Dinámica. Leyes de Newton

Dynamics and Newton’s Laws

Dynamics is the branch of classical mechanics that studies motion and the forces that cause or change it. Unlike kinematics, which describes motion without examining its causes, dynamics focuses on forces and their effects.

Force and Net Force

A force is an interaction that can change an object’s motion or deform it. Force is a vector, so it has both magnitude and direction.

The SI unit of force is the newton (N):

[ 1\ \text{N} = 1\ \text{kg·m/s}^2 ]

In the CGS system, the corresponding unit is the dyne:

[ 1\ \text{dyne} = 1\ \text{g·cm/s}^2 ]

When several forces act on a body, their net force is their vector sum. To calculate it in two dimensions:

1. Choose positive directions for the axes.
2. Add the horizontal components to find the net (x)-component.
3. Add the vertical components to find the net (y)-component.
4. Calculate the net force’s magnitude:

[ |\vec F_{\text{net}}| = \sqrt{F_x^2 + F_y^2} ]

Worked example: Forces of 22 N to the right and 12 N to the left give (F_x = 10) N to the right. Forces of 16 N upward and 10 N downward give (F_y = 6) N upward. The resulting magnitude is:

[ |\vec F_{\text{net}}| = \sqrt{10^2 + 6^2} \approx 11.7\ \text{N} ]

Newton’s First Law: Inertia

An object remains at rest or continues moving at constant velocity in a straight line unless a nonzero net external force changes its state.

Inertia is an object’s tendency to resist changes in its motion. For example, when a bicycle stops abruptly, the rider’s body tends to keep moving forward.

Skater example: A skater moving at 4.0 m/s at constant velocity has zero net force. The skater’s speed remains 4.0 m/s after two minutes. In one minute, the skater travels:

[ 4.0 \times 60 = 240\ \text{m} ]

Newton’s Second Law

For a body of constant mass, acceleration is proportional to the net force and inversely proportional to the mass:

[ \vec F_{\text{net}} = m\vec a ]

In magnitude form:

[ F_{\text{net}} = ma,\qquad a = \frac{F_{\text{net}}}{m},\qquad m = \frac{F_{\text{net}}}{a} ]

A constant net force on an object of constant mass produces constant acceleration.

Worked example: A body starts from rest and travels 35 m in 5.0 s under a constant 50 N force. Using (d = \tfrac12 at^2):

[ a = \frac{2d}{t^2} = 2.8\ \text{m/s}^2 ]

Its mass is therefore:

[ m = \frac{50}{2.8} \approx 17.9\ \text{kg} ]

Weight

Weight is the gravitational force exerted on an object. Near Earth’s surface, its magnitude is:

[ W = mg ]

Here, (g \approx 9.8\ \text{m/s}^2), directed toward Earth’s center. Weight is a vector and depends on gravitational conditions; mass is a scalar property of the object.

Worked example: An 82 kg man’s weight is:

[ 82 \times 9.8 \approx 803.6\ \text{N} ]

directed downward. If he is at rest on a sofa, he pushes down on it with that force, while the sofa pushes upward on him with an equal-magnitude force.

Newton’s Third Law: Action and Reaction

When one body exerts a force on another, the second body exerts a force of equal magnitude and opposite direction on the first. The forces act along the same line and occur simultaneously.

In the man-and-sofa example, the man pushes downward on the sofa, and the sofa pushes upward on the man.

The third law is not limited to situations where objects are in equilibrium or have zero acceleration. Action–reaction forces occur whether or not the objects are in equilibrium. They act on different bodies, so they do not cancel each other on a single object.

Centripetal Force and Circular Motion

An object moving along a circular path continually changes direction, even if its speed stays constant. A force directed toward the circle’s center is required to produce this motion.

The inward acceleration has magnitude:

[ a_c = \frac{v^2}{r} ]

The centripetal force has magnitude:

[ F_c = ma_c = \frac{mv^2}{r} ]

The acceleration and force vectors point toward the center, while their directions change as the object moves around the circle. Tension in a string is one example of a force that can provide the centripetal force.

Worked example: For a 150 g mass moving at 5.0 m/s in a circle of radius 1.2 m:

[ a_c \approx 20.8\ \text{m/s}^2 ]

Since (150\ \text{g} = 0.150\ \text{kg}), the centripetal force is approximately:

[ F_c \approx 3.1\ \text{N} ]

Speakers and Sources

• Instructor/narrator: The lesson’s presenter from the channel Prof. Osiris Decena.
• Isaac Newton: Cited as the originator of the laws discussed; he is not a speaking participant.
• Other audible element: Brief background music.