Subtopic: Design and selection of machine elements – gears, pulleys, chains and sprockets and belts
Question Statement:
A V-belt operates in a grooved pulley with a total included groove angle of $2\beta = 38^\circ$ (semi-groove angle $\beta = 19^\circ$). The coefficient of friction between the belt rubber and the steel sheave is $\mu = 0.25$. Taking into account the wedging action in the groove, the virtual (effective) coefficient of friction $\mu'$ is ________ (round off to 2 decimal places).
Answer Key:
0.77
Step-by-Step Solution & Explanation
```
V-Belt Groove Wedging Action:
\ / Included Angle 2*beta = 38 deg
\/ beta = 19 deg
Normal Reactions R_N on both side flanks
```
**Step-by-step Calculation:**
1. Resolving forces in the groove cross-section:
$R = 2 R_N \sin\beta \implies R_N = \frac{R}{2\sin\beta}$
2. Frictional force along the belt circumference:
$F_f = 2 \mu R_N = 2 \mu \left(\frac{R}{2\sin\beta}\right) = \left(\frac{\mu}{\sin\beta}\right) R$
3. The virtual coefficient of friction is defined as:
$\mu' = \frac{\mu}{\sin\beta}$
$\mu = 0.25, \quad \beta = 19^\circ$
$\sin 19^\circ = 0.32557$
$\mu' = \frac{0.25}{0.32557} = 0.76788 \approx 0.77$