A pendulum consists of a rod of mass 2 kg and length 1.4 m with a solid sphere at one end with mass 0.2 kg and radius 23 cm (see the following figure). If the pendulum is released from rest at an angle of 24°, what is the angular velocity at the lowest point? (Enter the magnitude in rad/s.)

To find the angular velocity of the pendulum at the lowest point, we can use the principle of conservation of mechanical energy. The mechanical energy at the highest point (where the pendulum is released) is equal to the mechanical energy at the lowest point.

### Step 1: Define the energies
- **Potential Energy at the highest point:**
  \[
  U_{\text{initial}} = m_{\text{eff}} \cdot g \cdot h
  \]
  where:
  - \( m_{\text{eff}} \) is the effective mass of the pendulum system.
  - \( g \) is the acceleration due to gravity (\( 9.81 \, \text{m/s}^2 \)).
  - \( h \) is the height difference between the highest and lowest points of the pendulum.

- **Kinetic Energy at the lowest point:**
  \[
  K_{\text{final}} = \frac{1}{2} I_{\text{total}} \omega^2
  \]
  where:
  - \( I_{\text{total}} \) is the moment of inertia of the pendulum system.
  - \( \omega \) is the angular velocity at the lowest point.

### Step 2: Calculate the effective mass (\( m_{\text{eff}} \))
Since the pendulum consists of a rod and a solid sphere, we need to calculate the center of mass and use it to determine the height \( h \).

#### Height difference \( h \):
The height difference can be calculated using trigonometry:
\[
h = L_{\text{cm}} \cdot (1 - \cos \theta)
\]
where:
- \( L_{\text{cm}} \) is the distance from the pivot to the center of mass.
- \( \theta \) is the angle of release (24°).

#### Center of Mass \( L_{\text{cm}} \):
\[
L_{\text{cm}} = \frac{m_{\text{rod}} \cdot \frac{L}{2} + m_{\text{sphere}} \cdot L_{\text{sphere}}}{m_{\text{rod}} + m_{\text{sphere}}}
\]
where:
- \( m_{\text{rod}} = 2 \, \text{kg} \)
- \( L = 1.4 \, \text{m} \) (length of the rod)
- \( m_{\text{sphere}} = 0.2 \, \text{kg} \)
- \( L_{\text{sphere}} = 1.4 \, \text{m} \) (since the sphere is at the end of the rod).

Now let's compute the height \( h \), moment of inertia \( I_{\text{total}} \), and finally the angular velocity \( \omega \).The angular velocity of the pendulum at the lowest point is approximately \(1.30 \, \text{rad/s}\).

Do you have any further questions or would you like more details?

Here are some related questions to explore:
1. How does changing the mass of the sphere affect the angular velocity?
2. What would happen to the angular velocity if the length of the rod were increased?
3. How does the release angle influence the speed at the lowest point?
4. How would the angular velocity change if the rod were replaced with a string?
5. What is the effect of gravity on the pendulum's motion in different environments?

**Tip:** Understanding the conservation of energy in pendulums is crucial for analyzing their motion and predicting outcomes based on changes in initial conditions.

Learn how to calculate the angular velocity of a pendulum released from rest at an angle of 24° using the principles of mechanical energy conservation. This problem involves determining effective mass, moment of inertia, and angular velocity. Suitable for advanced high school students.

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