# A Simple Problem on the Hardest Physics Olympiad (IPhO)

TLDRThis video explains a fascinating physics problem from the 1998 International Physics Olympiad, where the challenge is to determine the angle of inclination at which a hexagonal pencil begins to roll indefinitely. The video breaks down the forces at play, particularly focusing on the role of gravitational pull and kinetic energy. Through a mix of theory, geometry, and simulations, it reveals that at angles above 6.58 degrees, a hexagonal pencil, when slightly nudged, will continue rolling without stopping. The explanation is detailed, engaging, and accessible for learners of all levels.

###### Takeaways

- ๐๏ธ The video explores the physics behind the rolling motion of a hexagonal pencil on an inclined table.
- ๐ The main question addressed is determining the angle of inclination that allows the pencil to roll indefinitely.
- โ๏ธ The stability of the pencil is analyzed, with a focus on the role of gravity and the position of the center of mass.
- ๐ The pencil begins to roll when the gravitational pull is no longer aligned with the rotational axis, which occurs at a 30-degree inclination.
- โ๏ธ At angles less than 30 degrees, the pencil can still roll indefinitely if given a slight push, but it will eventually stop at very small angles (like 5 degrees) due to loss of kinetic energy.
- ๐ฌ The video compares a hexagonal pencil to a cylindrical pencil, noting that the latter rolls continuously due to its uniform center of mass.
- ๐ The hexagonal pencil's rolling motion is characterized by alternating gains and losses in kinetic and potential energy, particularly at each impact with the table.
- โ๏ธ The pencil loses about 58% of its kinetic energy with each impact, but it gains energy as its center of mass drops during each roll.
- ๐ A critical inclination angle of 6.58 degrees is identified as the threshold above which the pencil will roll indefinitely, assuming it has enough initial kinetic energy.
- ๐ The video highlights the approximations made in the problem, such as ignoring air resistance and assuming uniform material density in the pencil.

###### Q & A

### What is the main question that the video script aims to answer?

-The video script aims to answer the question: 'How steep does a table have to be in order for a hexagonal pencil to roll indefinitely?'

### Why is the pencil's stability an important factor in this problem?

-The pencil's stability is crucial because the point at which it becomes unstable is when it begins to roll. Understanding the conditions that lead to this instability helps determine the critical angle of inclination.

### What is the significance of the gravitational pull in the context of the pencil's stability?

-The gravitational pull acts directly downward from the pencil's center of mass. As long as this force remains within the supporting points, the pencil remains stable. Once it shifts beyond the rotational axis, the pencil becomes unstable and starts to roll.

### Why is 30 degrees mentioned as a significant angle in the video?

-The 30-degree angle is significant because it is identified as the maximum angle at which the pencil remains stable without any external force. Beyond this angle, the pencil will roll due to gravitational torque.

### What happens to the pencil's motion when the table is inclined at angles less than 30 degrees?

-At angles less than 30 degrees, if the pencil is given a slight initial push, it can start rolling and continue indefinitely until it falls off the table. However, at very small angles (e.g., 5 degrees), the pencil may start rolling but eventually stops due to energy losses.

### How does the rolling motion of a hexagonal pencil differ from that of a cylindrical pencil?

-A hexagonal pencil's center of mass moves up and down as it rolls, leading to energy losses at each impact with the table. In contrast, a cylindrical pencil's center of mass remains at a constant height, allowing it to roll smoothly without energy losses, even at smaller inclinations.

### What role does kinetic energy play in the rolling motion of the hexagonal pencil?

-Kinetic energy is crucial in overcoming the potential energy barriers as the pencil rolls. The pencil must have enough kinetic energy to reach the highest point of its trajectory in each rotational step; otherwise, it will eventually stop rolling.

### What is the significance of the equation K(n+1) = 0.42 * (K(n) + mgฮh)?

-This equation represents the relationship between the kinetic energies of consecutive rotational steps. It accounts for the gain in kinetic energy due to the drop in potential energy (mgฮh) and the loss of kinetic energy due to the impact (0.42 factor). This equation is key to understanding how the pencil's rolling motion evolves over time.

### What is the 'critical start kinetic energy' mentioned in the video?

-The 'critical start kinetic energy' is the minimum initial kinetic energy required for the pencil to reach the highest point in its trajectory. If the pencil starts with less than this energy, it will not be able to continue rolling indefinitely.

### What is the smallest angle of inclination at which a hexagonal pencil can roll indefinitely, and how is it determined?

-The smallest angle of inclination at which a hexagonal pencil can roll indefinitely is approximately 6.58 degrees. It is determined by balancing the stable kinetic energy required for continuous rolling with the potential energy differences as the pencil moves.

###### Outlines

##### ๐ The Physics of a Rolling Pencil

This paragraph introduces the concept of a pencil rolling down a slanted table and the underlying physics question of determining the critical angle of inclination. It sets the stage for a deeper dive into the problem, which was featured in the 1998 International Physics Olympiad. The focus is on the rotational motion and stability of a regular hexagonal pencil, and the video aims to guide viewers through the process of finding the angle at which the pencil can roll indefinitely.

##### ๐ Exploring Stability and the Role of Gravity

The second paragraph delves into the stability of the pencil on an inclined plane, highlighting the role of gravity as the primary force acting on the pencil. It discusses the importance of the center of mass and the rotational axis, and how these factors contribute to the pencil's stability or instability at various angles. The paragraph concludes with the observation that the pencil will start rolling when the gravitational pull is on the right side of the rotational axis, leading to a torque that initiates the rolling motion.

##### ๐ Understanding Hexagonal Pencil Dynamics

This paragraph explores the dynamics of a hexagonal pencil rolling down an inclined plane, contrasting it with a cylindrical pencil that rolls indefinitely even at smaller angles. The focus is on the center of mass and the impact stage of the hexagonal pencil's rolling motion, where kinetic energy is lost due to impacts with the table. The paragraph explains the concept of rotational steps and the energy transformations that occur during each step, including the loss of kinetic energy during impacts and the gain of kinetic energy as the pencil's center of mass drops.

##### ๐ Kinetic Energy and the Sustainability of Rolling Motion

The fourth paragraph discusses the kinetic energy involved in the rolling motion of the hexagonal pencil and the conditions under which the motion becomes sustainable indefinitely. It introduces a recurrent formula to calculate the kinetic energy after each impact and demonstrates through a graph how the initial kinetic energies for the rotational steps approach a stable value. The paragraph also explains the concept of critical start kinetic energy, which is the minimum energy required for the pencil to continue rolling over its trajectory, and concludes with the calculation of the critical angle of inclination for sustainable rolling motion.

###### Mindmap

###### Keywords

##### ๐กInclination Angle

##### ๐กRotational Motion

##### ๐กStability

##### ๐กHexagonal Pencil

##### ๐กCenter of Mass

##### ๐กPotential Energy

##### ๐กKinetic Energy

##### ๐กImpact Stage

##### ๐กSine Theta

##### ๐กCritical Kinetic Energy

###### Highlights

Introduction to the problem: determining the angle of inclination for a hexagonal pencil to roll indefinitely.

The concept of rotational motion and its relevance to the pencil's stability.

Exploring the stability of the pencil by analyzing different angles of inclination.

Explanation of gravitational pull and its impact on the pencil's stability and potential to roll.

Determining that 30 degrees is the maximum angle at which the pencil remains stable without rolling.

Observation that the pencil will roll indefinitely even at angles less than 30 degrees if given an initial push.

Comparison between the rolling behavior of a hexagonal pencil and a cylindrical pencil.

Understanding the importance of the center of mass in the pencil's rolling motion.

Description of the two distinct phases in the hexagonal pencil's rolling motion: rotation and impact stages.

Explanation of potential energy changes during the pencil's rolling motion and how they affect its kinetic energy.

Derivation of the general recurrence formula for the pencil's kinetic energy after each rotational step.

Introduction of the concept of stable kinetic energy (Ks) and its dependency on the angle of inclination.

Observation that regardless of the initial kinetic energy, the pencil's rolling motion stabilizes to a constant speed.

Calculation of the critical initial kinetic energy required for the pencil to continue rolling indefinitely.

Determination that an inclination angle of 6.58 degrees or above is necessary for the pencil to roll indefinitely.

###### Transcripts

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