Two masses M and m are connected at the two ends of an inextensible string. The string passes over a smooth frictionless pulley, calculate the acceleration of the masses and tension in the string, given M is greater than m.
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For mass m, the forces are given as
T - mg = ma ...... (1)
Similarly, for mass M, the forces are given as
Mg - T = Ma ...... (2)
From equation (1), we have
T = mg + ma ...... (3)
Substituting equation (3) in (2) we get
Mg - mg - ma = Ma
g(M - m) = a(M + m)
begin mathsize 12px style therefore straight a equals open parentheses fraction numerator straight M minus straight m over denominator straight M plus straight m end fraction close parentheses straight g end style
Substituting the value of a in equation 1,
begin mathsize 12px style straight T equals straight m cross times open parentheses fraction numerator straight M minus straight m over denominator straight M plus straight m end fraction close parentheses straight g plus mg
therefore straight T equals fraction numerator Mm minus straight m squared over denominator straight M plus straight m end fraction straight g plus mg
therefore straight T equals straight g open parentheses fraction numerator Mm minus straight m squared over denominator straight M plus straight m end fraction plus straight m close parentheses equals straight g open parentheses fraction numerator Mm minus straight m squared plus straight m squared plus Mm over denominator straight M plus straight m end fraction close parentheses equals straight g open parentheses fraction numerator 2 Mm over denominator straight M plus straight m end fraction close parentheses end style