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We are dealing with a problem that involves a puck of mass \( m = 1.00 \, \text{kg} \) sliding on a frictionless table in a circular path while connected to a hanging mass \( M = 10.0 \, \text{kg} \) by a cord. The goal is to solve two parts:

**A.** What is the tension in the string?

**B.** What is the puck's speed to keep the hanging cylinder at rest?

### Part A: Finding the Tension in the String
The key here is that the hanging cylinder of mass \( M \) is in equilibrium. Therefore, the tension \( T \) in the string must balance the weight of the hanging mass.

\[
T = M g
\]
Where:
- \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity,
- \( M = 10.0 \, \text{kg} \).

So the tension is:
\[
T = 10.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 98.0 \, \text{N}
\]

### Part B: Finding the Puck's Speed
For the puck to maintain a circular motion while keeping the cylinder at rest, the tension in the string also provides the centripetal force necessary for the puck's circular motion.

The centripetal force is given by:
\[
T = \frac{m v^2}{r}
\]
Where:
- \( T = 98.0 \, \text{N} \),
- \( m = 1.00 \, \text{kg} \),
- \( r = 0.30 \, \text{m} \) (converted from 30 cm),
- \( v \) is the speed of the puck.

Rearranging the formula to solve for \( v \):
\[
v = \sqrt{\frac{T r}{m}}
\]

Substituting the known values:
\[
v = \sqrt{\frac{98.0 \, \text{N} \times 0.30 \, \text{m}}{1.00 \, \text{kg}}}
\]

\[
v = \sqrt{29.4} \, \text{m/s} = 5.42 \, \text{m/s}
\]

Thus, the puck's speed must be **5.42 m/s** to keep the hanging cylinder at rest.

---

Would you like further details or clarifications on any part of the problem? Here are some related questions to think about:

1. How does the tension in the string affect the puck’s motion?
2. What would happen if the mass of the hanging cylinder increased?
3. How would the result change if there were friction on the table?
4. What forces are acting on the hanging mass and how do they balance out?
5. How is centripetal force connected to circular motion in this scenario?

**Tip**: Understanding how tension relates to both circular motion and equilibrium helps in solving more complex systems involving ropes or strings.

A puck of mass m = 1.00 kg slides in a circle of radius r = 30.0 cm on a frictionless table while attached to a hanging cylinder of mass M = 10.0 kg by a cord that extends through a hole in the table. What is the tension in the string and the puck's speed to keep the cylinder at rest?

This problem involves a puck sliding in a circle on a frictionless table, connected to a hanging mass by a cord. Learn how to calculate the tension in the string and the puck's speed required to keep the hanging cylinder at rest. Suitable for high school physics students (Grades 10-12).

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