A 12.85 kilogram object whose velocity is 12.9 meters per second moves
around a circular track with a radius of 1.8 meters. Calculate the following:
a. centripetal acceleration of the object
b. net force acting on the object

To solve this problem, we need to calculate the centripetal acceleration and the net force acting on the object. The formulas for these calculations are as follows:

### Given:
- Mass \( m = 12.85 \, \text{kg} \)
- Velocity \( v = 12.9 \, \text{m/s} \)
- Radius \( r = 1.8 \, \text{m} \)

### a. Centripetal Acceleration
The formula for centripetal acceleration \( a_c \) is:

\[
a_c = \frac{v^2}{r}
\]

Substituting the given values:

\[
a_c = \frac{(12.9 \, \text{m/s})^2}{1.8 \, \text{m}}
\]

\[
a_c = \frac{166.41 \, \text{m}^2/\text{s}^2}{1.8 \, \text{m}} \approx 92.45 \, \text{m/s}^2
\]

### b. Net Force Acting on the Object
The net force \( F \) acting on the object moving in a circle is the centripetal force, which is given by:

\[
F = m \times a_c
\]

Substituting the mass and the calculated centripetal acceleration:

\[
F = 12.85 \, \text{kg} \times 92.45 \, \text{m/s}^2 \approx 1187.08 \, \text{N}
\]

### Summary:
a. **Centripetal acceleration:** \( 92.45 \, \text{m/s}^2 \)  
b. **Net force acting on the object:** \( 1187.08 \, \text{N} \)

Do you want more details or have any questions?

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Here are 5 related questions for further exploration:

1. How would the centripetal acceleration change if the radius of the circular track was doubled?
2. What would be the net force if the mass of the object was increased to 20 kg while keeping the same velocity and radius?
3. How does the velocity affect the centripetal force if all other factors remain constant?
4. Can you calculate the period of the object’s motion around the track?
5. What would be the velocity required to produce a centripetal acceleration of 50 m/s² on the same track?

**Tip:** Centripetal force always points towards the center of the circular path, and it is essential for keeping the object in circular motion.

Learn how to calculate centripetal acceleration and net force in circular motion using mass, velocity, and radius. Solve problems step-by-step with detailed formulas and explanations.

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