Discussion
The figure shows a schematic of a simple Watt governor mechanism with the spindle O1O2 rotating at an angular velocity ω 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 O1 is denoted by h. The value of h = 400 mm at a certain ω. If ω is doubled, the value of h will be _________ mm.
Answer: B
The new height is 100 mm.
For a Watt governor with massless arms and sleeve, balancing the centrifugal force on a ball against its weight about the pivot gives the governor height
\[ h = \frac{g}{\omega^{2}} \]
independent of the arm lengths. Doubling \(\omega\) divides \(h\) by four: \(400/4 = 100\) mm.
200 mm assumes \(h\) falls in proportion to \(\omega\), not \(\omega^{2}\), and 50 mm divides by eight. 150 mm has no basis. Check with the data: \(\omega^{2} = 9.8/0.4 = 24.5\) rad2/s2 at first, and four times that gives \(h = 9.8/98 = 0.1\) m.
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