Two particles with masses m1 and m2 are joined by a light spring of force constant k . They vibr

Two particles with masses m1 and m2 are joined by a light...

Key Concepts

Center of Mass

The center of mass of a system is the point at which the total mass of the system may be considered to be concentrated for analysis of translational motion. In problems with internal forces, decomposing motion into center-of-mass and relative components allows a clearer understanding of the dynamics, especially when the center of mass is fixed.

Reduced Mass

The reduced mass is a concept in mechanics that simplifies the analysis of two-body systems by transforming them into an equivalent one-body problem. It is defined as ? = m?m?/(m?+m?) and enables the study of relative motion between the two masses while accounting for their individual contributions to the system's dynamics.

Relative Motion

Relative motion refers to the movement of one particle with respect to another. By focusing on the difference in coordinates or speeds between the two particles, the dynamics of the interaction can be addressed independent of the overall motion of the system. This is particularly useful in simplifying the energy expressions of interacting bodies.

Energy Partitioning

Energy partitioning in the context of two-body systems involves expressing the total kinetic energy as the sum of the kinetic energy associated with the center-of-mass motion and the kinetic energy corresponding to the relative motion. This separation facilitates the analysis of internal oscillations or vibrations within the system.

Harmonic Oscillator

The harmonic oscillator is a fundamental model in physics that describes systems experiencing a restoring force proportional to the displacement from equilibrium, such as a spring-mass system. Its potential energy is given by ½ kx², and it plays a crucial role in demonstrating how energy is stored and exchanged between kinetic and potential forms during oscillatory motion.

Transcript

00:01 Given in this question is two particles with different masses m1 and m2 connected by spring.

00:08 And we are given the total energy is as such.

00:11 It makes sense because it's just the kinetic energy, the first particle, kinetic energy of the second particle, plus the elastic potential energy that is actually stored inside the spring.

00:23 So x squared is probably the extension of the spring from its natural length.

00:28 Now you want to show that it can be actually rewritten in the form of half m u squared plus half kx square so there's a few terms new terms introduced here where we have mu which is the reduced mass m1 m2 over m1 plus m2 and u u square where u is given as absolute value of u1 plus absolute value of u2 now first of we want to make use of the fact that the center of mass is fixed in order to find the relationship between u1 and u2 right with the set of mass fixed it means is that m1 times u1 the momentum from the first particle plus the momentum from the second particle.

01:44 This is the total momentum of the system must be zero.

01:52 In order for the center of mass to not move, this is the condition.

02:00 Using this we can actually find, sorry this is u2, u1 as we goes to negative of m2 divided by m1 times u2.

02:21 Now what we can do is we take the square root, sorry not square root, we take the absolute value on both sides and remember that our u where here is given as u1 absolute value plus u2 absolute value right which means that u2 is actually u minus absolute value of u1 which we can actually bring to the left hand side and divide the factors over to get, sorry, this is m1, m2 for m1 times u.

03:39 On the left side we have m1, m2 over m1 times u1.

03:52 So if you have factorize out the u1 and divide the factor to the right -hand side.

04:20 Now we can further simplify this, just multiply m1 across.

04:27 Denominator and this is our expression for the absolute value of u1 now we can do the same thing for our u2 right we just substitute in rather than substituting u2 we substitute in u1 substitute as u minus u2 right to get an expression for u2 this should end up with and m1 u over m1 plus m2.

05:12 So we have another expression for in terms of the u.

05:22 Now we can start to substitute in what is our u square, which is just absolute u square.

05:43 So we substitute in this term as well as the u1 square term into the original energy expression.

06:02 So remember that the original energy is half m1 times u1 square so this half m2 times u2 square plus half kx square so this term we ignore now in order to combine these two terms together right we need to find out all right how we can combine them by putting these two fractions together because they are the same denominator of m1 plus m2 square what's left in the denominator sorry the numerator is um 1 times m2 square u square sorry u square also a common term so i'm going to bring that out plus m1 square times m2 so what we can do is actually take out the common term m1 and m2 right m1 times m2 is the common term is the common term and what we're left with is m2 plus m1 and we see that we can actually get rid of one of the terms below sorry, i'm confusing myself with these brackets, right? this on top here, if we put in u square, because the entire bracket is squared, so this will just be a u.

08:13 So if you are to divide the numerator and the denominator, you have the common factor m2 plus m1.

08:23 Gaff is half u square times m1, m2 divided by m1 plus m2.

08:32 Not forgetting of course our half kx square which we have left out for now...

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