<< Chapter < Page Chapter >> Page >

M = m 2 r 2 r 1 + m 2 = m 2 r 1 + r 2 r 1

m 2 = M r 1 r 1 + r 2

Substituting in the equation, involving angular velocity,

ω 2 = G M r 1 r 1 r 1 + r 2 3 = G M r 3

ω = G M r 3

This expression has identical form as for the case when a body revolves around another body at rest along a circular path (compare with “Earth – satellite” system). Here, combined mass “M” substitutes for the mass of heavier mass at the center and sum of the linear distance replaces the radius of rotation.

The linear velocity is equal to the product of the radius of circle and angular velocity. Hence,

v 1 = ω r 1

v 2 = ω r 2

Time period

We can easily find the expression for time period of revolution as :

T = 2 π ω = 2 π r 3 2 G M

This expression also has the same form as for the case when a body revolves around another body at rest along a circular path (compare with “Earth – satellite” system). Further squaring on either side, we have :

T 2 r 3

Moment of inertia

Here, we set out to find moment of inertia of the system about the common axis passing through center of mass and perpendicular to the plane of rotation. For this, we consider each of the bodies as point mass. Note that two bodies are rotating about a common axis with same angular velocity. Clearly, MI of the system is :

I = m 1 r 1 2 + m 2 r 2 2

We can express individual distance in terms of their sum using following two equations,

r = r 1 + r 2

m 1 r 1 = m 2 r 2

Substituting for “ r 1 ” in the equation or "r", we have :

r = m 2 r 2 m 1 + r 2 = r 2 m 1 + m 2 m 1

r 2 = r m 1 m 1 + m 2

Similarly, we can express, “ r 1 ” as :

r 1 = r m 2 m 1 + m 2

Substituting for “ r 1 ” and “ r 2 ” in the expression of moment of inertia,

I = m 1 m 2 2 r 2 m 1 + m 2 2 + m 2 m 1 2 r 2 m 1 + m 2 2

I = m 1 m 2 r 2 m 1 + m 2 2 X m 1 + m 2

I = m 1 m 2 r 2 m 1 + m 2

I = μ r 2

This expression is similar to the expression of momemnt of inertia of a particle about an axis at a perpendicualr distance, "r". It is, therefore, clear that “Two body” system orbiting around center of mass can be treated as “one body” system by using concepts of net distance “r” and reduced mass “μ”.

Angular momentum

The bodies move about the same axis with the same sense of rotation. The angular momentum of the system, therefore, is algebraic sum of individual angular momentums.

L = L 1 + L 2 = m 1 r 1 2 ω + m 2 r 2 2 ω

Substituting for “ r 1 ” and “ r 2 ” with expressions as obtained earlier,

L = m 1 m 2 2 r 2 ω m 1 + m 2 2 + m 2 m 1 2 r 2 ω m 1 + m 2 2

L = m 1 m 2 r 2 ω m 1 + m 2 2 X m 1 + m 2

L = m 1 m 2 r 2 ω m 1 + m 2

L = μ r 2 ω

This expression is similar to the expression of angular momemntum of a particle about an axis at a perpendicualr distance, "r". Once again, we see that “Two body” system orbiting around center of mass can be treated as “one body” system by using concepts of net distance “r” and reduced mass “μ”.

Kinetic energy

The kinetic energy of the system is equal to the algebraic sum of the kinetic energy of the individual body. We write expression of kinetic energy in terms of angular velocity – not in terms of linear velocity. It is so because angular velocity is same for two bodies and can, therefore, be used to simplify the expression for kinetic energy. Now, kinetic energy of the system is :

K = 1 2 m 1 r 1 2 ω 2 + 1 2 m 2 r 2 2 ω 2

Substituting for “ r 1 ” and “ r 2 ” with expressions as obtained earlier,

K = m 1 m 2 2 r 2 ω 2 2 m 1 + m 2 2 + m 2 m 1 2 r 2 ω 2 2 m 1 + m 2 2

K = m 1 m 2 r 2 ω 2 2 m 1 + m 2 2 X m 1 + m 2

K = m 1 m 2 r 2 ω 2 2 m 1 + m 2

K = 1 2 μ r 2 ω 2

This expression of kinetic energy is also similar to the expression of kinetic energy of a particle rotating about an axis at a perpendicualr distance, "r". Thus, this result also substantiates equivalence of “Two body” system as “one body” system, using concepts of net distance “r” and reduced mass “μ”.

Example

Problem 1 : In a binary star system, two stars of “m” and “M” move along two circular trajectories. If the distance between stars is “r”, then find the total mechanical energy of the system. Consider no other gravitational influence on the system.

Solution : Mechanical energy of the system comprises of potential and kinetic energy. Hence,

E = 1 2 μ r 2 w 2 G M m r

We know that angular velocity for “two body” system in circular motion is given by :

ω = { G M + m r 3 }

Also, reduced mass is given by :

μ = M m M + m

Putting in the expression of mechanical energy,

E = m M r 2 G m + M 2 m + M r 3 G M m r

E = G M m 2 r G M m r

E = G M m 2 r

Conclusions

Thus, we conclude the following :

1: Each body follows a circular path about center of mass.

2: The line joining centers of two bodies pass through center of mass.

3: The planes of two motions are in the same plane. In other words, two motions are coplanar.

4: The angular velocities of the two bodies are equal.

5: The linear distance between two bodies remains constant.

6: Magnitude of gravitational force is constant and same for two bodies.

7: Magnitude of centripetal force required for circular motion is constant and same for two bodies.

8: Since linear velocity is product of angular velocity and distance from the center of revolution, it may be different if the radii of revolutions are different.

9: We can treat two body system with an equivalent one body system by using concepts of (i) combined mass, “M”, (ii) net distance “r” and (iii) reduced mass “μ”.

Two body system - circular motion

Two body system as equivalent to one body system.

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, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics for k-12' conversation and receive update notifications?

Ask
Tess Armstrong
Start Quiz
George Turner
Start Quiz
Rylee Minllic
Start Quiz