To determine the angle $\theta$ that the ball makes with the vertical $y$-axis while it is moving in a horizontal circular path, we can use the following approach:

Step 1: Identify the forces acting on the ball

• The tension $T$ in the rope.
• The gravitational force $mg$ acting downward, where $m$ is the mass of the ball and $g$ is the acceleration due to gravity.
• The centripetal force $F_c$ that keeps the ball moving in a circular path, which is provided by the horizontal component of the tension.

Step 2: Analyze the components of the forces

Let the angle $\theta$ be the angle between the rope and the vertical axis (i.e., the $y$-axis).

1. Vertical component of the tension: $T_y = T \cos\theta = mg$

2. Horizontal component of the tension (providing the centripetal force): $T_x = T \sin\theta = \frac{mv^2}{r}$ Here, $v = 4 \, \text{m/s}$ is the speed of the ball, and $r = L \sin\theta$ is the horizontal distance from the pivot point to the ball, where $L = 1.8 \, \text{m}$ is the length of the rope.

Step 3: Use the equations to solve for $\theta$

We have the following system of equations:

1. $T \cos\theta = mg$
2. $T \sin\theta = \frac{mv^2}{L\sin\theta}$

Dividing these two equations to eliminate $T$: $\frac{\sin\theta}{\cos\theta} = \frac{v^2}{gL}$ $\tan\theta = \frac{v^2}{gL}$

Step 4: Calculate $\theta$

Now plug in the values: $\tan\theta = \frac{4^2}{9.81 \times 1.8} = \frac{16}{17.658} \approx 0.906$

Finally, find $\theta$ using the arctan (inverse tangent) function: $\theta = \arctan(0.906) \approx 42.17^\circ$

Final Answer

The ball is moving at an angle of approximately $42.17^\circ$ from the vertical $y$-axis.

Would you like further details or have any questions? Here are five related questions:

1. How would the angle change if the mass of the ball were doubled?
2. What would happen to the angle if the velocity of the ball were halved?
3. How does the length of the rope affect the angle for the same speed?
4. What would be the tension in the rope at this angle?
5. How would the motion change if the rope were replaced with a massless string?

Tip: Understanding the relationship between tension and the components of motion is crucial when dealing with circular motion problems.