M06Linear Momentum, Collisions, and Center of Mass

Linear Momentum

Momentum and Its Relation to Force

Linear momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is defined as the product of an object's mass and velocity.

• Definition: The linear momentum p of an object is given by $p = m v$, where m is mass and v is velocity.

• Relation to Force: The net force acting on an object is equal to the rate of change of its momentum: $F_{net} = \frac{dp}{dt}$.

• Impulse: The change in momentum of an object is called impulse, given by $J = F_{avg} \Delta t = \Delta p$.

• Example: A rocket sled or loaded cart on tracks demonstrates momentum when moving and slowing down due to friction.

Conservation of Momentum

The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it.

• Equation: $\sum p_{initial} = \sum p_{final}$

• Application: Used to analyze collisions and explosions.

• System Selection: Choose the system carefully to ensure all relevant forces are considered.

• Example: Two cars colliding and moving together after impact.

Collisions and Impulse

Collisions are events where two or more bodies exert forces on each other for a short time, resulting in changes in their momenta.

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

• Inelastic Collision: Momentum is conserved, but kinetic energy is not.

• Impulse-Momentum Theorem: $F_{avg} \Delta t = m v_f - m v_i$

• Example: A ball bouncing off a wall.

Center of Mass (CM)

Definition and Calculation

The center of mass is the point at which the mass of a system or body can be considered to be concentrated for the purpose of analyzing translational motion.

• Equation for Discrete Particles: $x_{CM} = \frac{\sum m_i x_i}{\sum m_i}$, $y_{CM} = \frac{\sum m_i y_i}{\sum m_i}$

• Equation for Continuous Mass Distribution: $x_{CM} = \frac{1}{M} \int x \, dm$

• Physical Meaning: The CM moves as if all mass and external forces act at that point.

• Example: A diver's CM follows a smooth path during a jump.

CM for the Human Body

For complex objects like the human body, the CM can be found by considering the CM of individual parts and their relative positions.

• Application: Used in biomechanics, sports science, and ergonomics.

• Example: The CM of a person changes position depending on body posture.

Distance of Joint from Floor (%)	Body Part	Center of Mass (%)	Percent Mass (%)
91.2%	Base of skull	Head and neck: 50%	6.9%
85.5%	Shoulder joint	Upper trunk: 46%	15.7%
25.8%	Hip joint	Lower trunk: 49%	43.2%
4.6%	Ankle joint	Lower leg: 43%	15.7%
*	Body CM	~54.8%	100%

Additional info: The table above summarizes the center of mass locations for major body parts as a percentage of height from the floor, along with their percent mass.

Center of Gravity

The center of gravity is the point where the gravitational force can be considered to act on an object. For most practical purposes, it coincides with the center of mass.

• Equation: $y_{CG} = \frac{\sum m_i y_i}{\sum m_i}$

• Application: Important in stability analysis and engineering design.

• Example: Balancing a wrench horizontally at its center of gravity.

Summary Table: Key Concepts

Concept	Definition	Key Equation
Linear Momentum	Product of mass and velocity	$p = m v$
Impulse	Change in momentum	$J = F_{avg} \Delta t$
Conservation of Momentum	Total momentum remains constant in closed system	$\sum p_{initial} = \sum p_{final}$
Center of Mass	Weighted average position of mass	$x_{CM} = \frac{\sum m_i x_i}{\sum m_i}$
Center of Gravity	Point where gravity acts	$y_{CG} = \frac{\sum m_i y_i}{\sum m_i}$