Theory

Newton's laws and forces

The three laws of dynamics, the forces that appear in mechanics problems (weight, constraint force, tension, friction, elastic force) and the method for writing and solving the equation of motion of a body or of a system of bodies, also in circular motion and in noninertial reference frames.

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Forces and Newton’s first law
Second law, mass and weight

Forces and Newton’s first law

1. Force

A force is a vector quantity that describes the action of one body on another, capable of changing its state of motion or of deforming it. It is specified by magnitude, direction, sense and point of application. In the International System the magnitude is measured in newtons, with $1 N = 1 kg \cdot m/ s^{2}$ .

In this topic bodies are treated as point particles, that is, their size and rotations are neglected and all forces are considered applied at the same point; the case in which the point of application matters is treated in the topic on rigid body dynamics.

2. Principle of superposition of forces

If the forces $F_{1}, F_{2}, \dots, F_{n}$ act simultaneously on a point particle, the effect on the motion is the same as that which the single force

R = F_{1} + F_{2} + \dots + F_{n},

called the net force, would produce. It is an experimental principle, and it states that forces behave as vectors and add by the parallelogram rule. In components $R_{x} = \sum_{i} F_{i x}$ and $R_{y} = \sum_{i} F_{i y}$ . For example a force of $3.0 N$ along $x$ and one of $4.0 N$ along $y$ have a net force of magnitude $3. 0^{2} + 4. 0^{2} N = 5.0 N$ , which forms with the $x$ axis the angle $arctan \frac{4}{3} \approx 5 3^{\circ}$ .

The forces

F_{1}

3 N

and

F_{2}

4 N

have net force

R

of magnitude

5 N

, the diagonal of the parallelogram

3. Inertial reference frame

A point particle is called isolated if the net force acting on it is zero. A reference frame is called inertial if in it every isolated point particle has constant velocity, that is, it remains at rest or moves in uniform straight-line motion.

A frame that translates with constant velocity relative to an inertial frame is also inertial, because by the Galilean transformations the accelerations in the two frames coincide. A frame attached to the ground is inertial to a good approximation, since the acceleration due to the Earth’s rotation is at most $0.034 m/ s^{2}$ , about $0.35%$ of $g$ .

4. Newton's first law

Inertial reference frames exist. In an inertial frame a point particle subject to zero net force remains at rest, if it was at rest, or moves in uniform straight-line motion.

This is the law of inertia, a principle of mechanics, that is, a statement assumed to be true because all its consequences are confirmed by experiments. It states that a force is needed to change the velocity of a body, while to keep it constant it is enough for the net force to be zero; indeed a hockey puck on ice slows down less the smaller the friction is. The first law also identifies the frames in which the second and third laws hold.

Second law, mass and weight

1. Newton's second law

In an inertial reference frame, a point particle of mass $m$ subject to the net force $R$ has acceleration $a$ such that

R = m a .

It is a principle of mechanics, confirmed by experiments. The inertial mass $m$ , measured in kilograms, expresses the resistance of the body to changing its velocity, and indeed for the same force the acceleration is inversely proportional to the mass. The vector equation is equivalent to one equation for each component, to be satisfied simultaneously:

{R_{x} = m a_{x} R_{y} = m a_{y}

For example a $3.0 kg$ crate on a smooth floor, pushed by a horizontal force of $12 N$ , has acceleration $a = \frac{12 N}{3.0 kg} = 4.0 m/ s^{2}$ in the direction and sense of the force.

2. Weight

Near the Earth’s surface every body of mass $m$ is subject to the weight

P = m g,

directed vertically downward, with $g = 9.81 m/ s^{2}$ . Mass is a scalar and does not depend on location; weight is a vector and depends on the local value of $g$ . A $70 kg$ person weighs $687 N$ on the Earth and $113 N$ on the Moon, where $g = 1.62 m/ s^{2}$ . Since weight is proportional to mass, in the absence of air resistance all bodies fall with the same acceleration $g$ ; the origin of weight is the subject of the topic on gravitation.

3. Equilibrium of a point particle

A point particle at rest in an inertial frame has zero acceleration, therefore by the second law the net force is zero. In components the equilibrium condition is

{R_{x} = 0 R_{y} = 0

The two equations make it possible to find unknown forces, such as tensions and constraint forces. A $2.0 kg$ body hanging at rest from a vertical string is subject to its weight and to the tension $T$ of the string, therefore equilibrium along the vertical gives $T - m g = 0$ , that is, $T = 19.6 N$ .

The theory continues with 8 more sections (third law and constraint forces, ideal strings and pulleys, sliding friction…), plus all worked examples, the solutions and the PDF.

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