To find the angle of the displacement of point $P$ during the interval when the wheel rolls through one-half of a revolution, let's first visualize and calculate the problem.

Given:

• Radius of the wheel: $R = 36$ cm
• Displacement during one-half revolution: $1.34$ m

Displacement Calculation:

When the wheel rolls through one-half of a revolution, the point $P$ initially at the contact point with the ground moves to the top of the wheel. The path traced by $P$ during this motion can be visualized as part of a circle and a horizontal displacement.

However, to answer your question, we need to determine the angle of the displacement vector relative to the horizontal axis.

Analysis:

1. Horizontal Displacement: When the wheel rolls without slipping, the horizontal distance covered by the point $P$ is equal to half the circumference of the wheel: $\text{Horizontal Displacement} = \frac{1}{2} \times 2\pi R = \pi R = \pi \times 36 \, \text{cm} = 113.04 \, \text{cm} = 1.1304 \, \text{m}$

2. Vertical Displacement: The vertical displacement is equal to the diameter of the wheel (from the bottom to the top): $\text{Vertical Displacement} = 2R = 72 \, \text{cm} = 0.72 \, \text{m}$

3. Angle of the Displacement: The displacement vector's angle $\theta$ relative to the horizontal axis can be calculated using trigonometry: $\tan(\theta) = \frac{\text{Vertical Displacement}}{\text{Horizontal Displacement}}$ $\tan(\theta) = \frac{0.72}{1.1304}$

   $\theta = \tan^{-1}\left(\frac{0.72}{1.1304}\right)$

Let's calculate the angle $\theta$.

Calculation:

$\theta = \tan^{-1}\left(\frac{0.72}{1.1304}\right) \approx \tan^{-1}(0.6369) \approx 32.45^\circ$

So, the angle of the displacement of $P$ during this interval is approximately 32.45 degrees.

Would you like more details on the calculations or have any questions?

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Here are 5 related questions you might consider:

1. What is the total distance traveled by point $P$ in one complete revolution?
2. How does the displacement change if the wheel's radius is doubled?
3. What would be the vertical and horizontal displacement if the wheel rolls through a quarter of a revolution?
4. How does the no-slip condition affect the displacement of point $P$?
5. What would be the displacement angle if the wheel were rolling up an incline?

Tip: Understanding the relationship between circular motion and linear displacement is key in problems involving rolling motion.