To calculate the centripetal acceleration $a_c$ of Titan as it orbits Saturn, we can use the formula:

$a_c = \frac{v^2}{r}$

Where:

• $v$ is the orbital velocity of Titan.
• $r$ is the radius of the orbit (the distance from Saturn to Titan).

We can find the orbital velocity $v$ using the formula:

$v = \frac{2\pi r}{T}$

Where:

• $T$ is the orbital period (time taken to complete one orbit).

Step 1: Convert the orbital period to seconds

Given that the period is $T = 16.0 \, \text{days}$, we first convert this to seconds:

$T = 16.0 \, \text{days} \times 24 \, \text{hours/day} \times 3600 \, \text{seconds/hour} = 1,382,400 \, \text{seconds}$

Step 2: Calculate the orbital velocity

Now, using the orbital radius $r = 1.22 \times 10^9 \, \text{m}$ and the orbital period $T$, we calculate the velocity $v$:

$v = \frac{2\pi (1.22 \times 10^9)}{1,382,400} \approx \frac{7.66 \times 10^9}{1,382,400} \approx 5544 \, \text{m/s}$

Step 3: Calculate the centripetal acceleration

Now, using the velocity $v$ and the radius $r$, we can find the centripetal acceleration $a_c$:

$a_c = \frac{(5544)^2}{1.22 \times 10^9} \approx \frac{3.07 \times 10^7}{1.22 \times 10^9} \approx 0.0251 \, \text{m/s}^2$

Thus, the centripetal acceleration of Titan is approximately:

$a_c \approx 0.0251 \, \text{m/s}^2$

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the centripetal force relate to centripetal acceleration in orbital motion?
2. What is the relationship between the period of orbit and the velocity in circular motion?
3. How does the mass of Titan affect its orbit around Saturn?
4. How would the centripetal acceleration change if Titan were further from Saturn?
5. What role does gravitational force play in maintaining Titan’s orbit?

Tip: In orbital motion, the centripetal acceleration is always directed toward the center of the orbiting body's path, in this case, Saturn.