Chapter 10: Problem 77
A 5.00-kg ball is dropped from a height of 12.0 m above one end of a uniform bar that pivots at its center. The bar has mass 8.00 kg and is 4.00 m in length. At the other end of the bar sits another 5.00-kg ball, unattached to the bar. The dropped ball sticks to the bar after the collision. How high will the other ball go after the collision?
Short Answer
Step by step solution
Calculate the Potential Energy of the Dropped Ball
Find the Moment of Inertia of the System
Determine Angular Velocity After Collision
Calculate the Maximum Height of the Other Ball
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gravitational Potential Energy
- \( PE = mgh \)
This energy becomes important when an object is elevated, as it represents the potential to do work when the object falls. In the problem, a ball of mass 5.00-kg is dropped from a height of 12.0 m, resulting in an initial potential energy of \( 588.6 \, \text{J} \).
This energy is converted into other forms of energy during the collision and subsequent motion of the system. Understanding this conversion is crucial for analyzing collision dynamics.
Moment of Inertia
- For a rod pivoting at its center: \[ I_{\text{bar}} = \frac{1}{12} ML^2 \]
- For a point mass at a distance \( r \) from the pivot: \[ I_{\text{point mass}} = mr^2 \]
This comprehensive value helps determine how the system will react to the forces applied during the collision.
Angular Momentum Conservation
For the initial angular momentum, the focus is on contributions from objects in motion. Here, only the falling ball has angular momentum:
- \[ L_i = mvr = m \sqrt{2gh} \cdot r \]
- \[ L_f = I_{\text{total}} \cdot \omega \]
Kinetic Energy Conversion
In the exercise, the rotational motion of the system leads to kinetic energy being partially converted back to potential energy as the unattached ball rises.
To find how high the unattached ball will go, we use
- the formula: \[ mgh = \frac{1}{2} mv^2 \]
- where \( v = \omega r \).
This calculation exemplifies the power of energy conservation in predicting the outcome of dynamic systems.