解：$(1)U_{1}=\frac 13U，$所以$R_{1}=\frac 13(R_{1}+R_{2})，$得$R_{2}=2R_{1}=10\ \mathrm {Ω}$
$(2)I_{ }=\frac {U_{ }}{R_{1 }+R_{ 2 }}=\frac {{ 3 }\ \text {V}}{{ 5 }Ω+{ 10 }Ω}={0.2 }\ \text {A}$
$Q_{ 1 }=I_{ }^2R_{ 1 }t_{ }=({ 0.2 }\ \text {A})^2×{ 5 }Ω×{ 60 }\ \text {s}={ 12 }\ \text {J}$
