To solve this problem, we can use the relationship between the tangential speed $v$, the radial distance $r$, and the angular speed $\omega$:

$v = r \cdot \omega$

Given:

• The angular speed $\omega = 0.70 \, \text{rad/s}$.
• The tangential speed of Point B $v_B = 5.5 \, \text{cm/s}$.
• The rotation time is $t = 6.8 \, \text{seconds}$.

Step 1: Find the radial distance $r$ of Point B

We can calculate the radial distance $r$ of Point B by rearranging the formula for tangential velocity:

$r = \frac{v_B}{\omega}$

Substituting the given values:

$r = \frac{5.5 \, \text{cm/s}}{0.70 \, \text{rad/s}} = 7.857 \, \text{cm}$

Thus, the radial distance of Point B is $7.857 \, \text{cm}$.

Step 2: Determine the radial distance of Point A

Points A, B, and C are equally spaced on the object, meaning the distance between adjacent points is the same. Since Point A is one of these equally spaced points, it must also be at the same radial distance from Point O as Point B.

Thus, the radial distance of Point A is the same as that of Point B:

$r_A = 7.857 \, \text{cm}$

Step 3: Find the tangential velocity of Point A

Now, using the same formula for tangential velocity $v = r \cdot \omega$ for Point A:

$v_A = r_A \cdot \omega = 7.857 \, \text{cm} \times 0.70 \, \text{rad/s} = 5.5 \, \text{cm/s}$

So, the magnitude of the tangential velocity of Point A is also $5.5 \, \text{cm/s}$.

Final Answer:

The magnitude of the tangential velocity of Point A is $\boxed{5.5 \, \text{cm/s}}$.

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Would you like more details or have any questions about the steps? Here are some related questions for further exploration:

1. How does the tangential velocity change if the angular speed changes?
2. What would happen to the tangential velocity of Point A if the radial distance were doubled?
3. How does the rotation time affect the overall motion of the object?
4. What is the relationship between angular velocity and linear velocity in circular motion?
5. How do you calculate the centripetal acceleration for Point A?

Tip: The tangential velocity is directly proportional to both the radial distance and the angular speed.