earth?
  7. The planned Blythe Solar Power Plant in California is expected to produce 968 MW. The Aswan Dam Power Plant in Egypt produces 2.1 gigawatts of power. How may solar power plants with similar power production as that of the Blythe Solar Power Plant would be needed to match the power production of the Aswan Dam Power Plant?
  8. An electric circuit consists of a 9.00 V battery in series with a load that has a resistance of 13.78 kΩ. (a) Find the power delivered by the battery. (b) Find the power absorbed by the load.
  9. The mass of the Earth is 5.97 x 10 24 kg, while that of the moon is 7.36 x 10 22 kg. The average distance between the Earth and the Moon is 384,403 km. What is he force exerted on the Moon by the Earth?
  10. Two charges are separated by 4.66 x 10 -8 m. Each has a positive charge of 3.6 x 10 -18 Coulombs. What is the repulsive force that they exert on one another?

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Math 1508 (laboratory) engineering applications of precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11337/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Math 1508 (laboratory) engineering applications of precalculus' conversation and receive update notifications?

Ask