Three uniform rods of lengths ℓ , 2 ℓ and 3 ℓ , masses m , 2 m and 3 m and coefficient of linear expansion α , 2 α and 3 α are placed in contact on a smooth horizontal surface as shown in figure. The

Question

Three uniform rods of lengths ℓ , 2 ℓ and 3 ℓ , masses m , 2 m and 3 m and coefficient of linear expansion α , 2 α and 3 α are placed in contact on a smooth horizontal surface as shown in figure. The temperature of the system is increased by T . By what distance (in mm) the point B will move. (Take ℓ α T = 29 12 ​ mm, ℓ = 1 m)

TestHub · Accepted Answer

Let e = ℓ α T . Expansions: rod 1: e ; rod 2: 2 ℓ ( 2 α ) T = 4 e rod 3: 3 ℓ ( 3 α ) T = 9 e No horizontal external force: the centre of mass does not move. Initial centre positions (A at 0, ℓ = 1 ): 0.5 , 2 , 4.5 . Let A move by s . New centres: s + 0.5 ( 1 + e ) s + 2 + 3 e s + 4.5 + 9.5 e CM unchanged: m ( s + 0.5 + 0.5 e ) + 2 m ( s + 2 + 3 e ) + 3 m ( s + 4.5 + 9.5 e ) = m ( 0.5 ) + 2 m ( 2 ) + 3 m ( 4.5 ) 6 s + 35 e = 0 ⇒ s = − 6 35 e ​ Point B (end of rod 1) moves by s + e ( 1 ) = e − 6 35 e ​ = − 6 29 e ​ i.e. 6 29 ​ × 29 12 ​ = 2 mm (towards A). Answer: 2.

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