The problem involves a wheel rolling without slipping, and you need to find the displacement of point $P$, which is marked on the rim of the wheel. Let's break this down step by step:

Given:

• Radius of the wheel, $r = 0.40 \, \text{m}$.
• The wheel rolls without slipping.
• Point $P$ is initially at the point of contact between the wheel and the floor at time $t_1$.
• By time $t_2$, the wheel has rolled through half a revolution.

Understanding the motion:

When the wheel rolls without slipping, the point on the rim (point $P$) traces a cycloidal path. At $t_1$, $P$ is in contact with the ground, and by $t_2$, after half a revolution, $P$ is at the highest point on the wheel.

In one-half revolution:

• The wheel moves horizontally by a distance equal to half the circumference of the wheel.

  • Circumference of the wheel = $2\pi r = 2\pi(0.40) = 2.513 \, \text{m}$.
  • Half the circumference = $\pi r = \pi(0.40) = 1.257 \, \text{m}$.

• Vertically, point $P$ moves from the ground to the top of the wheel, a vertical distance equal to the wheel's diameter (since point $P$ is now at the top of the wheel after half a revolution).

  • Vertical displacement = $2r = 2(0.40) = 0.80 \, \text{m}$.

Total displacement of point $P$:

The displacement of $P$ is the straight-line distance between its initial and final positions. This is a combination of horizontal and vertical displacements, which form a right triangle. The hypotenuse (total displacement) can be found using the Pythagorean theorem:

$d = \sqrt{(\text{horizontal displacement})^2 + (\text{vertical displacement})^2}$

$d = \sqrt{(1.257)^2 + (0.80)^2}$

$d = \sqrt{1.580 + 0.640} = \sqrt{2.22} \approx 1.49 \, \text{m}$

Final Answer:

The displacement of point $P$ after the wheel has rolled through half a revolution is approximately 1.49 meters.

Would you like further details or clarifications? Here are some related questions to deepen understanding:

1. How would the displacement change if the radius of the wheel were doubled?
2. What is the velocity of point $P$ when it is at the highest point of the wheel?
3. How would the displacement change if the wheel rolled through a full revolution?
4. Can you describe the cycloidal path traced by point $P$ in this scenario?
5. How does rolling without slipping differ from sliding motion in this context?

Tip: For rolling without slipping, the velocity of the point of contact with the ground is momentarily zero at the instant it touches the surface.