4.1 Force

Force and Its Role in Dynamics

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Concept of force in dynamics

A force is any push or pull that can change an object's motion or shape. It's measured in Newtons (N) in the SI system and represented as a vector quantity, meaning it has both magnitude and direction.

Forces can do three things to an object:

• Accelerate or decelerate it (a car speeding up or braking)
• Change its direction (a ball curving in flight)
• Deform it (squeezing a stress ball)

Newton's three laws of motion describe exactly how forces and motion are related:

• First Law (Law of Inertia): An object at rest stays at rest, and an object in motion stays in motion at constant velocity, unless a net external force acts on it. A book sitting on a table won't slide on its own; a satellite in orbit keeps moving because almost no net force slows it down. This law is really about inertia, which is an object's natural resistance to any change in its state of motion. The more mass an object has, the more inertia it has.
• Second Law: The acceleration of an object is directly proportional to the net force on it and inversely proportional to its mass:

$a = \frac{F_{net}}{m}$

This is why pushing a heavy box produces less acceleration than pushing a light one with the same force. You can also write this as $F_{net} = ma$, which is probably the most-used equation in all of mechanics.

• Third Law: For every action force, there is an equal and opposite reaction force. When you jump off a boat, you push the boat backward while the boat pushes you forward. Rocket propulsion works the same way: exhaust gases are pushed downward, and the rocket is pushed upward. The key detail here is that the action and reaction forces act on different objects, so they never cancel each other out.

Free-body diagrams for force analysis

A free-body diagram (FBD) is a visual tool that shows all the forces acting on a single object. It strips away the clutter of a real situation so you can focus on the physics of one object at a time.

Steps to create a free-body diagram:

1. Identify the object of interest and mentally isolate it from its surroundings.
2. Draw the object as a simple shape (a box or dot).
3. Identify all forces acting on that object. Ask yourself: What touches it? What fields act on it?
4. Draw arrows for each force, starting from the object. Each arrow's length should be proportional to the force's magnitude, and it should point in the direction the force acts.

Common forces you'll include in an FBD:

• Weight ($\vec{W}$ or $\vec{F_g}$): the gravitational pull on the object, always directed downward toward Earth's center
• Normal force ($\vec{N}$ or $\vec{F_N}$): the support force a surface exerts perpendicular to the surface
• Tension ($\vec{T}$): the pulling force transmitted through a rope, string, or cable
• Friction ($\vec{f}$): the force resisting sliding motion between two surfaces, directed parallel to the contact surface
• Applied forces ($\vec{F_{app}}$): any external push or pull, like a person pushing a box

Interpreting a free-body diagram:

The net force on the object is the vector sum of all forces in the diagram. If the net force is zero, the object is in equilibrium, meaning it's either at rest or moving at constant velocity (a book on a table, a car cruising at steady speed). If the net force is nonzero, the object accelerates in the direction of that net force (a ball in free fall, a car braking).

Contact forces vs. field forces

Forces fall into two broad categories based on how they're transmitted.

Contact forces require physical touching between two objects. Normal force, tension, friction, and applied forces are all contact forces. When you push a shopping cart or a rope pulls a crate, the force is transmitted through direct interaction between the particles at the surfaces of those objects.

Field forces act across empty space with no physical contact needed. Gravitational, electric, and magnetic forces are all field forces. The Earth pulls on the Moon through the gravitational field, and two magnets attract or repel through the magnetic field. These forces are mediated by invisible fields that extend through space and exert influence on objects within them.

Gravitational force is the attractive force between any two objects with mass. It depends on both masses and the distance between them:

$F_g = G \frac{m_1 m_2}{r^2}$

Here, $G$ is the universal gravitational constant ($6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2$). This force is always attractive and governs everything from falling apples to planetary orbits.

Electric force is the force between electrically charged objects. Unlike gravity, it can be either attractive (opposite charges) or repulsive (like charges):

$F_e = k \frac{|q_1 q_2|}{r^2}$

Here, $k$ is Coulomb's constant ($8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2$). Notice how both the gravitational and electric force equations follow an inverse-square relationship with distance. That pattern is worth remembering.

Magnetic force is exerted by magnetic fields on moving charges or on other magnets. It can be attractive or repulsive depending on the orientation of the fields (two bar magnets, an electromagnet lifting scrap metal). Unlike gravity and the electric force, the magnetic force on a moving charge depends on the charge's velocity and the angle at which it moves through the field.

Force-related concepts

These terms connect directly to forces and will come up throughout the course:

• Momentum ($\vec{p} = m\vec{v}$): the product of an object's mass and velocity. Newton's second law can actually be stated more generally as force equals the rate of change of momentum.
• Impulse ($\vec{J} = \vec{F} \Delta t$): the change in momentum caused by a force acting over a time interval. A larger force or a longer contact time produces a greater impulse.
• Work ($W = Fd\cos\theta$): the energy transferred when a force causes displacement. Only the component of force along the direction of motion does work.
• Energy: the capacity to do work. Forces can transfer energy between objects or convert it from one form to another (kinetic to potential, for example).
• Torque: the rotational equivalent of force. It describes how a force causes an object to rotate, and it depends on both the force's magnitude and how far from the pivot point it's applied.
• Pressure ($P = F/A$): force applied per unit area. This concept becomes central in fluid mechanics and gas behavior.