: A block of metal with mass 𝑀 M and initial temperature 𝑇 1 T 1 is placed into a perfectly insulated container holding a volume 𝑉 V of water at temperature 𝑇 2 T 2 ( 𝑇 2 > 𝑇 1 T 2 >T 1 ). After reaching thermal equilibrium, the final temperature is 𝑇 𝑓 T f . Assuming no heat loss to the surroundings, the specific heat capacity of the metal is 𝑐 𝑚 c m , and the specific heat capacity and density of water are 𝑐 𝑤 c w and 𝜌 𝑤 ρ w , respectively. Derive the expression for the final equilibrium temperature ( 𝑇 𝑓 T f ) in terms of the given variables ( 𝑀 , 𝑇 1 , 𝑉 , 𝑇 2 , 𝑐 𝑚 , 𝑐 𝑤 M,T 1 ,V,T 2 ,c m ,c w ).
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