Zinc reacts with H2SO4 to form Zincsulphate and hydrogen gas is also liberated . Zn(s) + H2SO4(aq) -----> ZnSO4(aq) + H2(g)
It is clear that Zn displaces Hydrogen from sulphuric acid. This happens beacause zinc is more reactive than hydrogen. This is a displacement reaction.
But copper does not react with H2SO4 as copper is less reactive than hydrogen. So, no reaction takes place.