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μ = m 1 m 2 m 1 + m 2

is known as “reduced mass”. It has the same unit as that of “mass”. It represents the “effect” of two bodies, if we want to treat “two body” system as “one body” system.

In the nutshell, we can treat motion of “two body” system along a straight line as “one body” system, which has a mass equal to “μ” and acceleration equal to relative acceleration, “ a r ”.

Velocity of approach

The two bodies are approaching towards each other. Hence, magnitude of velocity of approach is given by :

Velocity of approach

Two bodies are approaching each other along a straight line.

v r = v 1 + v 2

This is the expression of the magnitude of velocity of approach. We can write corresponding vector equation for velocity of approach in relation to reference direction. In this module, however, we will avoid vector notation or vector interpretation to keep the discussion simplified.

Kinetic energy

The kinetic energy of the “two body” system is given as the sum of kinetic energy of individual bodies,

K = 1 2 m 1 v 1 2 + 1 2 m 2 v 2 2

We can write this expression of kinetic energy in terms of relative velocity i.e. velocity of approach. For this, we need to express individual speeds in terms of relative speed as :

v r = v 1 + v 2

But,

m 1 v 1 = m 2 v 2

v 2 = m 1 v 1 m 2

Substituting in the expression of relative velocity,

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

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

Similarly,

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

Now, putting these expressions of individual speed in the equation of kinetic energy :

K = m 1 m 2 2 v r 2 m 1 + m 2 2 + m 2 m 1 2 v r 2 m 1 + m 2 2

K = m 1 m 2 v r 2 m 1 + m 2 2 m 1 + m 2

K = m 1 m 2 v r 2 m 1 + m 2

K = 1 2 μ v r 2

This result also indicates that we can treat “two body” system as “one body” system from the point of view of kinetic energy, as if the body has reduced mass of “μ” and speed equal to the magnitude of relative velocity, “ v r ”.

Example

Problem 1: Two masses “ m 1 ” and “ m 2 ” are initially at rest at a great distance. At a certain instant, they start moving towards each other, when released from their positions. Considering absence of any other gravitational field, calculate velocity of approach when they are at a distance “r” apart.

Solution : The bodies are at large distance in the beginning. There is no external gravitational field. Hence, we can consider initial gravitational energy of the system as zero (separated y infinite distance). Also, the bodies are at rest in the beginning. The initial kinetic energy is also zero. In turn, initial mechanical energy of the system is zero.

Let “ v r ” be the velocity of approach, when the bodies are at a distance “r” apart. Applying conservation of mechanical energy, we have :

K i + U i = K f + U f

0 + 0 = 1 2 μ v r 2 G m 1 m 2 r

v r 2 = 2 G m 1 m 2 μ r = 2 G m 1 m 2 m 1 + m 2 m 1 m 2 r

v r = { 2 G m 1 + m 2 r }

Conclusions

From the discussion above, we conclude the followings about the motion of “two body” system along a straight line :

1: Each body follows a straight line trajectory.

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: Magnitude of gravitational force is same for two bodies, but they vary as the distance between them changes.

5: We can treat “two body” system equivalent to “one body” system by using concepts of (i) reduced mass “μ” (ii) relative velocity, “ v r ” and (iii) relative acceleration “ a r ”.

Questions & Answers

which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad
what is the definition of resolution of forces
Atinuke Reply
what is energy?
James Reply
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
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Benjamin
can anyone tell who founded equations of motion !?
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n=a+b/T² find the linear express
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How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
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A balloon is released from the ground which rises vertically up with acceleration 1.4m/sec^2.a ball is released from the balloon 20 second after the balloon has left the ground. The maximum height reached by the ball from the ground is
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Torque
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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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