Mass m1 on the frictionless table of the figure is connected by a string through a hole in the table to a hanging mass m2. With what speed must  m1  rotate in a circle of radius  r  if  m2  is to remain hanging at rest? Express your answer in terms of some or all of the variables  m1 ,  m2 ,  r , and the free-fall acceleration  g .

To solve this problem, we need to analyze the forces acting on both masses \( m_1 \) and \( m_2 \).

1. **For the hanging mass \( m_2 \):**
   - The weight of \( m_2 \) exerts a force downwards, which is equal to \( m_2 g \), where \( g \) is the acceleration due to gravity.
   - The tension \( T \) in the string is the force holding \( m_2 \) at rest, so \( T = m_2 g \).

2. **For the mass \( m_1 \) on the table:**
   - Since \( m_1 \) is rotating in a circle, there is a centripetal force acting on it that keeps it in circular motion. This centripetal force is provided by the tension in the string.
   - The formula for the centripetal force is:
     \[
     F_{\text{centripetal}} = \frac{m_1 v^2}{r}
     \]
   - This force is equal to the tension in the string, so:
     \[
     \frac{m_1 v^2}{r} = T
     \]

3. **Equating the forces:**
   Since \( T = m_2 g \), we substitute this into the equation for the centripetal force:
   \[
   \frac{m_1 v^2}{r} = m_2 g
   \]

4. **Solving for \( v \) (the speed of \( m_1 \)):**
   Multiply both sides of the equation by \( r \) to isolate \( v^2 \):
   \[
   m_1 v^2 = m_2 g r
   \]
   Divide by \( m_1 \):
   \[
   v^2 = \frac{m_2 g r}{m_1}
   \]
   Take the square root to solve for \( v \):
   \[
   v = \sqrt{\frac{m_2 g r}{m_1}}
   \]

### Final Answer:
The speed at which \( m_1 \) must rotate for \( m_2 \) to remain hanging at rest is:
\[
v = \sqrt{\frac{m_2 g r}{m_1}}
\]

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Would you like further details or have any specific questions?

Here are 5 related questions for further exploration:
1. How would the speed change if \( m_1 \) were increased?
2. What happens if the radius \( r \) is doubled?
3. How does the free-fall acceleration \( g \) affect the system if it's not constant?
4. What would happen if there was friction on the table affecting \( m_1 \)'s motion?
5. Can the tension in the string ever be greater than \( m_2 g \)?

**Tip:** Always check the units of your variables in problems like this to ensure dimensional consistency when solving equations.

Mass m1 on the frictionless table of the figure is connected by a string through a hole in the table to a hanging mass m2. With what speed must m1 rotate in a circle of radius r if m2 is to remain hanging at rest? Express your answer in terms of some or all of the variables m1, m2, r, and the free-fall acceleration g.

Learn how to calculate the speed at which mass m1 must rotate in a circle for mass m2 to remain hanging at rest in a frictionless system. This physics problem covers key concepts such as circular motion, centripetal force, and gravitational force. Suitable for students in Grades 11-12.

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