A bowling ball weighing 71.2 N (16.0 lb) is attached to the ceiling by a 3.40 m rope. The ball is pulled to one side and released; it then swings back and forth as a pendulum. As the rope swings through the vertical, the speed of the bowling ball is 4.80 m/s.

At this instant, what is the acceleration of the bowling ball, in magnitude and direction?
Express your answer with the appropriate SI units. Enter positive value if the acceleration is upward and negative value if the acceleration is downward.

To solve for the acceleration of the bowling ball at the lowest point of its swing, we need to break down the forces acting on the ball and use principles from circular motion, since the bowling ball is essentially undergoing pendulum-like circular motion.

### Step 1: Identify forces at the lowest point
At the lowest point, two key forces act on the bowling ball:
1. **Tension in the rope**: Acts upward along the rope.
2. **Gravitational force (weight)**: Acts downward with a magnitude of \( mg \), where \( m \) is the mass of the bowling ball and \( g \) is the acceleration due to gravity.

Since the ball is moving in a circular path, it experiences centripetal acceleration toward the center of the circular path, which is directed upward along the rope at the lowest point. We can calculate the centripetal force using the following equation:

\[
F_{\text{centripetal}} = m \cdot a_{\text{centripetal}} = m \cdot \frac{v^2}{r}
\]

where:
- \( v = 4.80 \, \text{m/s} \) is the velocity at the lowest point,
- \( r = 3.40 \, \text{m} \) is the length of the rope (which is the radius of the circular path),
- \( m \) is the mass of the bowling ball.

### Step 2: Calculate the mass of the bowling ball
We know the weight of the ball is \( W = 71.2 \, \text{N} \), and weight is related to mass by \( W = m \cdot g \), where \( g = 9.81 \, \text{m/s}^2 \). Solving for \( m \):

\[
m = \frac{W}{g} = \frac{71.2 \, \text{N}}{9.81 \, \text{m/s}^2} = 7.26 \, \text{kg}
\]

### Step 3: Calculate the centripetal acceleration
Using the centripetal acceleration formula:

\[
a_{\text{centripetal}} = \frac{v^2}{r} = \frac{(4.80 \, \text{m/s})^2}{3.40 \, \text{m}} = 6.78 \, \text{m/s}^2
\]

This is the upward acceleration due to the circular motion of the ball.

### Step 4: Calculate the net vertical acceleration
At the lowest point, the total acceleration of the ball will be the net result of the gravitational force and the centripetal force. The gravitational force is acting downward with an acceleration of \( g = 9.81 \, \text{m/s}^2 \), and the centripetal acceleration is upward with a magnitude of \( 6.78 \, \text{m/s}^2 \).

The net vertical acceleration is the difference between the two, since they act in opposite directions:

\[
a_{\text{net}} = a_{\text{centripetal}} - g = 6.78 \, \text{m/s}^2 - 9.81 \, \text{m/s}^2 = -3.03 \, \text{m/s}^2
\]

### Final Answer:
The acceleration of the bowling ball at the lowest point is:

\[
a_{\text{net}} = -3.03 \, \text{m/s}^2
\]

The negative sign indicates that the acceleration is directed downward.

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Would you like more details on any step or another explanation? Here are 5 related questions to consider:

1. How does the tension in the rope change as the ball swings through its path?
2. What would be the ball's speed if the rope were longer?
3. How does the gravitational force affect the ball’s motion in pendulum-like swings?
4. How would the motion change if the weight of the ball were doubled?
5. What is the relationship between the angle of release and the ball's velocity at the lowest point?

**Tip:** In circular motion, the centripetal force is always directed toward the center of the circular path, which affects the object's acceleration and tension in connected systems like a pendulum.

A bowling ball weighing 71.2 N (16.0 lb) is attached to the ceiling by a 3.40 m rope. The ball is pulled to one side and released; it then swings back and forth as a pendulum. As the rope swings through the vertical, the speed of the bowling ball is 4.80 m/s. At this instant, what is the acceleration of the bowling ball, in magnitude and direction? Express your answer with the appropriate SI units. Enter positive value if the acceleration is upward and negative value if the acceleration is downward.

Explore how to calculate the acceleration of a 71.2 N bowling ball attached to a 3.40 m rope as it swings like a pendulum at 4.80 m/s. This solution uses concepts of circular motion, centripetal force, and gravity to determine the net acceleration. Suitable for students in Grades 11-12.

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