Constraint Relations — JEE Advanced Physics

Key Concepts

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Examples

Examples will be added here.

Introduction to Constraint Relations

Welcome to Constraint Relations. Content to be added.

Connected Systems and Constraint Motion

Systems of Connected Objects

Many problems involve multiple objects connected by strings, in contact, or constrained to move together. Two approaches:

1. System Approach: Treat all objects as one system, find common acceleration

2. Individual FBD Approach: Draw separate FBDs for each object, solve simultaneously

Often, you use both: system approach for acceleration, individual FBDs for internal forces (tension, contact forces).

Example 1: Two Blocks in Contact

Two blocks ( $m_1 = 2$ kg, $m_2 = 4$ kg) on smooth surface. Force $F = 30$ N pushes $m_1$ toward $m_2$ . Find acceleration and contact force.

Two blocks pushed together

System approach: Total mass $M = 6$ kg

$F = Ma \implies 30 = 6a \implies a = 5 \, \text{m/s}^2$

Contact force N (using FBD of $m_2$ ):

FBD of second block

$N = m_2 a = 4 \times 5 = 20 \, \text{N}$

In the two-block system, is the contact force between blocks less than the applied force?

Example 2: Two Blocks Connected by String

Two blocks ( $m_1 = 5$ kg, $m_2 = 3$ kg) connected by string on smooth surface. Force $F = 40$ N applied to $m_1$ . Find tension.

Two blocks connected by string

System: $M = 8$ kg, $a = F/M = 40/8 = 5 \, \text{m/s}^2$

Tension (using FBD of $m_2$ ):

FBD of second block

$T = m_2 a = 3 \times 5 = 15 \, \text{N}$

Atwood Machine

Classic problem: Two masses connected over a pulley.

Atwood machine setup

Example: $m_1 = 2$ kg, $m_2 = 3$ kg. Find acceleration and tension.

FBD of mass 1

FBD of mass 2

Solution: $m_2$ accelerates down, $m_1$ up, magnitude $a$

For $m_1$ (up positive): $T - 20 = 2a$ ... (1)

For $m_2$ (down positive): $30 - T = 3a$ ... (2)

Add: $10 = 5a \implies a = 2 \, \text{m/s}^2$

From (1): $T = 20 + 4 = 24 \, \text{N}$

In an Atwood machine, is the tension between the two weights (allowing heavier to accelerate down)?

General Atwood Machine Formula

For masses $m_1$ and $m_2$ (where $m_2 > m_1$ ):

Acceleration:

$a = \dfrac{(m_2 - m_1)g}{m_1 + m_2}$

Tension:

$T = \dfrac{2m_1 m_2 g}{m_1 + m_2}$

These formulas can be derived by solving the force equations as shown above.

Example: Three Blocks in Contact

Three blocks (1 kg, 2 kg, 3 kg) pushed by $F = 24$ N on smooth surface. Find accelerationand contact forces.

Three blocks pushed

System: $M = 6$ kg, $a = 24/6 = 4 \, \text{m/s}^2$

Contact forces:

FBD of third block

For $m_3$ : $N_{23} = m_3 a = 3 \times 4 = 12$ N

FBD of second block

For $m_2$ : $N_{12} - N_{32} = m_2 a \implies N_{12} = 12 + 8 = 20$ N

Contact forces: $N_{12} = 20$ N, $N_{23} = 12$ N

Key Concepts

Key concepts will be added here.

Examples

Examples will be added here.

Introduction to Constraint Relations

Welcome to Constraint Relations. Content to be added.

Connected Systems and Constraint Motion

Systems of Connected Objects

Many problems involve multiple objects connected by strings, in contact, or constrained to move together. Two approaches:

1. System Approach: Treat all objects as one system, find common acceleration

2. Individual FBD Approach: Draw separate FBDs for each object, solve simultaneously

Often, you use both: system approach for acceleration, individual FBDs for internal forces (tension, contact forces).

Example 1: Two Blocks in Contact

Two blocks ( $m_1 = 2$ kg, $m_2 = 4$ kg) on smooth surface. Force $F = 30$ N pushes $m_1$ toward $m_2$ . Find acceleration and contact force.

Two blocks pushed together

System approach: Total mass $M = 6$ kg

$F = Ma \implies 30 = 6a \implies a = 5 \, \text{m/s}^2$

Contact force N (using FBD of $m_2$ ):

FBD of second block

$N = m_2 a = 4 \times 5 = 20 \, \text{N}$

In the two-block system, is the contact force between blocks less than the applied force?

Example 2: Two Blocks Connected by String

Two blocks ( $m_1 = 5$ kg, $m_2 = 3$ kg) connected by string on smooth surface. Force $F = 40$ N applied to $m_1$ . Find tension.

Two blocks connected by string

System: $M = 8$ kg, $a = F/M = 40/8 = 5 \, \text{m/s}^2$

Tension (using FBD of $m_2$ ):

FBD of second block

$T = m_2 a = 3 \times 5 = 15 \, \text{N}$

Atwood Machine

Classic problem: Two masses connected over a pulley.

Atwood machine setup

Example: $m_1 = 2$ kg, $m_2 = 3$ kg. Find acceleration and tension.

FBD of mass 1

FBD of mass 2

Solution: $m_2$ accelerates down, $m_1$ up, magnitude $a$

For $m_1$ (up positive): $T - 20 = 2a$ ... (1)

For $m_2$ (down positive): $30 - T = 3a$ ... (2)

Add: $10 = 5a \implies a = 2 \, \text{m/s}^2$

From (1): $T = 20 + 4 = 24 \, \text{N}$

In an Atwood machine, is the tension between the two weights (allowing heavier to accelerate down)?

General Atwood Machine Formula

For masses $m_1$ and $m_2$ (where $m_2 > m_1$ ):

Acceleration:

$a = \dfrac{(m_2 - m_1)g}{m_1 + m_2}$

Tension:

$T = \dfrac{2m_1 m_2 g}{m_1 + m_2}$

These formulas can be derived by solving the force equations as shown above.

Example: Three Blocks in Contact

Three blocks (1 kg, 2 kg, 3 kg) pushed by $F = 24$ N on smooth surface. Find accelerationand contact forces.

Three blocks pushed

System: $M = 6$ kg, $a = 24/6 = 4 \, \text{m/s}^2$

Contact forces:

FBD of third block

For $m_3$ : $N_{23} = m_3 a = 3 \times 4 = 12$ N

FBD of second block

For $m_2$ : $N_{12} - N_{32} = m_2 a \implies N_{12} = 12 + 8 = 20$ N

Contact forces: $N_{12} = 20$ N, $N_{23} = 12$ N