Newton's Second Law: the Momentum Principle

Claimed by Arayna Saxena, Fall 2024

This page describes Newton's second law of motion, also known as the momentum principle, which relates net force to the change in Linear Momentum. This principle is used to predict the effects of forces on the motion of objects.

Newton's Second Law is fundamental to understanding motion and lays the groundwork for classical mechanics and modern physics applications.For instance, understanding the force needed to accelerate a car, or how a rocket changes velocity in space, relies on this principle.

The Main Idea

Newton's second law of motion, also known as the momentum principle, explains how forces cause the momentum of a system to change over time.

Linear Momentum ([math]\displaystyle{ \vec{p} }[/math]) is the product of an object's mass ([math]\displaystyle{ m }[/math]) and velocity ([math]\displaystyle{ \vec{v} }[/math]):

[math]\displaystyle{ \vec{p} = m\vec{v} }[/math].

Net Force ([math]\displaystyle{ \vec{F}_{net} }[/math]) is the total force acting on a system.

The principle states that:

[math]\displaystyle{ \vec{F}{net} = \frac{d\vec{p}{system}}{dt} }[/math]

where:

[math]\displaystyle{ \vec{p} }[/math] is the Linear Momentum of a system, [math]\displaystyle{ \vec{F}_{net} }[/math] is the external Net Force acting on the system from its surroundings, and [math]\displaystyle{ t }[/math] is time. Often, the system in question consists of a single particle whose motion we want to predict, although the law is also true for systems of particles and non-particle distributions of mass, such as disks. Note that both force and momentum are vector quantities, and that the change in momentum as a result of a force will always be in the direction of that force.

These examples illustrate how the momentum principle applies in different scenarios:

If a particle travels in a straight line at a constant speed, its momentum is constant, so there is no net force acting on the particle.
If a particle travels in a straight line while speeding up, its momentum is increasing, so there must be a net force acting in the same direction as the particle's momentum.
If a particle travels in a straight line while slowing down, its momentum is decreasing, so there must be a net force acting in the opposite direction as the particle's momentum.
If a particle travels along a curving path, its momentum changes direction, and a force not parallel to the object's momentum must be acting on it.

The momentum principle has no derivation, as it is considered the definition of force. The metric unit most commonly used for force is the Newton.

A Mathematical Model

The momentum principle states that:

[math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math]

where [math]\displaystyle{ \vec{p} }[/math] is the momentum of the system. Momentum is defined as:

[math]\displaystyle{ \vec{p} = m\vec{v} }[/math]

Applying the product rule to the time derivative of momentum:

[math]\displaystyle{ \frac{d\vec{p}}{dt} = m\frac{d\vec{v}}{dt} + \vec{v}\frac{dm}{dt} }[/math]

For systems where the mass of the particle or system is constant over time (e.g., most classical mechanics problems), the second term [math]\displaystyle{ \vec{v}\frac{dm}{dt} }[/math] becomes zero. This simplifies the momentum principle to:

[math]\displaystyle{ \vec{F}{net} = m\frac{d\vec{v}}{dt} }[/math] or equivalently [math]\displaystyle{ \vec{F}{net} = m\vec{a} }[/math],

which is the more familiar form of Newton's Second Law. However, the general form [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math] is preferred in many scenarios for the following reasons:

Changing Mass Systems: When the mass of a particle or system changes with time (e.g., a rocket expelling fuel or a snowball gaining mass as it rolls downhill), the term [math]\displaystyle{ \vec{v}\frac{dm}{dt} }[/math] is no longer zero. In such cases, [math]\displaystyle{ \vec{F}_{net} = m\vec{a} }[/math] is inaccurate, but the general form [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math] correctly accounts for these changes.
Relativistic Speeds: As a particle approaches the speed of light, classical mechanics fails to accurately describe the motion, and relativistic momentum must be used: [math]\displaystyle{ \vec{p} = \frac{m\vec{v}}{\sqrt{1 - \frac{v^2}{c^2}}} }[/math], where [math]\displaystyle{ c }[/math] is the speed of light. In these cases, [math]\displaystyle{ \vec{F}_{net} = m\vec{a} }[/math] is invalid, while [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math] remains accurate when relativistic momentum is substituted.
Rotational Analogues: The form [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math] provides a direct analogy to rotational motion, where **torque** ([math]\displaystyle{ \vec{\tau} }[/math]) is defined as the rate of change of **angular momentum** ([math]\displaystyle{ \vec{L} }[/math]): [math]\displaystyle{ \vec{\tau} = \frac{d\vec{L}}{dt} }[/math] Here: [math]\displaystyle{ \vec{\tau} }[/math] is the torque applied to a system, [math]\displaystyle{ \vec{L} }[/math] is the angular momentum ([math]\displaystyle{ \vec{L} = \vec{r} \times \vec{p} }[/math], where [math]\displaystyle{ \vec{r} }[/math] is the position vector). This analogy is valuable because many principles in linear motion (e.g., momentum, force) extend directly to rotational dynamics (e.g., angular momentum, torque). By learning linear dynamics in the general form [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math], students can more easily understand and apply analogous principles to rotational systems.

Despite these advantages, the relation [math]\displaystyle{ \vec{F}_{net} = m\vec{a} }[/math] is often sufficient for systems with constant mass moving at non-relativistic speeds. It is especially useful when combined with kinematic equations that utilize acceleration [math]\displaystyle{ \vec{a} }[/math] to predict motion.

Impulse and Momentum

The momentum principle also establishes the relationship between Impulse and Momentum:

[math]\displaystyle{ \vec{F}_{net} \Delta t = \Delta \vec{p} }[/math]

Here, [math]\displaystyle{ \vec{F}_{net} \Delta t }[/math] is the impulse applied to a system, and [math]\displaystyle{ \Delta \vec{p} }[/math] is the resulting change in momentum. This relationship is particularly useful for analyzing systems where forces act over short time intervals, such as collisions in sports or car crashes.

Conservation of Momentum

When the net external force on a system is zero (i.e., the system is isolated or closed), the momentum principle simplifies to:

[math]\displaystyle{ \frac{d\vec{p}}{dt} = 0 }[/math]

This implies that the system's total momentum remains constant:

[math]\displaystyle{ \vec{p}{initial} = \vec{p}{final} }[/math]

This principle is known as Conservation of Momentum, and it underpins many physical phenomena, including:

Elastic and inelastic collisions. Rocket propulsion in space, where the expelled fuel and rocket system together conserve momentum. By understanding these fundamental relationships, we gain insight into how forces interact with systems to produce motion and momentum changes under various conditions.

A Computational Model

The relationship between net force and change in momentum can be computationally modeled using the momentum update formula:

[math]\displaystyle{ p_2 = p_1 + F_{net} \cdot \Delta t }[/math].

This is a rewriting of [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math], where mass is assumed to be constant.

Steps in the Computational Model Define Momentum: Momentum is calculated as the object’s mass multiplied by its velocity in the initial conditions. Iterative Prediction: For each time step: Update [math]\displaystyle{ F_{net} }[/math] by summing all possible forces acting on the object during the simulation. Use the momentum principle to update the momentum: [math]\displaystyle{ p = p + F_{net} \cdot \Delta t }[/math]. Calculate the velocity from momentum: [math]\displaystyle{ \vec{v} = \frac{\vec{p}}{m} }[/math]. Update the position: [math]\displaystyle{ \vec{r}_2 = \vec{r}_1 + \vec{v} \cdot \Delta t }[/math].

Model Description

This simulation models the motion of a falling object under the force of gravity, using the momentum principle. The object's position and velocity are updated iteratively, and the output is visualized in real-time using graphs and arrows representing forces.

Key Features

Interactive Visualization: The simulation shows the position, velocity, and force vectors in real time. Graphing Tools: Position and velocity vs. time graphs are plotted, enabling analysis of motion dynamics. Customizability: Parameters like mass, initial position, initial velocity, and time step ([math]\displaystyle{ \Delta t }[/math]) can be adjusted for different scenarios. You can interact with the simulation on GlowScript here:

https://www.glowscript.org/#/user/Arayna%5fSaxena/folder/MyPrograms/program/Lab2onlygravity

Examples

1. (Simple)

At t=0, a 4kg particle is released from rest and a constant 12N force begins acting on it. How far has the particle moved after 5 seconds?

For this problem, it is easier to use [math]\displaystyle{ \vec{F}_{net} = m\vec{a} }[/math] than [math]\displaystyle{ \vec{F}_{net} = \frac{d\vec{p}}{dt} }[/math] because it allows us to solve for [math]\displaystyle{ \vec{a} }[/math] which can then be used in a kinematic equation. We know this form of Newton's second law is accurate for this problem because the particle travels at non-relativistic speeds and is of constant mass.

[math]\displaystyle{ \vec{F}_{net} = m\vec{a} }[/math] can be rearranged into [math]\displaystyle{ \vec{a} = \frac{\vec{F}_{net}}{m} }[/math]

[math]\displaystyle{ a = \frac{12}{4} = 3 }[/math] (direction doesn't matter for this problem.)

[math]\displaystyle{ x_f = \frac{1}{2}at^2 + v_it + x_i = \frac{1}{2}(3)(5^2) = \frac{75}{2} }[/math] = 37.5 m.

2. (Middling)

A tennis ball of mass mB = 0.8kg and an initial velocity of ViB = <7,0,0> m/s. The tennis ball collides head-on with a lump of mashed potatoes with a mass mP = 3kg that was initially at rest. The tennis ball becomes embedded in the mush. Determine the velocity of the tennis ball and mashed potatoes after the collision, assuming that this collision is totally inelastic. Calculate how much energy is lost and determine the tennis ball and mashed potatoes individual velocities after the collision if the collision was elastic.

If this collision is inelastic, then we know that kinetic energy is not conserved. However, momentum is conserved for all types of collisions (elastic, inelastic, partially elastic) due to the momentum principle. To start, write out the momentum principle. Since the tennis ball and mashed potatoes are stuck together after the collision, the final momentum can be written with their combined mass:

File:Momentumconserved.jpg

alt text

File:Momentumconserved.jpg

File:MCP2.jpg

alt text

3. (Difficult)

A cylindrical rocket ship of mass [math]\displaystyle{ M }[/math] and speed [math]\displaystyle{ v }[/math] is cruising through outer space when it encounters a stationary dust cloud. The dust cloud has a number density of [math]\displaystyle{ N }[/math] particles per unit volume, and each dust particle has a mass of [math]\displaystyle{ m }[/math]. The rocket ship has a sticky surface on its front face that allows it to accumulate all of the dust that it hits, absorbing it into its mass. The radius of the rocket ship's circular face is [math]\displaystyle{ R }[/math]. What force does the rocket's thruster need to exert on it in order to preserve its speed as it moves through the dust cloud? (Assume that the consumption of fuel has a negligible impact on the mass of the rocket.)

Advanced Note: in order to use [math]\displaystyle{ \frac{dm}{dt} }[/math] for the accumulation of mass, the accumulated mass (i.e. the dust) must initially be at rest, like in the above problem. Otherwise, one must treat the accumulation of dust particles as a series of collisions, or change one's frame of reference so that the accumulated mass is initially at rest.

4. (Difficult)

A particle of mass [math]\displaystyle{ m }[/math] and speed [math]\displaystyle{ v }[/math] moves in a circular path. Its angular frequency is [math]\displaystyle{ \Omega }[/math]. (Angular frequency is the rate at which the angle of the particle's position is changing in radians per unit time.) Using the momentum principle, show that

a) a nonzero net force is acting on the particle,

b) the magnitude of the force is given by [math]\displaystyle{ f = mv\Omega }[/math], and

c) the direction of the force is inwards towards the center of the circle.

a) The particle's momentum vector is changing over time; its magnitude is constant (mv), but its direction changes as its position along the circular path changes. It is always tangential to the circular path. Since [math]\displaystyle{ \vec{p} }[/math] changes over time, a nonzero net force must be acting on the particle. In fact, the only way a particle of constant mass can have a net force of 0 acting on it is if it is travelling along a straight line at a constant speed, or if it is at rest.

b)

c) As one can see from the diagram above, [math]\displaystyle{ d\vec{p} }[/math] is pointing towards the center of the circle. If the particle is exactly at the top of the circle, its momentum is to the right (or left if the particle is travelling counterclockwise), and as [math]\displaystyle{ d\theta }[/math] approaches 0, [math]\displaystyle{ d\vec{p} }[/math] approaches a downward direction, which is towards the center of the circle.

For more information, see Centripetal Force and Curving Motion. Our answer agrees with the formulas given on that page.

Connectedness

Newton's second law is applicable to any situation where a force is applied and a system's momentum changes. There are nearly limitless examples of such situations, as well as nearly limitless applications. A few can be found below.

Scenario: bungee jump

When a bungee jumper falls and their cord is pulled taut, the cord applies a force in the direction opposite to the jumper's direction of travel. When this force is greater in magnitude than the gravity force, the net force is upward, which cuses causes the jumper's momentum, which is initially downward, to change over time in the upward direction. Eventually, the jumper is pulled back to a high altitude, the cord becomes slack, and gravity is once again the dominant force. The net force is downward once more, and the jumper's momentum changes over time in the downward direction.

Application: automobile industry

Cars frequently undergo all sorts of acceleration; they speed up, turn, and slow down. Forces are responsible for all of those accelerations, and these forces ultimately come from the tires' contact with the road. This force is, in fact, a form of Static Friction. The magnitude of this static friction force can only be up to a certain quantity known as the maximum static friction force. This maximum is determined by the shape and materials of the tires, as well as the weight of the car. Automobile manufacturers must always be aware of this maximum friction force when designing their car, as it places a limit on the ability of the car to accelerate, turn, and brake without slipping.

History

In his 1687 work Principia Mathematica, Isaac Newton (1643-1727) published his three laws of motion. Although his first and third laws were inspired by other scientists' work, his second law was entirely original. Below was the law as it appeared in his book:

Original Latin:

“Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.”

This was translated closely in Motte's 1729:

“Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.”

In other words, the rate of change of momentum of a body is proportional to the force impressed on the body, and happens in the direction of that force.

This law was the first time force had ever been defined, and it is a definition that is still used today. As a result of Newton's contribution to the science of forces, the unit of force most commonly still used today is named after him.