A conical heap of wheat is formed on a flat horizontal concrete floor by discharging grain from an overhead chute. The vertical height of the cone is $h = 1.35\text{ m}$ and the base diameter is $D = 4.50\text{ m}$. The filling angle of repose of the wheat grain is ________ degrees (round off to two decimal places).
Answer Key:
30.96
Step-by-Step Solution & Explanation
Multi-Approach Solution:
**Method 1: Trigonometric Definition of Angle of Repose**
For a conical pile:
Base radius $R = \frac{D}{2} = \frac{4.50}{2} = 2.25\text{ m}$.
Height $h = 1.35\text{ m}$.
The filling angle of repose ($\theta$) is:
$\tan\theta = \frac{h}{R} = \frac{1.35}{2.25} = 0.60$
$\theta = \tan^{-1}(0.60) \approx 30.9638^\circ \approx 30.96^\circ$
**Method 2: Static Friction Coefficient Relation**
At the verge of sliding down the conical face, the gravitational component along the slope equals the static frictional force:
$m g \sin\theta = \mu_s m g \cos\theta \implies \mu_s = \tan\theta$
$\mu_s = \frac{2 h}{D} = \frac{2 \times 1.35}{4.50} = 0.60 \implies \theta = 30.96^\circ$