4.5 Normal, Tension, and Other Examples of Forces

Types of Forces

Normal and tension forces

The normal force acts perpendicular to whatever surface an object is touching. It's what prevents objects from passing through each other. A book sitting on a table, for instance, has a normal force pushing upward from the table that exactly balances the book's weight. The normal force adjusts its magnitude to match whatever is pushing the object into the surface, so it's not always equal to the object's weight (think about someone pushing down on the book, or the book sitting on a ramp).

The tension force is the pulling force transmitted through a rope, string, cable, or chain when it's pulled taut. Tension always pulls along the length of the rope, directed away from the object it's attached to. A key assumption in many physics problems is that the rope is "massless and inextensible," which means the tension is the same everywhere along the rope.

Friction opposes the relative motion (or attempted motion) between two surfaces in contact. Static friction keeps an object from starting to move, while kinetic friction acts on an object that's already sliding. Both depend on the normal force between the surfaces:

• Static friction: $f_s \leq \mu_s N$
• Kinetic friction: $f_k = \mu_k N$

Here, $\mu_s$ and $\mu_k$ are the coefficients of static and kinetic friction, and $N$ is the normal force. Notice that static friction is an inequality: it can take any value up to its maximum, adjusting to whatever is needed to prevent motion.

Applying Newton's Laws

Applications of Newton's laws

Newton's three laws work together to let you analyze almost any force problem:

• First law: An object stays at rest or moves at constant velocity unless an unbalanced (net) force acts on it.
• Second law: $F_{net} = ma$. The net force on an object equals its mass times its acceleration. Net force is the vector sum of all individual forces acting on the object.
• Third law: Forces always come in equal and opposite pairs. If object A pushes on object B, then B pushes back on A with the same magnitude but in the opposite direction. These two forces act on different objects, so they never cancel each other in a free-body diagram.

Problem-solving steps

1. Identify all forces acting on each object (gravity, normal, tension, friction, applied forces).
2. Draw a free-body diagram for each object, showing every force as a vector arrow starting from the object.
3. Choose a coordinate system. For inclined planes, tilt your axes so one axis runs along the surface. This simplifies the math considerably.
4. Decompose any forces not aligned with your axes into components.
5. Apply $F_{net} = ma$ along each axis separately.
6. Solve the resulting system of equations for the unknowns (acceleration, tension, normal force, etc.).

Weight components in force problems

Weight is the gravitational force on an object: $W = mg$, where $m$ is mass and $g$ is the acceleration due to gravity (approximately $9.8 \, m/s^2$ near Earth's surface). Weight always points straight down, toward the center of the Earth.

On an inclined plane at angle $\theta$ from the horizontal, you need to break the weight into two components relative to the surface:

• Parallel to the incline: $W_\parallel = mg \sin \theta$. This is the component that tries to slide the object down the ramp.
• Perpendicular to the incline: $W_\perp = mg \cos \theta$. This component pushes the object into the surface and is balanced by the normal force.

So on a frictionless ramp, the acceleration down the incline is simply $a = g \sin \theta$. As a quick check: when $\theta = 0°$ (flat surface), $\sin \theta = 0$ and there's no acceleration. When $\theta = 90°$ (vertical drop), $\sin \theta = 1$ and the object is in free fall. That makes sense.

Equilibrium and Force Balance

Equilibrium means the net force on an object is zero ($F_{net} = 0$), which means the object has zero acceleration.

• Static equilibrium: The object is at rest and stays at rest. Example: a lamp hanging from a ceiling by two cables.
• Dynamic equilibrium: The object moves with constant velocity. Example: a car cruising at a steady speed on a straight highway.

In both cases, you set up the problem the same way: draw the free-body diagram, resolve forces into components, and set the sum of forces equal to zero along each axis. For a hanging object supported by two angled cables, for instance, the vertical components of the two tension forces must add up to the object's weight, and the horizontal components must cancel each other out.