NEET Class neet Answered

Find the acceleration a and tension T, T1 T2, T3 in terms of M1, m2 ,M3 and g

Asked by prakriti12oct | 17 Apr, 2020, 11:46: PM

Expert Answer

Figure shows the forces acting on masses m₁ , m₂ and m₃ . Tension forces acting on pulley also shown in figure.

Since masses m₁ and m₂ are moving downward with acceleration a , by applying Newtons second law,

we get the following equations.

( m₁ + m₂ ) g - T₁ = m₁ a ......................(1)

T₁ - T₂ = m₂ a ........................(2)

since masses m₂ and m₃ are connected by same string , Tension forces T₂ and T₃ are same

T₂ = T₃ ............................(3)

By applying Newton's second law on block of mass m₃ , we get

T₃ - m₃ g = m₃ a ........................(4)

By adding all the above equations (1), (2), (3) and (4), we get

( m₁ + m₂ - m₃ )g = ( m₁ + m₂ + m₃ ) a ..................... (5)

Hence acceleration a is obtained from above equation as ,

begin mathsize 14px style a space equals space open parentheses fraction numerator m subscript 1 space plus space m subscript 2 space minus space m subscript 3 over denominator m subscript 1 space plus space m subscript 2 space plus space m subscript 3 end fraction close parentheses space g end style

.....................(6)

Tension force T₁ is obtained from (1) as , T₁ = ( m₁ + m₂ ) g- m₁ a ....................(7)

By substituting for acceleration a in eqn.(7) , we get T₁ after simplification as

begin mathsize 14px style T subscript 1 space equals space open parentheses fraction numerator m subscript 1 m subscript 2 space plus space m subscript 2 superscript 2 space plus space 2 space m subscript 1 space m subscript 3 space plus space m subscript 2 m subscript 3 over denominator m subscript 1 plus m subscript 2 plus m subscript 3 end fraction close parentheses space g end style

Tension force T₃ is obtained from (4) as , T₃ = m₃(a+g) ....................(8)

By substituting for acceleration a in eqn.(8) , we get T₃ after simplification as

begin mathsize 14px style T subscript 3 space equals space 2 open parentheses fraction numerator left parenthesis space m subscript 1 space plus space m subscript 2 space right parenthesis space m subscript 3 over denominator m subscript 1 plus m subscript 2 plus m subscript 3 end fraction close parentheses space g end style

...................(9)

T = T₂ + T₃ and T₂ = T₃ . Hence we get T as

begin mathsize 14px style T space equals space 4 space open parentheses fraction numerator left parenthesis space m subscript 1 space plus space m subscript 2 space right parenthesis space m subscript 3 over denominator m subscript 1 plus m subscript 2 plus m subscript 3 end fraction close parentheses space g end style

Answered by Thiyagarajan K | 18 Apr, 2020, 05:21: PM

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