Forces, Free Fall, and a Beautiful Oscillation

​In Chapter 9-5 of The Feynman Lectures on Physics, Richard Feynman turns his attention to a crucial practical question in dynamics: if Newton’s laws tell us to look for forces, how do we actually find them? To apply Newton’s sweeping principles of motion, one must know the specific agencies that cause objects to accelerate. This chapter is Feynman’s elegant introduction to the art of identifying those forces and understanding how they give rise to different kinds of motion.

Feynman begins by emphasising Newton’s insight that whenever an object accelerates, some cause must be at work. It is not enough to describe the motion; one must point to the force that produces it. This becomes the guiding programme for all future dynamics: discover the laws governing the forces themselves.

Newton, he reminds us, had already made remarkable progress in this direction. For gravity, Newton provided a precise rule describing how two masses influence each other. For other interactions he offered clues, most famously the idea—explored in the next chapter—that forces between two bodies come in equal and opposite pairs. This framework is the starting point for Feynman’s exploration.

Having established the need for specific force laws, Feynman returns to familiar ground: objects falling close to the Earth’s surface. Here, the force of gravity acts vertically and is proportional to the object’s mass, while remaining almost completely unchanged for heights small compared with the size of the Earth.

The key consequence is that every object experiences the same acceleration, regardless of its mass. This leads to the classic behaviour of free fall: a steady, constant increase in vertical velocity, while horizontal motion proceeds undisturbed at a constant rate. Feynman uses this simple scenario to highlight one of the most beautiful aspects of Newtonian physics: even complicated motions—such as a stone thrown into the air—can be understood as the combination of two independent components, one influenced by gravity and one left entirely alone.

This part of the chapter serves as a bridge between everyday experience—dropping objects, throwing balls, watching fountains—and the abstract logic of Newton’s laws. Feynman is showing us that the mathematics of motion is not an invention but a faithful echo of the world around us.

Feynman then changes direction by introducing a second force law, one very different from gravity: the restoring force of a spring. Here his teaching shines. He imagines a carefully designed device in which a mass is attached to a spring that pulls harder the further it is stretched or compressed. The force always acts to restore the mass towards an equilibrium point, and the size of that force is directly proportional to how far the mass has been displaced.

If the mass is pulled down, the spring pulls up; if pushed up, it pulls down. Crucially, the further the displacement, the stronger the pull. In this simple setup, gravity’s steady effect is assumed to have been cancelled by the way the device is mounted, allowing Feynman to focus solely on the added or “excess” forces produced by the spring.

Anyone who has played with a spring-mass toy knows what happens next: a gentle, rhythmic up-and-down motion, endlessly repeating if no friction is present. Feynman calls the motion “rather beautiful,” and he means it. The mass does not rush off in one direction or topple into chaos; instead it glides smoothly, reversing direction at the top and bottom of its travel.

But the real question—posed with the trademark Feynman twinkle—is this:

Does Newton’s law describe this graceful oscillation correctly?

To find out, he writes an equation relating the rate of change of the mass’s velocity to its position. The result is a wonderfully simple relationship: one quantity is always proportional to the negative of the other. In other words, when the mass is far from equilibrium, its velocity is changing rapidly, and always in such a way as to push it back towards the centre.

Feynman then simplifies the situation still further by adjusting scales and units so that the constants fall away, leaving only the pure essence of the motion: the change in velocity is the negative of the displacement. From here, he points out that the velocity is itself simply the rate of change of position. These two linked ideas—how position changes and how velocity changes—form the heart of the oscillation.

Although he does not yet solve the full motion in this chapter, Feynman has laid all the conceptual groundwork. The resulting motion is periodic, elegant, and entirely predicted by Newton’s laws once we know the correct rule for the force.

What ties the chapter together is Feynman’s larger message: motion becomes intelligible once we know the forces. Whether we are studying a falling apple or a vibrating spring, the behaviour flows from the specific way each force acts.

Gravity produces constant acceleration; a spring produces an acceleration that shifts and reverses according to position. These differences in physical behaviour emerge naturally from the differences in the underlying forces.

In this chapter, Feynman teaches us not only how to analyse motion but how to listen to nature’s structure. Newton’s laws provide the grammar; the force laws supply the vocabulary. Together they compose the story of why objects move as they do.