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Given Data:

• Mass of the rotor (m): 150 kg

• Radius of gyration (k): 250 mm = 0.25 m

• Initial speed (N₁): 1500 RPM

• Final speed (N₂): Same as driving member (1500 RPM)

• Time to attain speed (t): 40 s

• Radius ratio (r₂/r₁): 2

• Permissible pressure (P): 1 MPa = $1\times {10}^6$ Pa

• Coefficient of friction (μ): 0.2

• Cone angle (α): Assumed to be small (typically 10°–15°), but not given. For simplicity, we assume $\sin \alpha \approx \alpha$ (if α is small).

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Step 1: Moment of Inertia (I)

The moment of inertia of the rotor is:

$$\mathrm{<span\; style="background-color:\; white;">I</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}{\mathrm{<span\; style="background-color:\; white;">k</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">150</span>}<span\; style="background-color:\; white;">\times </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">0.25</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">9.375</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}$$

Step 2: Angular Acceleration (α)

The driven member accelerates from rest to 1500 RPM in 40 s.

$${\mathrm{<span\; style="background-color:\; white;">\omega </span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">rad/s</span>}<span\; style="background-color:\; white;">,</span>{\mathrm{<span\; style="background-color:\; white;">\omega </span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">N</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{\mathrm{<span\; style="background-color:\; white;">60</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">1500</span>}}{\mathrm{<span\; style="background-color:\; white;">60</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">157.08</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">rad/s</span>}$$$$\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">\omega </span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">\omega </span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}}{\mathrm{<span\; style="background-color:\; white;">t</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">157.08</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">0</span>}}{\mathrm{<span\; style="background-color:\; white;">40</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3.927</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">rad/s</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}$$

Step 3: Torque Required (T)

The torque required to accelerate the rotor:

$$\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">I</span>}\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">9.375</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3.927</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">36.82</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}$$

Step 4: Dimensions of the Clutch

For a cone clutch, the torque transmitted is:

$$\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">\mu </span>}\mathrm{<span\; style="background-color:\; white;">F</span>}}{\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\text{<span style="background-color: white;">avg</span>}}$$

Where:

• $F$ = Axial force (N)

• $r_{\text{avg}}=\frac{r_1+r_2}{2}$

• Given $r_2\mathrm{/}{r}_1=2$, let $r_1=r$, then $r_2=2r$, so $r_{\text{avg}}=1.5r$

The permissible pressure $P$ limits the axial force:

$$\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">P</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">A</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}$$

Where $A=2\pi r_{\text{avg}}b$ (contact area), and $b$ is the width of the friction surface.

But for simplicity, we assume uniform pressure and use:

$$\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">P</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">)</span>$$$$\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">6</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">(</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">r</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">6</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">3</span>}{\mathrm{<span\; style="background-color:\; white;">r</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">r</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">6</span>}}$$

Now, substitute $F$ into the torque equation:

$$\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">\mu </span>}\mathrm{<span\; style="background-color:\; white;">F</span>}}{\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\text{<span style="background-color: white;">avg</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">0.2</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">r</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">6</span>}}}{\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1.5</span>}\mathrm{<span\; style="background-color:\; white;">r</span>}$$$$\mathrm{<span\; style="background-color:\; white;">36.82</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">0.9</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">r</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">6</span>}}}{\mathrm{<span\; style="background-color:\; white;">sin</span>}<span\; style="background-color:\; white;"></span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}}$$

Assuming $\sin \alpha \approx 0.1736$ (for $\alpha ={10}^{\circ }$):

$$\mathrm{<span\; style="background-color:\; white;">36.82</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">0.9</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">r</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">6</span>}}}{\mathrm{<span\; style="background-color:\; white;">0.1736</span>}}$$$${\mathrm{<span\; style="background-color:\; white;">r</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">36.82</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">0.1736</span>}}{\mathrm{<span\; style="background-color:\; white;">0.9</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">6</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2.25</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}}$$$$\mathrm{<span\; style="background-color:\; white;">r</span>}<span\; style="background-color:\; white;">=</span>\sqrt[\mathrm{<span\; style="background-color:\; white;">3</span>}]{\mathrm{<span\; style="background-color:\; white;">2.25</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">6</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0.0131</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">m</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">13.1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">mm</span>}$$$${\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">13.1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">mm</span>}<span\; style="background-color:\; white;">,</span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">26.2</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">mm</span>}$$

Step 5: Axial Force (F)

Using $F=3\pi r^2\times {10}^6$:

$$\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">0.0131</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">6</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1618.5</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">N</span>}$$

Step 6: Heat Generated (Q)

The heat generated during engagement is equal to the work done to overcome friction:

$$\mathrm{<span\; style="background-color:\; white;">Q</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">\theta </span>}$$

Where $\theta$ is the angle rotated during engagement:

$$\mathrm{<span\; style="background-color:\; white;">\theta </span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">\omega </span>}}_{\text{<span style="background-color: white;">avg</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">\omega </span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">\omega </span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">0</span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">157.08</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">40</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">3141.6</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">rad</span>}$$$$\mathrm{<span\; style="background-color:\; white;">Q</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">36.82</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">3141.6</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">115</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">674</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">J</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">115.67</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">kJ</span>}$$

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Final Answers:

1. Dimensions of the friction surface:

   • Inner radius ($r_1$) = 13.1 mm

   • Outer radius ($r_2$) = 26.2 mm

2. Axial force required for engagement ($F$): 1618.5 N

3. Heat generated during engagement ($Q$): 115.67 kJ