4.5.2 Unstable Equilibrium

Have you ever tried to balance a pencil on its tip? The slightest disturbance causes it to fall over. This is the essence of unstable equilibrium. An object in unstable equilibrium is in a precarious state—it will not return to its original position if disturbed; instead, it will move further away and topple. Understanding unstable equilibrium is crucial for identifying situations where objects are prone to falling or collapsing.

Definition

Unstable Equilibrium is a state in which a body, when slightly displaced and then released, moves further away from its original position.

Key Condition: In unstable equilibrium, the center of mass (or center of gravity) of the object falls when it is displaced.

Main Idea

The main idea is that an object in unstable equilibrium has a high center of gravity and a narrow base of support, making it prone to toppling. When displaced, the center of gravity moves lower, and gravity creates a torque that pushes the object further away from its original position. This is the opposite of stable equilibrium, where the object returns to its original position.

Core Concept

The Mechanism of Unstable Equilibrium The text explains: "A body is in unstable equilibrium if it moves further away from its previous position when slightly displaced and released. Its centre of mass falls when it is displaced, because there is a moment which increases the displacement."

• When the object is displaced, its center of mass moves to a lower position.
• Gravity pulls down on the center of mass, creating a torque about the point of contact (pivot).
• This torque acts in a direction that increases the displacement, causing the object to rotate further away from its original position.

Example from Text The text provides a simple example: "For example Take a pencil and try to keep it in the vertical position on its tip. Whenever you leave it, the pencil topples over about its tip and falls down. This is called an unstable equilibrium."

When you try to balance a pencil on its tip, its center of mass is very high, and its base of support is tiny (just the tip). The slightest disturbance causes the center of mass to move outside the base of support, and gravity pulls it down, causing the pencil to fall.

Detailed Explanation

Unstable equilibrium is the opposite of stable equilibrium. In stable equilibrium, a displacement raises the center of mass and creates a restoring torque. In unstable equilibrium, a displacement lowers the center of mass and creates a torque that increases the displacement.

The Role of the Center of Mass For an object to be in unstable equilibrium, its center of mass must be high relative to its base of support. When displaced, the center of mass falls, and the object moves further away from its original position.

The Cone Example The text uses the example of a cone balanced on its tip (Figure 4.12). When the cone is upright on its tip, its center of mass is high. If it is slightly displaced, the center of mass falls, and gravity pulls it further away from the upright position. This is a classic example of unstable equilibrium.

• High center of mass: The higher the center of mass, the more unstable the object.
• Narrow base of support: A small base means the center of mass can easily move outside the base when displaced.
• Center of mass falls when displaced: This is the defining characteristic of unstable equilibrium.

Comparison with Stable Equilibrium | Feature | Stable Equilibrium | Unstable Equilibrium | | :--- | :--- | :--- | | Center of Mass | Rises when displaced | Falls when displaced | | Restoring Torque | Brings object back | Pushes object further away | | Base of Support | Wide | Narrow | | Center of Gravity | Low | High | | Example | Book on table, cone in bowl | Pencil on tip, cone on tip |

Intermediate Level

In physics, unstable equilibrium is characterized by a maximum in potential energy. When an object is displaced from an unstable equilibrium position, its potential energy decreases (because its center of mass falls). The force of gravity then acts to increase the displacement, causing the object to move further away from the equilibrium position.

The stability of an object can be increased by lowering its center of mass and widening its base of support. This is why tall, narrow objects are more unstable than short, wide objects. For example, a pencil on its tip is unstable, while a pencil lying flat on a table is stable. The position of the center of mass relative to the base of support is the key factor in determining stability.

Important Points

• - Unstable Equilibrium: Object moves further away from original position when displaced.
• - Key Condition: Center of mass falls when displaced.
• - Mechanism: Gravity creates a torque that increases the displacement.
• - Examples: Pencil on its tip, cone balanced on its tip, tall narrow vase.
• - Cause: High center of mass, narrow base of support.

Remember

• - Unstable Equilibrium: Moves further away.
• - CG falls when displaced.
• - Torque increases displacement.
• - Examples: Pencil on tip, cone on tip.
• - Cause: High CG, narrow base.

Common Mistakes

• - Thinking unstable means it cannot be balanced: An object in unstable equilibrium can be balanced momentarily, but any small disturbance causes it to fall.
• - Confusing unstable with neutral equilibrium: In neutral equilibrium, the object stays in its new position (e.g., a ball on a flat table). In unstable equilibrium, it moves further away.
• - Forgetting the role of the center of mass: The key to instability is the height of the center of mass relative to the base of support.

Quick Revision

Unstable equilibrium is a state in which an object, when slightly displaced and released, moves further away from its original position. This occurs because displacing the object lowers its center of mass, and gravity creates a torque that increases the displacement. Objects with a high center of mass and a narrow base of support are typically in unstable equilibrium.

60-Second Revision

• CG falls when displaced.
• Torque increases displacement.
• Examples: Pencil on tip, cone on tip, tall narrow vase.
• Cause: High CG, narrow base.

Frequently Asked Questions (FAQs)

• Q1. What is unstable equilibrium?
• Answer: Unstable equilibrium is a state in which a body, when slightly displaced and then released, moves further away from its original position.

• Q2. What happens to the center of mass when an object in unstable equilibrium is displaced?
• Answer: The center of mass falls when the object is displaced.

Q3. Why does an object in unstable equilibrium move further away when displaced? Answer: Because displacing the object lowers its center of mass. Gravity then creates a torque about the point of contact that acts to increase the displacement, pushing the object further away from its original position.

• Q4. Give an example of unstable equilibrium.
• Answer: A pencil standing on its tip is a classic example. The slightest disturbance causes it to topple over. A cone balanced on its tip is another example.

Q5. What are the conditions for unstable equilibrium? Answer: Unstable equilibrium typically occurs when an object has a high center of mass and a narrow base of support. The defining condition is that the center of mass falls when the object is displaced.

Q6. How is unstable equilibrium different from stable equilibrium? Answer: In stable equilibrium, the object returns to its original position when displaced (center of mass rises). In unstable equilibrium, the object moves further away when displaced (center of mass falls).

Multiple Choice Questions (MCQs)

• Q1. In unstable equilibrium, when an object is displaced, its center of mass:
• A. Falls
• B. Rises
• C. Remains at the same height
• D. Becomes zero

• Correct Answer: A
• Explanation: The text states that in unstable equilibrium, the center of mass falls when it is displaced.

• Q2. A body in unstable equilibrium, when slightly displaced and released, will:
• A. Return to its original position
• B. Stay in the new position
• C. Move further away from its original position
• D. Fall over completely

Correct Answer: C Explanation: The text defines unstable equilibrium as the state where the body moves further away from its previous position when slightly displaced and released.

• Q3. Which of the following is an example of unstable equilibrium?
• A. A book lying on a table
• B. A ball on a flat horizontal surface
• C. A pencil standing on its tip
• D. A cone resting at the bottom of a bowl

• Correct Answer: C
• Explanation: The text gives a pencil standing on its tip as an example of unstable equilibrium.

• Q4. The torque in unstable equilibrium:
• A. Decreases the displacement
• B. Increases the displacement
• C. Has no effect on displacement
• D. Brings the object back to equilibrium

• Correct Answer: B
• Explanation: The text states that in unstable equilibrium, there is a moment which increases the displacement.

• Q5. An object with a high center of mass and a narrow base is likely to be in:
• A. Stable equilibrium
• B. Unstable equilibrium
• C. Neutral equilibrium
• D. Dynamic equilibrium

• Correct Answer: B
• Explanation: A high center of mass and narrow base make an object prone to toppling, which is characteristic of unstable equilibrium.

• Q6. A cone balanced on its tip is in:
• A. Stable equilibrium
• B. Unstable equilibrium
• C. Neutral equilibrium
• D. Dynamic equilibrium

• Correct Answer: B
• Explanation: The text uses the cone on its tip illustration to show unstable equilibrium (the cone falls over when displaced).

• Q7. When an object in unstable equilibrium is displaced, it loses:
• A. Kinetic energy
• B. Potential energy
• C. Momentum
• D. Force

• Correct Answer: B
• Explanation: Since the center of mass falls, the object loses gravitational potential energy.

• Q8. The condition for unstable equilibrium is that the center of mass must:
• A. Rise when displaced
• B. Fall when displaced
• C. Remain constant when displaced
• D. Move outside the base

• Correct Answer: B
• Explanation: The text states that in unstable equilibrium, the center of mass falls when it is displaced.

• Q9. A tall, narrow vase is more likely to be in:
• A. Stable equilibrium
• B. Unstable equilibrium
• C. Neutral equilibrium
• D. Dynamic equilibrium

• Correct Answer: B
• Explanation: A tall, narrow vase has a high center of mass and a narrow base, making it unstable.

• Q10. If the center of mass of an object is lowered when it is tilted, the object is in:
• A. Stable equilibrium
• B. Unstable equilibrium
• C. Neutral equilibrium
• D. No equilibrium

• Correct Answer: B
• Explanation: The fall of the center of mass upon displacement is the defining characteristic of unstable equilibrium.

Frequently Asked Questions (FAQs)

Q1. What is the difference between stable and unstable equilibrium? Answer: In stable equilibrium, the object returns to its original position when displaced (center of mass rises). In unstable equilibrium, the object moves further away when displaced (center of mass falls).

Q2. What is the difference between unstable and neutral equilibrium? Answer: In unstable equilibrium, the object moves further away when displaced. In neutral equilibrium, the object stays in its new position when displaced (center of mass remains at the same height).

Q3. Why does a pencil standing on its tip fall over easily? Answer: A pencil standing on its tip has a very small base of support and a high center of mass. The slightest displacement causes the center of mass to fall outside the base, creating a torque that makes the pencil fall over.

• Q4. Can an object in unstable equilibrium be balanced?
• Answer: Yes, but only momentarily. Any small disturbance will cause it to move away from its balanced position. It is a precarious state.

Q5. How can you make an unstable object more stable? Answer: You can make it more stable by lowering its center of mass (e.g., adding weight at the bottom) or increasing its base of support (e.g., making it wider).

Q6. Why are tall vehicles more prone to overturning than short vehicles? Answer: Tall vehicles have a higher center of mass, making them more unstable. When they turn a corner, the center of mass can easily shift outside the base of support, causing them to overturn.

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