1.7 - Newton's Second Law

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Introduction

Newton's Second Law of Motion is the cornerstone of classical mechanics. It quantifies the relationship between force, mass, and acceleration, providing the mathematical framework for understanding how objects respond to forces. This law allows us to predict and calculate the motion of objects when forces act upon them, making it one of the most powerful and widely used principles in physics.

Newton's Second Law of Motion:

Newton's original formulation states that the net force $\mathbf{F}_\text{net}$ acting on an object equals the time rate of change of its momentum $\dot{\mathbf{p}}$:

$$\mathbf{F}_\text{net} = \dot{\mathbf{p}} $$

where $\mathbf{p} = m\mathbf{v}$ is the momentum of the object. For objects with constant mass, this reduces to the more familiar form:

$$\mathbf{F}_\text{net} = m\mathbf{a}$$

where $m$ is the mass of the object and $\mathbf{a}$ is its acceleration.

Units:
  • Force $\mathbf{F}_\text{net}$ is measured in Newtons (N)
  • Mass $m$ is measured in kilograms (kg)
  • Acceleration $\mathbf{a}$ is measured in meters per second squared (m/s²)

Both equations are vector equations, meaning both the magnitude and direction of the net force determine the magnitude and direction of the change in momentum (or acceleration).

Relationship between the Momentum and Acceleration Forms of Newton's Second Law

The momentum form of Newton's Second Law is more general than the acceleration form. For systems where mass changes (like rockets expelling fuel), we must treat the mass as a time-dependent quantity. We can use the product rule for derivatives to write the momentum form as:

$$\mathbf{F}_\text{net} = \dot{\mathbf{p}} = \frac{d(m\mathbf{v})}{dt} = m\dot{\mathbf{v}} + \dot{m}\mathbf{v}$$

where $\dot{m}$ is the time rate of change of the mass. Only when the mass is constant (i.e. $\dot{m} = 0$) does this reduce to the acceleration form $\mathbf{F}_\text{net} = m\dot{\mathbf{v}}= m\mathbf{a}$.

Momentum vs. Acceleration Form of Newton's Second Law:
  • Use $\mathbf{F}_\text{net} = m\mathbf{a}$ when:
    • The object's mass is constant
    • You need to find acceleration directly
    • Working with rigid bodies or particles
    • Solving typical mechanics problems
  • Use $\mathbf{F}_\text{net} = \dot{\mathbf{p}}$ when:
    • The object's mass changes with time (rockets, conveyor belts)
    • Working with systems that gain or lose mass
    • You need to analyze momentum changes directly
    • Dealing with relativistic mechanics (with appropriate momentum definition)

Component Form of Newton's Second Law

Since Newton's Second Law is a vector equation, we can write it in component form. In Cartesian coordinates:

$$F_{\text{net},x} = ma_x \quad \quad F_{\text{net},y} = ma_y \quad \quad F_{\text{net},z} = ma_z$$

If the $x$, $y$, and $z$ force components are independent of each other we can analyze motion in each direction independently. The net force in the $x$-direction determines the acceleration in the $x$-direction, and similarly for the $y$- and $z$-directions.

Relationship to Newton's First Law

Newton's first law Newton's First Law is actually a special case of the Second Law. When $\mathbf{F}_\text{net} = \mathbf{0}$, we have:

$$\dot{\mathbf{p}} = \mathbf{0} \quad \Rightarrow \quad \mathbf{p} = \text{constant} \quad \Rightarrow \quad \mathbf{v} = \text{constant}$$

(For constant mass, this gives $m\mathbf{a} = \mathbf{0} \Rightarrow \mathbf{a} = \mathbf{0} \Rightarrow \mathbf{v} = \text{constant}$.)

This is exactly what the First Law states: an object with no net force maintains constant momentum (and thus constant velocity for constant mass, or remains at rest). The Second Law is more general and applies to all situations, including when forces are present.

Newton's Second Law as a System of Differential Equations

The acceleration form of Newton's Second Law can be written as a vector differential equation by substituting $\mathbf{a} = \ddot{\mathbf{r}}$:

$${\mathbf{a}} = \frac{1}{m} \mathbf{F}_\text{net}$$
becomes
$$\ddot{\mathbf{r}} = \frac{1}{m} \mathbf{F}_\text{net}.$$

In one dimension, Newton's Second Law becomes a single, second-order differential equation:

$$\ddot{x} = \frac{1}{m} F_\text{net}(x, v, t)$$
where the net force $F_\text{net}$ can, in general, be a function of the position $x$, the velocity $v$, and the time $t$. The details of the physics problem will determine the specific form of the net force. From there, one often solves the differential equation (either analytically or numerically) to find the position $x(t)$ of the object as a function of time. Sometimes, we may want to solve for $v(t)$ or $v(x)$ instead.

In two dimensions, Newton's Second Law becomes a pair of second-order differential equations:

$$\ddot{x} = \frac{1}{m} F_\text{net,x}(x,y, v_x, v_y, t)$$ $$\ddot{y} = \frac{1}{m} F_\text{net,y}(x,y, v_x, v_y, t)$$
where the net force $F_\text{net,x}$ and $F_\text{net,y}$ can, in general, be a function of the vector position $(x,y)$, the vector velocity $(v_x, v_y)$, and the time $t$. If the $F_\text{net,x}$ only depends on $x$ and $v_x$, and the $F_\text{net,y}$ only depends on $y$ and $v_y$, then we can solve the two differential equations independently, greatly simplifying the problem.

In three dimensions, Newton's Second Law becomes a system of three second-order differential equations that have a similar form to the two-dimensional case.

Examples and Applications

Example 1: Pushing a Box

A person pushes a box of mass $m = 10 \text{ kg}$ with a constant horizontal force of $F = 50 \text{ N}$ on a frictionless surface. What is the acceleration of the box?

Solution: Using Newton's Second Law:

$$a = \frac{F_\text{net}}{m} = \frac{50 \text{ N}}{10 \text{ kg}} = 5 \text{ m/s}^2$$

The box accelerates at $5 \text{ m/s}^2$ in the direction of the applied force.

Example 2: Falling Object

An object of mass $m$ falls freely near Earth's surface. Ignoring air resistance, what is its acceleration?

Solution: The only force acting is gravity: $\mathbf{F}_\text{net} = m\mathbf{g}$. Applying Newton's Second Law:

$$m\mathbf{g} = m\mathbf{a} \quad \Rightarrow \quad \mathbf{a} = \mathbf{g}$$

The acceleration is $\mathbf{g}$, approximately $9.8 \text{ m/s}^2$ downward, independent of the object's mass. This is why all objects fall with the same acceleration in the absence of air resistance.

Example 3: Object on an Inclined Plane

A block of mass $m$ slides down a frictionless inclined plane that makes an angle $\theta$ with the horizontal. What is the acceleration of the block?

Solution: The net force parallel to the incline is $mg\sin\theta$ (the component of gravity along the incline). Using Newton's Second Law:

$$F_\text{net} = mg\sin\theta = ma \quad \Rightarrow \quad a = g\sin\theta$$

The acceleration is $g\sin\theta$, which is less than $g$ and depends on the angle of the incline.

Example 4: Two Forces Acting on an Object

An object of mass $m = 5 \text{ kg}$ experiences two forces: $\mathbf{F}_1 = (20, 0) \text{ N}$ and $\mathbf{F}_2 = (-10, 15) \text{ N}$. What is the acceleration of the object?

Solution: First, find the net force:

$$\mathbf{F}_\text{net} = \mathbf{F}_1 + \mathbf{F}_2 = (20, 0) + (-10, 15) = (10, 15) \text{ N}$$

Then apply Newton's Second Law:

$$\mathbf{a} = \frac{\mathbf{F}_\text{net}}{m} = \frac{(10, 15)}{5} = (2, 3) \text{ m/s}^2$$

The acceleration is $(2, 3) \text{ m/s}^2$, meaning $2 \text{ m/s}^2$ in the $x$-direction and $3 \text{ m/s}^2$ in the $y$-direction.

Example 5: Rocket Propulsion (Constant Mass Approximation)

A rocket of mass $m = 1000 \text{ kg}$ produces a thrust force of $F_\text{thrust} = 5000 \text{ N}$ upward. If the rocket is near Earth's surface, what is its acceleration? (Assume the mass remains constant for this example.)

Solution: The net force is the thrust minus the weight:

$$F_\text{net} = F_\text{thrust} - mg = 5000 - (1000)(9.8) = 5000 - 9800 = -4800 \text{ N}$$

Wait—this gives a negative net force, meaning the rocket would accelerate downward! This suggests the thrust is insufficient to lift the rocket. For the rocket to accelerate upward, we need $F_\text{thrust} > mg$.

Example 6: Rocket with Variable Mass (Requires Momentum Form)

A rocket expels exhaust gases, causing its mass to decrease. The rocket's mass at time $t$ is $m(t)$, and it expels exhaust at a rate $\dot{m}_\text{exhaust} = -\frac{dm}{dt} > 0$ with exhaust velocity $\mathbf{v}_\text{exhaust}$ relative to the rocket. What is the rocket's acceleration?

Solution: This problem requires the momentum form $\mathbf{F}_\text{net} = \dot{\mathbf{p}}$ because mass is changing. The net force includes both external forces (like gravity) and the reaction force from expelling exhaust. The full analysis involves:

$$\mathbf{F}_\text{net} = \dot{\mathbf{p}} = \frac{d(m\mathbf{v})}{dt} = m\dot{\mathbf{v}} + \dot{m}\mathbf{v}$$

For a rocket, this leads to the rocket equation, which cannot be derived using $\mathbf{F}_\text{net} = m\mathbf{a}$ alone because that form assumes constant mass. This example demonstrates why the momentum form is essential for variable mass systems.

Common Misconceptions

Important Clarifications:
  • "Force causes velocity": This is incorrect. Force causes acceleration (change in velocity), not velocity itself. An object can have a large velocity with no net force (constant velocity), or zero velocity with a large net force (about to accelerate).
  • "More massive objects fall faster": This is false. In the absence of air resistance, all objects fall with the same acceleration $g$, regardless of mass. Heavier objects have more weight, but they also have more mass, and these effects cancel out.
  • "Mass and weight are the same": Mass is a measure of matter (inertia), while weight is a force. An object has the same mass everywhere, but its weight depends on the local gravitational field.
  • "The equation $\mathbf{F} = m\mathbf{a}$ means force equals mass times acceleration": While mathematically true, it's better to think of it as: net force causes acceleration, and the amount of acceleration depends on both the force and the mass. The equation describes a relationship, not just a calculation.
  • "If velocity is zero, acceleration must be zero": This is false. An object at rest can have a non-zero acceleration if a net force acts on it. For example, a ball at the top of its trajectory has zero velocity but is accelerating downward due to gravity.

Solving Problems with Newton's Second Law

When solving problems involving Newton's Second Law, follow these steps:

Problem-Solving Strategy:
  1. Identify the object of interest: Clearly define what object you're analyzing.
  2. Draw a free-body diagram: Sketch the object and all forces acting on it. This helps visualize the problem and identify all forces.
  3. Choose a coordinate system: Select axes that simplify the problem (e.g., align one axis with the direction of motion or along an incline).
  4. Find the net force: Sum all forces vectorially to find $\mathbf{F}_\text{net}$.
  5. Apply Newton's Second Law: Use $\mathbf{F}_\text{net} = m\mathbf{a}$ or its component forms.
  6. Solve for the unknown: Calculate the desired quantity (acceleration, force, or mass).
  7. Check your answer: Verify that units are correct and the result makes physical sense.

Practical Applications

Newton's Second Law is fundamental to countless applications:

Limitations and Extensions

The acceleration form $\mathbf{F}_\text{net} = m\mathbf{a}$ applies only to objects with constant mass. For systems where mass changes (like rockets expelling fuel), we must use the momentum form:

$$\mathbf{F}_\text{net} = \dot{\mathbf{p}} = \frac{d(m\mathbf{v})}{dt} = \frac{d\mathbf{p}}{dt}$$

where $\mathbf{p} = m\mathbf{v}$ is the momentum. For constant mass, this reduces to $\mathbf{F}_\text{net} = m\mathbf{a}$.

Newton's Second Law (in either form) is valid only in inertial reference frames. In accelerating or rotating frames, additional "fictitious" forces must be included.

Relativistic Considerations:

At speeds approaching the speed of light, Newton's Second Law must be modified according to Einstein's theory of special relativity. However, for everyday speeds (much less than the speed of light), Newton's Second Law is extremely accurate and widely applicable.

Summary

Key Takeaways:
  • Newton's original formulation: $\mathbf{F}_\text{net} = \dot{\mathbf{p}} = \frac{d\mathbf{p}}{dt}$ is the most general form, stating that net force equals the time rate of change of momentum.
  • Constant mass form: $\mathbf{F}_\text{net} = m\mathbf{a}$ is valid only when mass is constant, and is a special case of the momentum form.
  • Domain of applicability:
    • Use $\mathbf{F}_\text{net} = \dot{\mathbf{p}}$ for variable mass systems (rockets, conveyor belts) and relativistic mechanics
    • Use $\mathbf{F}_\text{net} = m\mathbf{a}$ for constant mass systems (most everyday situations)
  • Force and acceleration are directly proportional: For constant mass, larger forces produce larger accelerations.
  • Mass and acceleration are inversely proportional: For constant mass and given force, more massive objects accelerate less.
  • Acceleration is in the direction of the net force: The acceleration vector points in the same direction as the net force vector (for constant mass).
  • Mass is not the same as weight: Mass is a measure of inertia, while weight is a gravitational force.
  • Newton's Second Law is a vector equation and can be written in component form for each coordinate direction.
  • Newton's First Law is a special case of the Second Law when the net force is zero (constant momentum).
  • The law applies in inertial reference frames. The momentum form is more general and applies to variable mass systems, while the acceleration form is simpler but limited to constant mass.