To find the angular velocity of the ball at the highest point of its journey about the point of projection, we can follow these steps:
### Step 1: Determine the initial velocity components
The ball is projected with an initial velocity \( u = 20\sqrt{2} \, \text{m/s} \) at an angle \( \theta = 45^\circ \) with the horizontal. We can find the horizontal and vertical components of the initial velocity.
\[
u_x = u \cos \theta = 20\sqrt{2} \cos 45^\circ = 20\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 20 \, \text{m/s}
\]
\[
u_y = u \sin \theta = 20\sqrt{2} \sin 45^\circ = 20\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 20 \, \text{m/s}
\]
### Step 2: Find the velocity at the highest point
At the highest point of its trajectory, the vertical component of the velocity becomes zero, and only the horizontal component remains.
\[
v = u_x = 20 \, \text{m/s}
\]
### Step 3: Calculate the maximum height
The maximum height \( H \) can be calculated using the formula:
\[
H = \frac{u_y^2}{2g}
\]
Substituting \( u_y = 20 \, \text{m/s} \) and \( g = 10 \, \text{m/s}^2 \):
\[
H = \frac{(20)^2}{2 \cdot 10} = \frac{400}{20} = 20 \, \text{m}
\]
### Step 4: Determine the radius of rotation
At the highest point, the radius \( R \) from the point of projection to the ball is equal to the horizontal distance traveled when the ball reaches the maximum height. Since the ball is projected at \( 45^\circ \), the horizontal range \( R \) can be calculated as:
\[
R = \frac{u^2 \sin 2\theta}{g} = \frac{(20\sqrt{2})^2 \sin 90^\circ}{10} = \frac{800}{10} = 80 \, \text{m}
\]
At the highest point, the horizontal distance from the point of projection is \( R/2 = 40 \, \text{m} \).
### Step 5: Calculate the angular velocity
The angular velocity \( \omega \) can be calculated using the formula:
\[
\omega = \frac{v}{R}
\]
Substituting \( v = 20 \, \text{m/s} \) and \( R = 40 \, \text{m} \):
\[
\omega = \frac{20}{40} = 0.5 \, \text{rad/s}
\]
### Final Answer
The angular velocity of the particle at the highest point of its journey about the point of projection is \( 0.5 \, \text{rad/s} \).
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