To calculate the angle through which wheel B rotates, we need to find the length through which wheel A travels at 180°.
We are given
[tex]2P_A=P_B[/tex]
We can calculate the perimeter of the wheel as
[tex]P=2\pi r[/tex]
If
[tex]\begin{gathered} P_A=2\pi r \\ P_B=2\pi R \end{gathered}[/tex]
we will have
[tex]\begin{gathered} 2(2\pi r)=2\pi R \\ 4\pi r=2\pi R \\ \text{Therefore} \\ R=2r \end{gathered}[/tex]
The length travelled by wheel A in 180° is given as
[tex]\begin{gathered} L=\frac{\theta}{360}\times2\pi r \\ L=\frac{180}{360}\times2\pi r \\ \text{Therefore,} \\ L=\pi r \end{gathered}[/tex]
Since wheel B will travel the same length as wheel A, we can write out the expression as
[tex]L=\frac{\theta}{360}\times2\pi R[/tex]
Putting the value of L and R as gotten above, we have
[tex]\pi r=\frac{\theta}{360}\times2\times\pi\times2r[/tex]
Dividing through by πr and solving for θ, we have
[tex]\begin{gathered} 1=\frac{\theta}{360}\times4 \\ \theta=\frac{360}{4} \\ \theta=90 \end{gathered}[/tex]
The wheel B is rotated 90°.
