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Centre of Mass

Two blocks A ...

Question

Two blocks A and B of mass m and 2m are connected by a massless spring of force constant k. They are placed on a smooth horizontal plane. Spring is stretched by an amount x and then released. The relative velocity of the blocks when the spring comes to its natural length is

(a) [√(3k/2m)]x (b) [√(2k/3m)]x

(c) √(2kx/m) (d) √(3km/x)

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Solution

In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem.
So the two masses in the problem are converted to a single mass which is given by
$straight mu equals begin inline style fraction numerator straight m subscript 1 straight m subscript 2 over denominator straight m subscript 1 plus straight m subscript 2 end fraction end style Here space straight m subscript 1 equals straight m$
$straight m subscript 2 equals 2 straight m So space straight mu equals begin inline style fraction numerator straight m left parenthesis 2 straight m right parenthesis over denominator straight m plus 2 straight m end fraction end style equals begin inline style fraction numerator 2 straight m squared over denominator 3 straight m end fraction end style equals begin inline style 2 over 3 end style straight m$
$Let space straight V subscript straight r space end subscript be space the space relative space velocity space of space two space blocks. By space conservation space of space mechanical space energy comma begin inline style 1 half end style kx squared equals begin inline style 1 half end style μV subscript straight r superscript 2 kx squared equals μV subscript straight r superscript 2 kx squared equals begin inline style 2 over 3 end style mV subscript straight r superscript 2 straight V subscript straight r superscript 2 equals begin inline style fraction numerator 3 kx squared over denominator 2 straight m end fraction end style straight V subscript straight r equals square root of begin inline style fraction numerator 3 straight k over denominator 2 straight m end fraction end style end root space space straight x$

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In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. So the two masses in the problem are converted to a single mass which is given by