Newton's Laws and Momentum (Advanced)

Newton’s Second Law Of Motion

Building upon the concept of inertia introduced by the First Law, Newton's Second Law provides the quantitative relationship between force and the resulting change in motion. Unlike the First Law which describes the condition for no change in velocity (zero net force), the Second Law describes what happens when a non-zero net force acts on an object.

The Second Law introduces the concept of momentum, which is a measure of the "quantity of motion" an object possesses. For a single particle of mass $ m $ moving with velocity $ \vec{v} $, its linear momentum $ \vec{p} $ is defined as:

$ \vec{p} = m\vec{v} $

Momentum is a vector quantity, and its direction is the same as the velocity vector. The standard SI unit for momentum is kilogram-metre per second (kg m/s).

Statement of the Second Law

Newton's Second Law of Motion is formally stated in terms of momentum:

"The rate of change of the linear momentum of a body is directly proportional to the applied external unbalanced force acting on the body and takes place in the direction of the force."

This means that the faster an object's momentum changes, the larger the net force acting on it. Also, the change in momentum occurs in the direction of the net force.

Mathematical Formulation Of Second Law Of Motion

Mathematically, we can express this proportionality as:

$ \vec{F}_{net} \propto \frac{d\vec{p}}{dt} $

where $ \vec{F}_{net} $ is the net external force acting on the object, and $ \frac{d\vec{p}}{dt} $ is the instantaneous rate of change of momentum.

By choosing appropriate units for force, mass, and time (like the SI system), the constant of proportionality can be made equal to 1. Thus, the equation becomes:

$ \vec{F}_{net} = \frac{d\vec{p}}{dt} $

This is the fundamental form of Newton's Second Law. Now, let's substitute the definition of momentum, $ \vec{p} = m\vec{v} $:

$ \vec{F}_{net} = \frac{d(m\vec{v})}{dt} $

Using the product rule for differentiation, we get:

$ \vec{F}_{net} = m \frac{d\vec{v}}{dt} + \vec{v} \frac{dm}{dt} $

This is the general form of the Second Law, applicable even when the mass of the object changes over time (e.g., a rocket expelling fuel, a raindrop growing in size). The term $ \frac{d\vec{v}}{dt} $ is the acceleration $ \vec{a} $, and $ \frac{dm}{dt} $ is the rate of change of mass.

$ \vec{F}_{net} = m\vec{a} + \vec{v} \frac{dm}{dt} $

Special Case: Constant Mass

For the vast majority of problems in basic mechanics, the mass of the object remains constant during its motion ($ m = \text{constant} $). In this case, the rate of change of mass $ \frac{dm}{dt} = 0 $. The general equation simplifies to:

$ \vec{F}_{net} = m \frac{d\vec{v}}{dt} + \vec{v} (0) $

$ \vec{F}_{net} = m \frac{d\vec{v}}{dt} $

Since $ \vec{a} = \frac{d\vec{v}}{dt} $, we get the familiar form:

$ \vec{F}_{net} = m\vec{a} $

This simplified form is valid only when the mass of the object is constant. It states that the net force is directly proportional to the acceleration and the constant of proportionality is the mass. The direction of the acceleration is always the same as the direction of the net force.

Relation to First Law:

The First Law is a special case of the Second Law. If the net external force $ \vec{F}_{net} = 0 $, then from $ \vec{F}_{net} = m\vec{a} $ (for constant mass), we get $ 0 = m\vec{a} $. Since mass $ m $ cannot be zero for a physical object, the acceleration $ \vec{a} $ must be zero ($ \vec{a} = 0 $). Zero acceleration means the velocity $ \vec{v} $ is constant. This is precisely the statement of the First Law: an object maintains constant velocity (rest or uniform motion) if the net force is zero.

Impulse and Momentum:

Integrating the fundamental form of the Second Law $ \vec{F}_{net} = \frac{d\vec{p}}{dt} $ over a time interval from $ t_1 $ to $ t_2 $ gives:

$ \int_{t_1}^{t_2} \vec{F}_{net} \, dt = \int_{t_1}^{t_2} \frac{d\vec{p}}{dt} \, dt $

$ \int_{t_1}^{t_2} \vec{F}_{net} \, dt = \vec{p}(t_2) - \vec{p}(t_1) = \Delta \vec{p} $

The integral of the net force over time is called the Impulse ($ \vec{I} $) of the net force.

$ \vec{I} = \int_{t_1}^{t_2} \vec{F}_{net} \, dt $

For a constant net force, the impulse is simply $ \vec{I} = \vec{F}_{net} \Delta t $.

The equation $ \vec{I} = \Delta \vec{p} $ is known as the Impulse-Momentum Theorem. It states that the impulse of the net force acting on an object is equal to the change in its momentum. This theorem is particularly useful in analysing situations involving forces that act for a very short duration, such as impacts and collisions, where the force magnitude may vary rapidly with time.

Example (Impulse-Momentum Theorem):

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Newton’s Third Law Of Motion

Newton's Third Law addresses the nature of force as an interaction between two objects. It states that forces always occur in pairs, and these pairs have specific properties.

Statement of the Third Law

The formal statement of Newton's Third Law is:

"To every action, there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts."

In modern terms, this is usually expressed as:

"For every action force, there is an equal and opposite reaction force."

If object A exerts a force $ \vec{F}_{AB} $ on object B, then object B simultaneously exerts a force $ \vec{F}_{BA} $ on object A, such that:

$ \vec{F}_{AB} = - \vec{F}_{BA} $

Here, the magnitude of the force exerted by A on B is equal to the magnitude of the force exerted by B on A ($ |\vec{F}_{AB}| = |\vec{F}_{BA}| $), and these forces act in opposite directions.

Detailed Analysis of Action-Reaction Pairs

It is critical to understand that action and reaction forces have the following defining characteristics:

• Act on Different Bodies: This is the most crucial point. If the action force is on body B (exerted by A), the reaction force is on body A (exerted by B). They never act on the same body. This is why action-reaction pairs do not cancel each other out when considering the net force on a single body. They only "cancel" in the sense that their vector sum for the *entire system* of interacting bodies might be zero if they are the only forces.
• Equal in Magnitude: The strength of the action force is precisely equal to the strength of the reaction force.
• Opposite in Direction: The direction of the reaction force is exactly opposite to the direction of the action force.
• Simultaneous: The pair of forces exists only during the interaction. They appear and disappear together. One does not cause the other; they are two aspects of a single interaction.
• Same Nature: Both forces in the pair are of the same fundamental type (e.g., both are gravitational, both are electromagnetic, both are contact forces).

Example (Identifying Action-Reaction Pairs):

Answer:

The Third Law is fundamental to understanding how objects interact and forms the basis for the conservation of momentum in isolated systems.

Conservation Of Momentum

The Law of Conservation of Linear Momentum is a fundamental principle that arises directly from Newton's Laws, particularly the Third Law. It is an extremely useful concept for analyzing systems of interacting objects, especially during events like collisions and explosions.

Momentum of a System

For a system of multiple particles, the total linear momentum $ \vec{P}_{total} $ is the vector sum of the individual momenta of all the particles within the system. If a system consists of $ N $ particles with momenta $ \vec{p}_1, \vec{p}_2, \ldots, \vec{p}_N $, the total momentum is:

$ \vec{P}_{total} = \sum_{i=1}^{N} \vec{p}_i = \vec{p}_1 + \vec{p}_2 + \ldots + \vec{p}_N $

$ \vec{P}_{total} = \sum_{i=1}^{N} m_i \vec{v}_i = m_1 \vec{v}_1 + m_2 \vec{v}_2 + \ldots + m_N \vec{v}_N $

Isolated System

The Law of Conservation of Momentum applies to an isolated system. An isolated system is a system where the net external force acting on it is zero ($ \vec{F}_{ext, net} = 0 $).

Note that this does not mean there are no forces acting within the system (internal forces). Objects within the system can exert forces on each other. However, these internal forces always occur in action-reaction pairs, which, according to Newton's Third Law, are equal in magnitude and opposite in direction. As we will see, this property of internal forces is key to the conservation of momentum.

Statement of Conservation of Momentum

The Law of Conservation of Linear Momentum states:

"If the net external force acting on a system of particles is zero, then the total linear momentum of the system remains constant."

Mathematically, this can be expressed as: If $ \vec{F}_{ext, net} = 0 $, then $ \frac{d\vec{P}_{total}}{dt} = 0 $, which implies $ \vec{P}_{total} = \text{constant} $.

This means the total momentum of the system before an interaction (like a collision or explosion) is equal to the total momentum of the system after the interaction, provided no external forces acted during the interaction (or the net external force was zero).

$ \vec{P}_{total, \text{initial}} = \vec{P}_{total, \text{final}} $

Derivation from Newton's Laws

Consider a system of $ N $ particles. The rate of change of the total momentum of the system is given by:

$ \frac{d\vec{P}_{total}}{dt} = \frac{d}{dt} \left( \sum_{i=1}^{N} \vec{p}_i \right) = \sum_{i=1}^{N} \frac{d\vec{p}_i}{dt} $

According to Newton's Second Law, the rate of change of momentum of the $ i^{th} $ particle is equal to the net force acting on that particle:

$ \frac{d\vec{p}_i}{dt} = \vec{F}_{i, net} $

The net force on the $ i^{th} $ particle can be divided into two parts: external forces ($ \vec{F}_{i, ext} $) exerted by agents outside the system, and internal forces ($ \vec{F}_{i, int} $) exerted by other particles within the system.

$ \vec{F}_{i, net} = \vec{F}_{i, ext} + \vec{F}_{i, int} $

So, $ \frac{d\vec{P}_{total}}{dt} = \sum_{i=1}^{N} (\vec{F}_{i, ext} + \vec{F}_{i, int}) = \sum_{i=1}^{N} \vec{F}_{i, ext} + \sum_{i=1}^{N} \vec{F}_{i, int} $.

The sum of all external forces on all particles is the net external force on the system:

$ \vec{F}_{ext, net} = \sum_{i=1}^{N} \vec{F}_{i, ext} $

Now consider the sum of all internal forces. Internal forces always occur in action-reaction pairs between particles within the system. For any pair of particles $ i $ and $ j $, the force exerted by $ i $ on $ j $ ($ \vec{F}_{ji} $) and the force exerted by $ j $ on $ i $ ($ \vec{F}_{ij} $) satisfy Newton's Third Law:

$ \vec{F}_{ij} = - \vec{F}_{ji} $

When we sum all the internal forces in the system, every action force ($ \vec{F}_{ij} $) has a corresponding equal and opposite reaction force ($ \vec{F}_{ji} $), which is also an internal force within the system. Therefore, the sum of all internal forces is zero:

$ \sum_{i=1}^{N} \vec{F}_{i, int} = 0 $

Substituting these back into the equation for the rate of change of total momentum:

$ \frac{d\vec{P}_{total}}{dt} = \vec{F}_{ext, net} + 0 $

$ \frac{d\vec{P}_{total}}{dt} = \vec{F}_{ext, net} $

This equation is essentially Newton's Second Law applied to the entire system of particles: the rate of change of the system's total momentum is equal to the net external force acting on the system.

From this equation, if the net external force $ \vec{F}_{ext, net} = 0 $, then $ \frac{d\vec{P}_{total}}{dt} = 0 $, which means the total momentum $ \vec{P}_{total} $ is constant. This proves the Law of Conservation of Linear Momentum from Newton's Second and Third Laws.

Conservation of Momentum in Different Dimensions:

Momentum is a vector. In vector form, the conservation of momentum equation is $ \vec{P}_{initial} = \vec{P}_{final} $. This vector equation is equivalent to three separate scalar equations, one for each dimension (x, y, z). If the net external force is zero, then the total momentum is conserved along each coordinate axis independently:

$ P_{total, x, initial} = P_{total, x, final} $ $ P_{total, y, initial} = P_{total, y, final} $ $ P_{total, z, initial} = P_{total, z, final} $

This means if the net external force in, say, the x-direction is zero, the total momentum in the x-direction is conserved, even if there is a net external force (and thus momentum change) in the y or z directions.

Examples (More Advanced):

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Answer:

The Law of Conservation of Momentum is a very general law and is valid in situations where energy may or may not be conserved (e.g., inelastic collisions). Its application relies on identifying the system correctly and determining if the net external force on that system is zero.