4.3 Newton's Second Law of Motion

Newton's Second Law of Motion

Newton's Second Law connects force, mass, and acceleration into one equation that lets you predict how any object will move when forces act on it. It's the workhorse of mechanics: once you know the net force on an object and its mass, you can find its acceleration (and from there, its entire motion).

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Calculations Using Newton's Second Law

The core equation is:

$F_{net} = ma$

where $F_{net}$ is the net force in Newtons (N), $m$ is mass in kilograms (kg), and $a$ is acceleration in meters per second squared (m/s²).

You can rearrange this equation three ways depending on what you're solving for:

1. Find net force: Multiply mass by acceleration.

   • A 5 kg object accelerating at 2 m/s² experiences a net force of $F = 5 \times 2 = 10 \text{ N}$

2. Find mass: Divide net force by acceleration.

   • An object with 20 N of net force accelerating at 4 m/s² has a mass of $m = \frac{20}{4} = 5 \text{ kg}$

3. Find acceleration: Divide net force by mass.

   • A 10 kg object with 50 N of net force accelerates at $a = \frac{50}{10} = 5 \text{ m/s}^2$

Always use net force here, not just any single force. If multiple forces act on an object, you need to add them as vectors first.

Force, Mass, and Acceleration Relationship

The equation $F_{net} = ma$ contains two proportionality relationships worth understanding separately.

Acceleration is directly proportional to net force (when mass is constant). If you double the net force on an object, its acceleration doubles. Triple the force, triple the acceleration. This is intuitive: push harder, speed up faster.

Acceleration is inversely proportional to mass (when net force is constant). If you double the mass, acceleration is cut in half. This is why a loaded truck accelerates more slowly than an empty one given the same engine force.

A few things to keep straight:

• Force and acceleration are both vector quantities, meaning they have magnitude and direction. The acceleration always points in the same direction as the net force.
• Mass is a scalar. It has magnitude only and is always positive.
• An object with zero net force has zero acceleration. It's either at rest or moving at constant velocity.

Mass vs. Weight in Gravity

These two terms get mixed up constantly, but they describe different things.

Mass is a measure of how much matter an object contains, or equivalently, how much it resists acceleration (its inertia). Mass is an intrinsic property: it stays the same no matter where you are. It's measured in kilograms (kg).

Weight is the gravitational force acting on an object. It depends on where you are because gravitational field strength varies by location. Weight is measured in Newtons (N) and calculated using:

$W = mg$

where $g$ is the local acceleration due to gravity.

Here's where the distinction really shows up. Take a 10 kg object:

• On Earth ($g = 9.8 \text{ m/s}^2$): $W = 10 \times 9.8 = 98 \text{ N}$
• On the Moon ($g = 1.7 \text{ m/s}^2$): $W = 10 \times 1.7 = 17 \text{ N}$
• In deep space ($g \approx 0$): $W \approx 0 \text{ N}$

The mass is 10 kg in all three cases. The weight changes dramatically.

Analyzing Forces and Motion

To apply Newton's Second Law to real problems, you'll almost always start with a free-body diagram: a sketch showing the object as a dot with arrows representing every force acting on it. Each arrow's direction shows the force's direction, and its length represents the force's relative magnitude.

When the net force on an object is zero, the object is in equilibrium. That doesn't necessarily mean it's sitting still; it means there's no acceleration. An object moving at constant velocity is also in equilibrium.

The study of how forces cause motion is called dynamics. Newton's Second Law is the central tool of dynamics. It also connects to momentum: the net force on an object equals the rate of change of its momentum over time, written as $F_{net} = \frac{\Delta p}{\Delta t}$. For constant mass, this reduces right back to $F_{net} = ma$. You'll use this momentum form more in later units, but it's worth knowing that $F = ma$ is actually a special case of a broader principle.