The figure shows a schematic of a simple Watt governor mechanism with the spindle \( O_1O_2 \) rotating at an angular velocity \( \omega \) about a vertical axis. The balls at P and S have equal mass. Assume that there is no friction anywhere and all other components are massless and rigid. The vertical distance between the horizontal plane of rotation of the balls and the pivot \( O_1 \) is denoted by \( h \). The value of \( h = 400 \, \text{mm} \) at a certain \( \omega \). If \( \omega \) is doubled, the value of \( h \) will be ________________ mm.
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In a Watt governor mechanism, the vertical distance between the horizontal plane of rotation and the pivot changes inversely with the square of the angular velocity.
Let’s analyze the system. The force acting on each ball in the governor mechanism is due to the centrifugal force, which causes the balls to move outward as the angular velocity increases. The centrifugal force is given by:
\[
F = m\omega^2r
\]
where \( m \) is the mass of the ball, \( \omega \) is the angular velocity, and \( r \) is the distance of the ball from the axis of rotation.
Since \( P \) and \( S \) have equal mass, the centrifugal force on both balls is equal. This leads to an increase in the vertical distance \( h \) when the angular velocity \( \omega \) is increased.
Now, let's consider the relationship between \( h \) and \( \omega \). The centrifugal force causes a displacement of the balls outward, and this displacement is proportional to \( \omega^2 \). If the angular velocity is doubled, the centrifugal force will quadruple, resulting in a change in the height \( h \). Since the height is inversely proportional to the square root of the centrifugal force, we have:
\[
h_2 = h_1 \times \left( \frac{\omega_1}{\omega_2} \right)^2
\]
Given that \( \omega_2 = 2\omega_1 \), we can substitute this into the equation:
\[
h_2 = h_1 \times \left( \frac{1}{2} \right)^2 = h_1 \times \frac{1}{4}
\]
Substituting the given value of \( h_1 = 400 \, \text{mm} \):
\[
h_2 = 400 \times \frac{1}{4} = 100 \, \text{mm}
\]
Thus, when \( \omega \) is doubled, the value of \( h \) will become 100 mm.
