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The problem involves a ball swinging in a vertical circle at the end of a rope with a length of 1.50 m. At a specific moment when the ball is 36.9° past the lowest point on its way up, the total acceleration is given as \(\vec{a} = (-22.5 \hat{i} + 20.2 \hat{j}) \, \text{m/s}^2\).

Let's break down the parts of the problem:

### (a) Sketch a vector diagram showing the components of its acceleration
The vector \(\vec{a} = (-22.5 \hat{i} + 20.2 \hat{j}) \, \text{m/s}^2\) can be split into two components:
- **Radial acceleration**: This points toward the center of the circle (centripetal).
- **Tangential acceleration**: This is perpendicular to the radial acceleration and along the path of the ball.

To sketch the vector diagram:
1. The \(i\)-component is negative, meaning the radial acceleration points to the left (towards the center).
2. The \(j\)-component is positive, indicating that the tangential acceleration points upward.

We can use trigonometry to relate these to the radial and tangential components, given the angle of 36.9°.

### (b) Determine the magnitude of the radial acceleration
The radial acceleration can be found using the radial component of the total acceleration. The radial direction points towards the center of the circular path. The magnitude of radial acceleration is related to the centripetal acceleration, which is:
\[
a_r = \frac{v^2}{r}
\]
We can also use the vector information to isolate the radial part.

### (c) Determine the speed and velocity of the ball
The speed can be determined from the centripetal acceleration:
\[
v = \sqrt{a_r \cdot r}
\]
The velocity would have both magnitude and direction. 

### Next steps:
1. Let's first calculate the components of the total acceleration and then determine the radial and tangential accelerations.
2. Do you need further details on how to start each calculation?

Here are 5 relative questions:
1. What is the difference between tangential and radial acceleration?
2. How does the angle of 36.9° relate to the components of acceleration?
3. Why does the radial acceleration contribute to the centripetal force?
4. What are the implications of a negative \(i\)-component for motion in this context?
5. Can we find the angular velocity of the ball using this information?

Tip: Always remember that radial acceleration always points toward the center of a circular path, and the tangential acceleration is responsible for changing the speed of the object.

A ball swings counterclockwise in a vertical circle at the end of a rope 1.50 m long. When the ball is 36.9° past the lowest point on its way up, its total acceleration is (-22.5 î + 20.2 ĵ) m/s². For that instant, (a) sketch a vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball.

Learn how to analyze the acceleration of a ball swinging in a vertical circle at 36.9° from the lowest point. We break down the total acceleration vector and determine radial and tangential components, speed, and velocity. Suitable for advanced high school students.

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