# Solution to Hartnell Governor Problem Let's address each part step by step. --- ## **Given Data** - **Mass of each ball, \( m \) = 4 kg** - **Mean radius of rotation, \( r \) = 150 mm = .15 m** - **Spring stiffness, \( k \) = 18 kN/m = 18,000 N/m** - **Initial spring compression, \( x_ \) = 20 mm = .02 m** --- ## **1. Calculate the Equilibrium Speed (\( N \))** ### **Hartnell Governor Principle** The centrifugal force exerted by the balls is balanced by the spring force: \[ F_c = m \omega^2 r \] \[ \text{Total spring force} = k x_ \] \[ \Rightarrow 2 m \omega^2 r = k x_ \] - \( \omega \): Angular speed in rad/s - \( N \): Speed in rpm \[ \omega = 2\pi N / 60 \] ### **Calculation** \[ 2 \times 4 \times \omega^2 \times .15 = 18,000 \times .02 \] \[ 1.2 \omega^2 = 360 \] \[ \omega^2 = \frac{360}{1.2} = 300 \] \[ \omega = \sqrt{300} = 17.32 \text{ rad/s} \] Convert to \( N \) (rpm): \[ \omega = \frac{2\pi N}{60} \] \[ 17.32 = \frac{2\pi N}{60} \] \[ N = \frac{17.32 \times 60}{2\pi} \] \[ N = \frac{1039.2}{6.2832} \] \[ N \approx 165.4 \text{ rpm} \] --- ## **2. Determine the Range of Speed** ### **Range of Speed Formula** The range is based on the **maximum and minimum compression** of the spring. Let \( x_1 \) be the maximum compression and \( x_2 \) be the minimum compression. #### **Let’s Assume:** - **Maximum additional compression (\( \Delta x \)) = 10 mm = .01 m** (as a typical value for illustration) So, - **Maximum compression**: \( x_{max} = x_ + \Delta x = .02 + .01 = .03 \) m - **Minimum compression**: \( x_{min} = x_ - \Delta x = .02 - .01 = .01 \) m #### **Maximum Speed (\( N_{max} \)):** \[ 2 m \omega_{max}^2 r = k x_{max} \] \[ 1.2 \omega_{max}^2 = 18,000 \times .03 = 540 \] \[ \omega_{max}^2 = 540 / 1.2 = 450 \] \[ \omega_{max} = 21.21 \text{ rad/s} \] \[ N_{max} = \frac{21.21 \times 60}{2\pi} = \frac{1272.6}{6.2832} = 202.5 \text{ rpm} \] #### **Minimum Speed (\( N_{min} \)):** \[ 1.2 \omega_{min}^2 = 18,000 \times .01 = 180 \] \[ \omega_{min}^2 = 180 / 1.2 = 150 \] \[ \omega_{min} = 12.25 \text{ rad/s} \] \[ N_{min} = \frac{12.25 \times 60}{2\pi} = \frac{735}{6.2832} = 117 \text{ rpm} \] #### **Range of Speed** \[ \text{Range} = N_{max} - N_{min} = 202.5 - 117 = 85.5 \text{ rpm} \] --- ## **3. Advantage of Hartnell Governor over Watt Governor** **Hartnell governors can operate at higher speeds and have a more sensitive and quicker response to speed changes than Watt governors. They are also more compact and can be adjusted for different springs and loads.** --- ## **Summary Table** | Quantity | Value | |---------------------|---------------| | Equilibrium Speed | 165.4 rpm | | Maximum Speed | 202.5 rpm | | Minimum Speed | 117 rpm | | Range of Speed | 85.5 rpm | | Advantage | More sensitive and compact; suitable for higher speeds | --- ### **If you need calculations with a different range or more details, let me know!**