Dynamics of a particle
Newton does not say what a force is: he says what it does. And what it does is change velocity. The whole craft lies in drawing the free-body diagram correctly before writing a single equation.
01The three laws, stated precisely
A body on which the net force is zero keeps its velocity: if it was at rest it stays at rest, and if it was moving it keeps moving in a straight line at constant speed. What needs explaining is not the motion, but the change of motion.
The first law also defines the frames in which the other two hold: the inertial frames.
It is a vector equation: it holds component by component. The form with momentum p⃗ = m·v⃗ is the most general, because it also works when the mass changes, as in a rocket.
If A exerts a force on B, B exerts on A a force of equal magnitude and opposite direction. The key point: they act on different bodies, so they never cancel each other in the same diagram. If they did cancel, nothing could ever accelerate.
02The free-body diagram
It is the method, not a decorative preliminary. It consists of mentally isolating one body and drawing only the forces that others exert on it.
- Choose the body and isolate it.
- Draw the weight, always vertical and pointing down.
- Draw one force for each contact: normal force, friction, rope tension, thrust.
- Choose convenient axes. On an inclined plane it helps to put one axis parallel to the plane.
- Write ΣF = m·a along each axis and only then substitute numbers.
It equals m·g only when the surface is horizontal and there are no other vertical forces. On an inclined plane it is m·g·cosθ; in an elevator accelerating upward, m·(g + a); and in free fall, zero. That feeling of lightness in an elevator that starts going down is exactly the normal force decreasing.
03The forces that always appear
| Force | Direction | Magnitude |
|---|---|---|
| Weight | Vertical, downward | \( P = m\cdot g \) |
| Normal | Perpendicular to the surface | Whatever it takes to keep the body from sinking in |
| Static friction | Opposite to the impending sliding | \( f \le \mu_e \, N \) |
| Kinetic friction | Opposite to the velocity | \( f = \mu_c \, N \) |
| Tension | Along the rope, pulling | The same throughout an ideal rope |
| Elastic | Opposite to the deformation | \( F = -k \cdot x \) (Hooke’s law) |
It is the only one in the list without a fixed value: it takes whatever value is needed to prevent sliding, up to a maximum of μe·N (static coefficient). Beyond that limit, the body starts moving and the force becomes μc·N (kinetic coefficient), which is somewhat smaller. That is why it is harder to start pushing a piece of furniture than to keep pushing it.
Change the mass, the incline, the friction and the applied force, and watch the free-body diagram and the resulting acceleration.
04The inclined plane, solved
With the axes rotated —one parallel to the plane, the other perpendicular— the weight is resolved into components and we get:
The body starts to slide when tanθ exceeds μe. This critical angle makes it possible to measure the friction coefficient with a board and a protractor: tilt it until the object starts moving and note the angle.
05Systems of particles
When several bodies are linked —two blocks joined by a rope, an Atwood machine— there are two approaches:
- Treat the whole as a single body to find the acceleration: the internal forces cancel in pairs by the third law.
- Isolate each body if you also need the internal forces, such as the tension in the rope that joins them.
The center of mass of the system moves as if all the mass were there and only the external forces acted. This is why a wrench tumbling through the air traces, at its center of mass, a perfect parabola while the rest of the body spins around it.
06From Newton to electronics
- Motors and actuators. Sizing a stepper motor requires calculating the necessary torque from the mass to be moved, the friction and the required acceleration.
- MEMS accelerometers. They literally measure the second law: a proof mass on microscopic springs is displaced and the displacement changes a capacitance, which is converted into a voltage. This is what detects the orientation of a phone.
- Vibration and mounting. Onboard equipment is designed for the force m·a that its components must withstand, not for their weight at rest.
- Analogy with circuits. Mass behaves like an inductance, viscous friction like a resistance and the spring like a capacitor. This equivalence makes it possible to simulate a mechanical system in a circuit simulator.
07In the lab
Tilt a board until a block starts to slide. μe = tanθ. Repeat with different surfaces —wood, rubber, glass— and with the larger and smaller face of the same block, to check that the contact area has no effect.
A cart pulled by a hanging weight. Measure the acceleration with two optical gates and compare with a = m₂g/(m₁+m₂). Repeat while changing the hanging mass and plot a as a function of F: the inverse slope is the total mass.
With an MPU-6050 connected over I²C to a microcontroller, record the acceleration at rest —it should read 9.8 m/s² on the vertical axis— and during a brief fall, where all three components approach zero.
08Common mistakes
- Drawing the force “of the motion”. It does not exist: a body moving by inertia has no force acting in its favor.
- Canceling action and reaction in the same diagram. They are applied to different bodies.
- Using N = m·g on an inclined plane.
- Taking static friction to be always μe·N, when that is only its maximum value.
- Confusing mass with weight. Mass is a scalar in kilograms and does not change with location; weight is a force in newtons and depends on g.
- Forgetting that the second law is a vector equation and adding magnitudes of forces with different directions.
09Self-assessment
A 4 kg block is at rest on a table. What is the reaction to its weight?
The attraction the block exerts on the Earth, 39.2 N upward. It is not the normal force: that is the reaction to the force the block exerts on the table.
At what angle does a body with μe = 0.5 start to slide?
When tanθ > 0.5, that is θ > 26.6°. It does not depend on the mass.
An elevator goes up accelerating at 2 m/s². What does a scale read with a 70 kg person on it?
N = m(g + a) = 70·11.8 = 826 N, equivalent to a reading of 84 kg.
Two blocks of 2 and 3 kg joined by a rope are pushed with 10 N. What is the tension?
a = 10/5 = 2 m/s². Isolating the rear block: T = 3·2 = 6 N if the force is applied to the 2 kg block.
Why does the acceleration on a frictionless plane not depend on the mass?
Because the driving force is proportional to the mass —m·g·sinθ— and the second law divides it by that same mass.
What does an accelerometer at rest on a table really measure?
The normal force per unit mass, which at rest equals 9.8 m/s² upward. That is why in free fall it reads zero, even though the real acceleration is maximum.
10Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics, Volume 1. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. Chapters 5 and 6, with a very complete collection of free-body diagrams.
- Paul A. Tipler and Gene Mosca. Physics for Scientists and Engineers, Volume 1. 6th ed., W. H. Freeman, 2008 (Spanish edition, “Física para la ciencia y la tecnología”, Reverté, 2010). Good treatment of linked systems and of the center of mass.
- Richard P. Feynman. The Feynman Lectures on Physics, Volume I. Addison-Wesley, 1963 (Spanish edition, “Física, volumen I”, Addison-Wesley / Pearson). Chapter 12 discusses what a force is and what it is not, with unusual honesty.
- Marcelo Alonso and Edward J. Finn. Physics. Addison-Wesley, 1992 (Spanish edition, “Física”, Addison-Wesley Iberoamericana, 1995). Builds dynamics starting from the conservation of momentum, a more modern approach.