Momentum, Collisions, and Center of Mass – Study Notes

Chapter 8: Momentum

Introduction to Momentum

Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is a vector quantity, meaning it has both magnitude and direction, and is conserved in isolated systems.

• Definition: The momentum p of an object is defined as the product of its mass m and velocity v:

$p = m v$

• Conservation of Momentum: In the absence of external forces, the total momentum of a system remains constant.

• Applications: Momentum is crucial in analyzing collisions, explosions, and the motion of systems of particles.

Collisions

Types of Collisions

Collisions are interactions between two or more bodies where momentum is transferred. They are classified based on energy conservation:

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Momentum is conserved, but kinetic energy is not. Some energy is transformed into other forms (e.g., heat, deformation).

• Completely Inelastic Collision: The colliding objects stick together after the collision, moving as a single mass.

Head-On Collisions (One-Dimensional)

In head-on (1D) collisions, all motion and momentum are along a single axis. Conservation of momentum allows us to solve for unknown velocities after the collision.

• Conservation of Momentum Equation:

$m_A v_{A,i} + m_B v_{B,i} = m_A v_{A,f} + m_B v_{B,f}$

• Example: Two blocks, A (0.50 kg) and B (0.30 kg), move toward each other with velocities of 2.0 m/s and -2.0 m/s, respectively. After an elastic collision, their velocities are reversed.

Completely Inelastic Collisions

In a completely inelastic collision, the objects stick together after impact. The final velocity can be found using conservation of momentum:

• Equation:

$m_A v_{A,i} + m_B v_{B,i} = (m_A + m_B) v_f$

• Example: Two blocks (0.50 kg and 0.30 kg) collide and stick together. Their combined final velocity is determined by the above equation.

Collisions in Two Dimensions (Horizontal Plane)

Collisions can also occur in a plane, requiring conservation of momentum in both the x and y directions.

• Conservation Equations:

$x$-direction: $m_A v_{A,i,x} + m_B v_{B,i,x} = m_A v_{A,f,x} + m_B v_{B,f,x}$ $y$-direction: $m_A v_{A,i,y} + m_B v_{B,i,y} = m_A v_{A,f,y} + m_B v_{B,f,y}$

• Example: Two spheres collide at an angle, and their final velocities are determined by solving the above equations for both axes.

Center of Mass

Definition and Properties

The center of mass of an object or system is the average position of all the mass that makes up the object. It is an imaginary point in space and may not be located on the object itself.

• Center of Gravity: If the acceleration due to gravity is the same everywhere, the center of mass coincides with the center of gravity.

• Importance: The center of mass is useful for analyzing the motion of extended bodies and systems of particles.

Center of Mass of Common Objects

For homogeneous objects with regular shapes, the center of mass is located at the geometric center. For objects with an axis of symmetry, the center of mass lies along that axis.

Object	Center of Mass Location
Cube	Geometric center
Sphere	Geometric center
Cylinder	Geometric center
Disk	Center of symmetry
Donut (Torus)	Center of symmetry (may not be within the object)

Mathematical Definition of Center of Mass

For a system of particles, the center of mass coordinates $(x_{cm}, y_{cm})$ are given by:

$x_{cm} = \frac{m_A x_A + m_B x_B + m_C x_C + \cdots}{m_A + m_B + m_C + \cdots}$ $y_{cm} = \frac{m_A y_A + m_B y_B + m_C y_C + \cdots}{m_A + m_B + m_C + \cdots}$

• Units: $x_{cm}$ and $y_{cm}$ have units of meters.

• Extension: The velocity and acceleration of the center of mass can be found by differentiating these expressions with respect to time.

Acceleration of the Center of Mass

When a system is acted on by external forces, the center of mass moves as if all the mass were concentrated at that point and acted on by the resultant external force.

$\sum \vec{F}_{ext} = M \vec{a}_{cm}$

• Interpretation: The motion of the center of mass reflects the net effect of all external forces on the system.

Explosions and the Center of Mass

When an object explodes (e.g., a stationary bomb), the center of mass of the system continues to move according to the initial momentum, even though the individual fragments may move in different directions.

• Example: After a shell explodes, the center of mass of the fragments follows the trajectory the shell would have taken if it had not exploded.

Worked Example: Spring-Loaded Blocks

Problem Statement

Block A (1 kg) and block B (3 kg) are compressed together with a spring between them and released from rest on a frictionless surface. The spring is not attached and drops away after expansion. Block B acquires a speed of 2 m/s. Find:

• Final speed of block A

• Potential energy stored in the spring

Solution Outline

• Conservation of Momentum: Since the system starts from rest, total initial momentum is zero.

$0 = m_A v_{A,f} + m_B v_{B,f}$ $\Rightarrow v_{A,f} = -\frac{m_B}{m_A} v_{B,f}$ $= -\frac{3}{1} \times 2 = -6 \text{ m/s}$

• Conservation of Energy: The potential energy stored in the spring is converted into the kinetic energy of both blocks.

$U = \frac{1}{2} m_A v_{A,f}^2 + \frac{1}{2} m_B v_{B,f}^2$ $= \frac{1}{2} \times 1 \times 36 + \frac{1}{2} \times 3 \times 4 = 18 + 6 = 24 \text{ J}$

• Interpretation: The negative sign for $v_{A,f}$ indicates block A moves in the opposite direction to block B.

Additional info: These notes expand on the original slides by providing full definitions, equations, and worked examples for clarity and completeness.