Let's address each part of the problem step by step.

Part (i): Find the $X$ and $Y$ distances of the object at $t = 8$ seconds

Given:

• Radius of the circle, $r = 10$ cm
• Angular velocity, $\omega = 0.2$ radians/second
• The object starts at the position $(10, 0)$ at $t = 0$

Step 1: Find the angle covered by the object at $t = 8$ seconds

The angle $\theta$ covered by the object after time $t$ is given by: $\theta = \omega t$

Substituting the given values: $\theta = 0.2 \times 8 = 1.6 \text{ radians}$

Step 2: Determine the $X$ and $Y$ coordinates at $t = 8$ seconds

The position of the object in Cartesian coordinates $(X, Y)$ can be found using the parametric equations for circular motion: $X = r \cos(\theta)$ $Y = r \sin(\theta)$

Substitute $r = 10$ cm and $\theta = 1.6$ radians: $X = 10 \cos(1.6)$ $Y = 10 \sin(1.6)$

Calculating the values: $X \approx 10 \times (-0.0292) \approx -0.292 \text{ cm}$ $Y \approx 10 \times 0.9996 \approx 9.996 \text{ cm}$

So, the coordinates at $t = 8$ seconds are approximately: $(X, Y) \approx (-0.292, 9.996) \text{ cm}$

Part (ii): Find the $(X, Y)$ location of the object at $t = 20$ seconds

Step 1: Find the angle covered by the object at $t = 20$ seconds

$\theta = \omega t = 0.2 \times 20 = 4 \text{ radians}$

Step 2: Determine the $X$ and $Y$ coordinates at $t = 20$ seconds

Substitute $r = 10$ cm and $\theta = 4$ radians into the parametric equations:

$X = 10 \cos(4)$ $Y = 10 \sin(4)$

Calculating the values: $X \approx 10 \times (-0.6536) \approx -6.536 \text{ cm}$ $Y \approx 10 \times (-0.7568) \approx -7.568 \text{ cm}$

So, the coordinates at $t = 20$ seconds are approximately: $(X, Y) \approx (-6.536, -7.568) \text{ cm}$

Part (iii): Find the time taken to complete one full rotation

To complete one full rotation, the object needs to cover an angle of $2\pi$ radians.

The time $T$ to complete one full rotation can be calculated as: $T = \frac{2\pi}{\omega}$

Substituting $\omega = 0.2$ radians/second: $T = \frac{2\pi}{0.2} = \frac{2\pi}{1/5} = 10\pi \text{ seconds}$

Calculating the value: $T \approx 10 \times 3.1416 \approx 31.416 \text{ seconds}$

Summary of the Answers:

1. The $(X, Y)$ coordinates at $t = 8$ seconds are approximately $(-0.292, 9.996)$ cm.
2. The $(X, Y)$ coordinates at $t = 20$ seconds are approximately $(-6.536, -7.568)$ cm.
3. The time taken to complete one full rotation is approximately $31.416$ seconds.

Would you like more details or have any questions about the steps?

Here are 5 related questions:

1. What would be the $(X, Y)$ coordinates at $t = 15$ seconds?
2. How would the solution change if the object had a different initial position?
3. What is the physical significance of angular velocity in circular motion?
4. How can we determine the speed of the object in its circular path?
5. How does the radius affect the time taken to complete one full rotation?

Tip: In circular motion problems, always start by finding the angle covered over the time interval, as it directly determines the position of the object on the circle.