Answer:
The angular velocity of the wheel is [tex]w = 0.1661 \ rad/ sec[/tex]
Explanation:
From the question we are told that
The mass of the toy train is [tex]m[/tex]
The speed of the train is [tex]v_t = 0.15 m/s[/tex]
The radius of the wheel is [tex]r = 0.43 \ m[/tex]
The mass of the wheel is [tex]m_w = 1.1 * m[/tex]
According to the law of conservation of momentum
[tex]L_i = L_f[/tex]
Where [tex]L _i[/tex] is the initial angular momentum which is mathematically represented as
[tex]L_i = rmv[/tex]
and
[tex]L_f[/tex] is the final angular momentum which is mathematically represented as
[tex]L_f = I * w[/tex]
Where I is the moment of inertia of the wheel which is mathematically represented as
[tex]I = m_w * r^2[/tex]
So
[tex]rmv = m_w r^2 w[/tex]
[tex]r * m * 0.15 = 1.1 * m * r^2 * w[/tex]
[tex]v = 1.1 * r * w[/tex]
But we know the train is moving relative to the wheel so
[tex]v = v_t - wr[/tex]
Where wr is the linear velocity component of the wheel so
Substituting values
[tex]0.15 - (w * 0.43) = 1.1 * 0.43 * w[/tex]
=> [tex]0.15 - (w * 0.43) = 0.473 * w[/tex]
[tex]0.15 = 0.903w[/tex]
[tex]w = 0.1661 \ rad/ sec[/tex]
