We will have the following:
First, we write down the values given:
![\begin{cases}M_1=1.0\operatorname{kg} \\ r=50.0cm=0.5m \\ M_2=20.0g=0.02\operatorname{kg} \\ I=0.310\operatorname{kg}\cdot m^2\end{cases}]()
Then, from definition of intertia, we will have that:
Here "n" is the number of spokes the wheel has, so:
![I_{\text{rim}}=M_1\cdot r^2\Rightarrow I_{\text{rim}}=(1.0\operatorname{kg})(0.5m)^2\Rightarrow I_{\text{rim}}=0.25\operatorname{kg}\cdot m^2]()
&
![I_{\text{spoke}}=(1)/(3)\cdot M_2\cdot r^2\Rightarrow I_{\text{spoke}}=(1)/(3)(0.02\operatorname{kg})(0.5m)^2\Rightarrow I_{\text{spoke}}=\frac{_{}1}{600}kg\cdot m^2]()
Now, replacing the values, we will have that:
![0.310\operatorname{kg}\cdot m^2=0.25\operatorname{kg}\cdot m^2+n((1)/(600)kg\cdot m^2)\Rightarrow0.06\operatorname{kg}\cdot m^2=n((1)/(600)kg\cdot m^2)]()
So, the number of spokes is 36.
Now, we calculate the mass of the wheel:
Here, we will have that:
Where "Mw" is the mass of the wheel. So, we replace the values:
![M_w=(1.00\operatorname{kg})+36(0.02\operatorname{kg})\Rightarrow M_w=(43)/(25)kg]()
![\Rightarrow M_w=1.72\operatorname{kg}]()
So, the mass of the wheel is 1.72 kg.