4.4 Equilibrium

In our daily lives, we see many objects that are stable and not moving—a book on a table, a building, a bridge. We also see objects moving at a constant speed in a straight line, like a car on a highway with cruise control. In both cases, the forces acting on the objects are balanced. This state of balance is called equilibrium. Understanding equilibrium is essential for analyzing why objects stay at rest or move uniformly.

Definition

Equilibrium is the state in which all the individual forces and moments of force (torques) exerted upon an object are balanced. This means that the net force and the net torque on the object are zero.

Main Idea

• The net force on the object must be zero (translational equilibrium).
• The net torque on the object must be zero (rotational equilibrium).

When both conditions are satisfied, the object will either remain at rest or move with constant velocity in a straight line, and it will not rotate (or will rotate with constant angular velocity).

Core Concept

The Two Conditions of Equilibrium The text states: "For complete equilibrium two conditions must be met."

A. First Condition of Equilibrium (Translational Equilibrium): When the vector sum of all the forces acting on the body is zero, the first condition of equilibrium is satisfied. Mathematically: F_net = F₁ + F₂ + F₃ + … + F_n = 0 This means the object will either be at rest or move with uniform velocity in a straight line.

B. Second Condition of Equilibrium (Rotational Equilibrium): The second condition of equilibrium is also called the principle of moments. It states that for an object in equilibrium, the sum of the clockwise moments about the pivot must be equal to the sum of the anticlockwise moments about the same pivot. Mathematically: Sum of clockwise moments = Sum of anticlockwise moments or τ_net = 0

Types of Equilibrium The text divides equilibrium into two types:

A. Static Equilibrium: When a body is at rest under the action of several forces and torques acting together, it is said to be in static equilibrium. For example, a book resting on a table is in static equilibrium.

• Dynamic Translational Equilibrium: When a body is moving with uniform linear velocity. For example, a paratrooper falling with constant velocity (terminal velocity) is in dynamic translational equilibrium.
• Dynamic Rotational Equilibrium: When a body is rotating with uniform angular velocity.

• Anticlockwise moment: τ₁ = 4 N × 0.2 m = 0.8 N·m (positive)
• Clockwise moment: τ₂ = 2 N × 0.4 m = 0.8 N·m (negative)
• Net moment: τ_net = 0.8 - 0.8 = 0 N·m

Since the net moment is zero, the meter stick is in rotational equilibrium.

Detailed Explanation

Equilibrium is a fundamental concept in mechanics. It describes the state of an object when all the forces and torques acting on it cancel out.

The First Condition: Net Force is Zero The first condition ensures that the object does not accelerate in a straight line. If the net force is zero, the object will not speed up, slow down, or change direction. It will either stay at rest or continue moving with constant velocity. This condition is sufficient for translational equilibrium but not for complete equilibrium, because the object could still rotate.

The Second Condition: Net Torque is Zero The second condition ensures that the object does not rotate. If the net torque is zero, the sum of clockwise moments equals the sum of anticlockwise moments. This condition is necessary for an object to be in complete equilibrium. For example, a couple (two equal and opposite forces acting at different points) produces a net torque but no net force. The object would rotate. To prevent rotation, the net torque must be zero.

• Static equilibrium: The object is at rest. All forces and torques balance. Examples include a book on a table, a bridge, a building.
• Dynamic equilibrium: The object is moving with constant velocity (and/or constant angular velocity). The net force and net torque are still zero. Examples include a car moving at a constant speed on a straight road, a satellite in a circular orbit (dynamic translational equilibrium), a ceiling fan rotating at a constant speed (dynamic rotational equilibrium).

Intermediate Level

• Translational equilibrium: The vector sum of all forces is zero (∑ F = 0). This means the object's center of mass has zero acceleration. It is either at rest or moving with constant velocity.
• Rotational equilibrium: The sum of all torques about any point is zero (∑ τ = 0). This means the object has zero angular acceleration. It is either not rotating or rotating with constant angular velocity.

For complete equilibrium, both conditions must be satisfied. For example, a ladder leaning against a wall is in static equilibrium. The forces acting on it are gravity (downwards), the normal force from the ground (upwards and possibly sideways), and the normal force from the wall (horizontal). These forces must sum to zero. Also, the torques about any point (e.g., the base of the ladder) must sum to zero to prevent rotation. This is why the ladder doesn't slip or fall.

The principle of moments is a direct consequence of rotational equilibrium. It states that for an object in equilibrium, the total clockwise moment about a pivot equals the total anticlockwise moment.

Important Points

• - Equilibrium: The state where net force and net torque are zero.
• - First Condition: Net force = 0. (Translational equilibrium)
• - Second Condition: Net torque = 0. (Rotational equilibrium)
• - Principle of Moments: Clockwise moments = Anticlockwise moments.
• - Static Equilibrium: Object is at rest.
• - Dynamic Equilibrium: Object is moving with constant velocity (and/or constant angular velocity).
• - For complete equilibrium, both conditions must be met.

Remember

• - Equilibrium: Balance.
• - Two Conditions:
• 1. Net force = 0.
• 2. Net torque = 0.
• - Static: At rest.
• - Dynamic: Constant velocity.
• - Principle of Moments: τ_clockwise = τ_{anticlockwise}.

Common Mistakes

• - Thinking one condition is enough: An object can have zero net force but still rotate (e.g., a couple). Both conditions are needed for complete equilibrium.
• - Forgetting about torque: For objects that can rotate, the second condition (net torque = 0) is crucial.
• - Confusing static and dynamic equilibrium: Static means at rest; dynamic means moving with constant velocity.

Quick Revision

Equilibrium is the state in which all forces and torques acting on an object are balanced. For complete equilibrium, two conditions must be met: the net force must be zero (first condition) and the net torque must be zero (second condition). Static equilibrium occurs when the object is at rest, while dynamic equilibrium occurs when the object is moving with constant velocity. The principle of moments states that for rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.

60-Second Revision

• First Condition: ∑ F = 0 (translational equilibrium).
• Second Condition: ∑ τ = 0 (rotational equilibrium).
• Principle of Moments: Clockwise moments = Anticlockwise moments.
• Static: At rest.
• Dynamic: Constant velocity.

Frequently Asked Questions (FAQs)

Q1. What is equilibrium? Answer: Equilibrium is the state in which all the individual forces and torques acting on an object are balanced. The net force and net torque on the object are zero.

• First Condition: The vector sum of all forces acting on the object must be zero (F_net = 0).
• Second Condition: The sum of all torques (moments) acting on the object must be zero (τ_net = 0).

Q3. What is the difference between static and dynamic equilibrium? Answer: Static equilibrium occurs when an object is at rest (e.g., a book on a table). Dynamic equilibrium occurs when an object is moving with constant velocity (e.g., a car moving at a constant speed on a straight road) or rotating with constant angular velocity.

Q4. What is the principle of moments? Answer: The principle of moments states that for an object in rotational equilibrium, the sum of the clockwise moments about a pivot is equal to the sum of the anticlockwise moments about the same pivot.

Q5. Why are two conditions needed for complete equilibrium? Answer: The first condition ensures there is no translational acceleration. The second condition ensures there is no rotational acceleration. An object could have zero net force but still rotate (e.g., a couple). Both conditions are needed to ensure the object is completely at rest or moving with uniform velocity without rotation.

Q6. What is the first condition of equilibrium? Answer: The first condition of equilibrium states that the vector sum of all forces acting on a body is zero. This ensures that the object has no translational acceleration.

Q7. What is the second condition of equilibrium? Answer: The second condition of equilibrium states that the sum of all torques (moments) acting on a body is zero. This ensures that the object has no rotational acceleration.

Multiple Choice Questions (MCQs)

• Q1. The state in which all forces and torques acting on an object are balanced is called:
• A. Motion
• B. Equilibrium
• C. Acceleration
• D. Inertia

• Correct Answer: B
• Explanation: The text defines equilibrium as the state in which all forces and torques are balanced.

• Q2. The first condition of equilibrium states that the:
• A. Net torque is zero
• B. Net force is zero
• C. Net momentum is zero
• D. Net acceleration is zero

• Correct Answer: B
• Explanation: The first condition of equilibrium is that the vector sum of all forces is zero.

• Q3. The second condition of equilibrium is also called the:
• A. Principle of moments
• B. Principle of inertia
• C. Principle of force
• D. Principle of motion

• Correct Answer: A
• Explanation: The text states that the second condition of equilibrium is also called the principle of moments.

• Q4. When a body is at rest under the action of forces, it is in:
• A. Dynamic equilibrium
• B. Static equilibrium
• C. Translational equilibrium
• D. Rotational equilibrium

• Correct Answer: B
• Explanation: The text states that when a body is at rest, it is in static equilibrium.

• Q5. When a body is moving with uniform velocity, it is in:
• A. Static equilibrium
• B. Dynamic equilibrium
• C. Unstable equilibrium
• D. Neutral equilibrium

• Correct Answer: B
• Explanation: The text states that when a body is moving with uniform velocity, it is in dynamic equilibrium.

• Q6. A paratrooper falling with constant velocity is an example of:
• A. Static equilibrium
• B. Dynamic translational equilibrium
• C. Dynamic rotational equilibrium
• D. Unstable equilibrium

• Correct Answer: B
• Explanation: The text gives a paratrooper falling with constant velocity as an example of dynamic translational equilibrium.

• Q7. For an object to be in complete equilibrium, the net force and net torque must be:
• A. Maximum
• B. Minimum
• C. Zero
• D. Infinite

• Correct Answer: C
• Explanation: The text states that for complete equilibrium, the net force and torque on the object are zero.

• Q8. The principle of moments states that for an object in equilibrium, the sum of clockwise moments equals the sum of:
• A. Anticlockwise moments
• B. Forces
• C. Torques
• D. Weights

• Correct Answer: A
• Explanation: The text states that the sum of clockwise moments must be equal to the sum of anticlockwise moments.

• Q9. Which of the following is an example of static equilibrium?
• A. A car moving at constant speed
• B. A book resting on a table
• C. A rotating fan
• D. A satellite in orbit

• Correct Answer: B
• Explanation: A book at rest is in static equilibrium.

• Q10. Which of the following is an example of dynamic equilibrium?
• A. A book on a table
• B. A building
• C. A car moving at constant speed on a straight road
• D. A pendulum at its highest point

• Correct Answer: C
• Explanation: A car moving at constant speed on a straight road is in dynamic translational equilibrium.

Frequently Asked Questions (FAQs)

Q1. What is the difference between static and dynamic equilibrium? Answer: Static equilibrium is when the object is at rest (zero velocity). Dynamic equilibrium is when the object is moving with constant velocity (zero acceleration).

Q2. What is the principle of moments? Answer: The principle of moments states that for an object in rotational equilibrium, the sum of the clockwise moments about a pivot is equal to the sum of the anticlockwise moments about the same pivot.

Q3. Why are two conditions needed for complete equilibrium? Answer: The first condition ensures there is no translational acceleration. The second condition ensures there is no rotational acceleration. Both are needed for complete equilibrium.

Q4. What is a couple? Answer: A couple is a pair of equal and opposite parallel forces whose lines of action do not coincide. It produces a net torque but no net force, causing rotation without translation.

Q5. How is equilibrium related to Newton's laws? Answer: Equilibrium is a direct consequence of Newton's first law. When the net force is zero, an object remains at rest or moves with constant velocity (translational equilibrium). When the net torque is zero, an object does not rotate or rotates with constant angular velocity (rotational equilibrium).

Q6. What is the first condition of equilibrium? Answer: The first condition of equilibrium states that the vector sum of all forces acting on an object is zero. This ensures that the object has no translational acceleration.

Q7. What is the second condition of equilibrium? Answer: The second condition of equilibrium states that the sum of all torques acting on an object is zero. This ensures that the object has no rotational acceleration.

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