4.2 Newton's Laws of Motion

Newton's Laws of Motion

Newton's laws of motion form the foundation of classical mechanics. These three principles explain how forces affect objects, from a book resting on a table to a rocket launching into space. Understanding these laws lets you predict and analyze motion in nearly every situation you'll encounter in this course.

Newton's Laws of Motion

Newton's three laws of motion

First Law (Law of Inertia): An object stays at rest or moves at constant velocity in a straight line unless a net external force acts on it. Inertia is the tendency to resist changes in motion, and it's directly proportional to an object's mass. A more massive object is harder to start moving or stop. Think of passengers lurching forward when a bus brakes suddenly: the bus slows down, but your body wants to keep moving at the original speed.

Second Law (Law of Force and Acceleration): The net force on an object equals its mass times its acceleration:

$F_{net} = ma$

This tells you two things at once. Doubling the net force on an object doubles its acceleration. Doubling the object's mass (while keeping force the same) cuts the acceleration in half. A light, empty shopping cart accelerates easily with a small push; a fully loaded one needs much more force for the same acceleration.

Third Law (Law of Action-Reaction): Forces always come in pairs that are equal in magnitude and opposite in direction. The crucial detail: the two forces in a pair act on different objects, so they never cancel each other out. When a gun fires, the bullet is pushed forward and the gun is pushed backward (recoil). Both forces are the same size, but they act on different objects.

Applications of Newton's first law

Objects at rest remain stationary whenever the forces on them are balanced (net force equals zero). A book sitting on a table stays put because gravity pulling it down is exactly balanced by the normal force pushing it up.

Objects in motion keep moving in a straight line at constant speed when no net force acts on them. A spacecraft coasting in deep space, far from any gravitational pull, will travel at the same velocity indefinitely. An air hockey puck glides across the table with very little friction, so it barely slows down.

Practical applications show up everywhere:

• Seat belts restrain passengers whose bodies would otherwise keep moving forward (due to inertia) during a sudden stop.
• Objects slide much farther on ice than on rough pavement because reduced friction means less net force to change the object's motion.
• Tightrope walkers carry long poles to increase their rotational inertia, making it easier to maintain balance by resisting tipping.

Calculations with Newton's second law

You can rearrange $F_{net} = ma$ depending on what you need to find:

• Acceleration: $a = \frac{F_{net}}{m}$
• Net force: $F_{net} = ma$
• Mass: $m = \frac{F_{net}}{a}$

The standard units are: force in newtons (N), mass in kilograms (kg), and acceleration in $\text{m/s}^2$. One newton is the force needed to accelerate a 1 kg mass at $1 \text{ m/s}^2$.

Steps for solving a Newton's second law problem:

1. Draw a free-body diagram showing all forces on the object (gravity, normal force, friction, applied forces, tension, etc.).
2. Choose a coordinate system. For inclined planes, tilt your axes so one axis runs along the surface.
3. Break any angled forces into components along your axes.
4. Sum the force components along each axis to find the net force. Remember that forces in opposite directions subtract.
5. Apply $a = \frac{F_{net}}{m}$ along the direction of motion to find acceleration.

Quick example: A 5 kg block is pushed with a 30 N horizontal force across a floor with 10 N of friction.

• Net force: $F_{net} = 30 \text{ N} - 10 \text{ N} = 20 \text{ N}$
• Acceleration: $a = \frac{20 \text{ N}}{5 \text{ kg}} = 4 \text{ m/s}^2$

For a block on an inclined plane, you'd resolve the gravitational force into a component parallel to the slope ($mg \sin\theta$) and one perpendicular to it ($mg \cos\theta$), then apply the same process.

Force pairs in Newton's third law

Every action-reaction pair shares three characteristics:

• Equal in magnitude
• Opposite in direction
• Acts on two different objects

Because the two forces act on separate objects, they don't cancel. This is one of the most common points of confusion. The gravitational pull of Earth on you and your gravitational pull on Earth are an action-reaction pair, but they act on different bodies (you and Earth), so they don't produce zero net force on either one.

Common examples:

• Pushing a wall: Your hands exert a force on the wall; the wall exerts an equal force back on your hands. You feel this as resistance.
• Rocket propulsion: The rocket pushes exhaust gases backward; the gases push the rocket forward. This works even in the vacuum of space because the interaction is between the rocket and its exhaust, not between the rocket and air.
• Walking: Your foot pushes backward on the ground; the ground pushes your foot forward via friction. Without friction (like on ice), your foot can't push backward effectively, and you can't walk.

Analyzing force diagrams with third-law pairs:

When drawing free-body diagrams for interacting objects, isolate each object and draw only the forces acting on that object. For a book on a table, the book's free-body diagram shows gravity (Earth pulling the book down) and the normal force (table pushing the book up). The third-law partner of the normal force is the book pushing down on the table, which appears on the table's free-body diagram, not the book's.

Applications in sports:

• A swimmer's arms push water backward; the water pushes the swimmer forward.
• When you jump, your legs push down on the ground, and the ground pushes you upward. The harder you push, the greater the upward force and the higher you jump.