To find the displacement of point $P$ on the wheel when it turns through an angle of $210^\circ$, we can follow these steps:

1. Convert the Angle to Radians: Displacement in a circular path is typically calculated using radians. $\theta = 210^\circ \times \left(\frac{\pi \text{ radians}}{180^\circ}\right) = \frac{7\pi}{6} \text{ radians}$

2. Calculate the Arc Length: The arc length $s$, which represents the distance traveled along the edge of the wheel, is given by: $s = R \times \theta$ Substituting the values: $s = 4 \times \frac{7\pi}{6} \approx 14.66 \text{ meters}$

3. Displacement Vector: Displacement in this context refers to the straight-line distance between the initial and final positions of point $P$. For a wheel turning less than $360^\circ$, this can be found using the chord length formula: $\text{Displacement} = 2R \sin\left(\frac{\theta}{2}\right)$ $\text{Displacement} = 2 \times 4 \times \sin\left(\frac{7\pi}{12}\right)$ Calculate the sine value: $\sin\left(\frac{7\pi}{12}\right) \approx \sin(105^\circ) \approx 0.966$ $\text{Displacement} \approx 2 \times 4 \times 0.966 \approx 7.73 \text{ meters}$

Therefore, the displacement of point $P$ is approximately $7.73$ meters.

Would you like more details or have any questions about this problem?

Here are 5 related questions to expand on this topic:

1. How does the displacement change if the wheel turns by a different angle?
2. What is the relationship between the radius and the displacement for a fixed angle of rotation?
3. How would the displacement be affected if the wheel had a different radius?
4. What is the significance of the chord length in circular motion problems?
5. How does the arc length compare to the displacement for the same angle of rotation?

Tip: When dealing with rotational motion, always convert angles to radians for accurate calculations in physics and engineering.