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