5.2 Tension, Normal Force, and Other Contact Forces

Contact Forces in Physics

Contact forces show up any time objects physically touch each other. Tension, normal force, and friction are the three you'll encounter most often in this course, and nearly every Newton's Laws problem involves at least one of them. Getting comfortable with how they work (and how to represent them in free-body diagrams) is what makes force problems solvable.

Types of Contact Forces

Tension is the pulling force transmitted through a string, rope, cable, or chain when it's pulled taut. It always acts along the length of the rope, directed away from the object it's attached to. In a tug-of-war, for example, the rope exerts tension on both teams, pulling each toward the center.

A few key properties of tension:

• Tension pulls; it can never push. A rope goes slack rather than pushing an object.
• For an ideal (massless) rope, the tension is the same at every point along its length.
• If a rope passes over a frictionless pulley, the tension stays the same on both sides.

Normal force is the support force a surface exerts on an object resting against it. It always acts perpendicular to the surface (that's what "normal" means in math: perpendicular). When a book sits on a table, the table pushes up on the book with a normal force that balances the book's weight.

• The normal force adjusts its magnitude to match whatever is pressing the object into the surface. It's a response force, not a fixed value.
• On a flat, horizontal surface with no other vertical forces, $N = mg$. But this isn't always true; if someone pushes down on the book, the normal force increases to compensate.

Friction is the force that resists relative motion (or attempted motion) between two surfaces in contact. It acts parallel to the contact surface.

• Static friction keeps stationary objects from sliding. It can range from zero up to a maximum value: $f_s \leq \mu_s N$, where $\mu_s$ is the coefficient of static friction. A box on the floor won't budge until your push exceeds $\mu_s N$.
• Kinetic friction opposes the motion of objects already sliding. Its magnitude is $f_k = \mu_k N$, where $\mu_k$ is the coefficient of kinetic friction. Typically $\mu_k < \mu_s$, which is why it's harder to start sliding a heavy box than to keep it sliding.

Free-Body Diagrams for Multiple Forces

A free-body diagram (FBD) isolates a single object and shows every force acting on it. This is the single most important step in solving any force problem. If your FBD is wrong, everything after it will be wrong too.

Steps to draw one:

1. Isolate the object. Represent it as a dot or simple box. Don't include other objects.
2. Identify every force acting on that object: gravity, normal force, tension, friction, applied forces, etc.
3. Draw each force as an arrow starting from the object, pointing in the direction the force acts. Make the arrow lengths roughly proportional to the force magnitudes if you know them.
4. Label every arrow with the force type ($T$, $N$, $f_k$, $mg$, etc.) and its value if given.
5. Set up a coordinate system. Choose x- and y-axes that simplify the math. For inclined planes, tilt your axes so x runs along the surface and y runs perpendicular to it.

A common mistake is including forces the object exerts on other things. Your FBD should only show forces acting on the object you're analyzing.

Newton's Laws in Force Problems

First Law: An object stays at rest or moves at constant velocity unless a net external force acts on it. A hockey puck gliding on frictionless ice keeps going at the same speed in the same direction because the net force on it is zero.

Second Law: $\Sigma F = ma$. The net force on an object equals its mass times its acceleration. This is the equation you'll use to actually solve problems. You apply it separately along each axis:

$\Sigma F_x = ma_x$

$\Sigma F_y = ma_y$

Third Law: When object A pushes on object B, object B pushes back on A with a force equal in magnitude and opposite in direction. When you walk, your foot pushes backward on the ground, and the ground pushes forward on your foot. These action-reaction pairs always act on different objects, so they never cancel each other out on a single FBD.

Problem-solving process:

1. Read the problem and identify what you're solving for.
2. Draw a free-body diagram for each object of interest.
3. Choose a coordinate system (align axes to simplify, especially on inclines).
4. Write $\Sigma F = ma$ for each axis.
5. Substitute known values and solve the resulting equations for the unknowns.
6. Check your answer: Does the sign make sense? Are the units correct? Is the magnitude reasonable?

Inclined Planes and Normal Force

On an inclined plane, gravity doesn't line up neatly with the surface, so you need to break the weight into components. Tilt your coordinate system so the x-axis runs along the slope and the y-axis is perpendicular to it.

• The component of gravity perpendicular to the surface: $mg \cos \theta$
• The component of gravity parallel to the surface (pulling the object downhill): $mg \sin \theta$

Since the object typically doesn't accelerate through the surface, the normal force balances the perpendicular component:

$N = mg \cos \theta$

Notice what happens at the extremes:

• At $\theta = 0°$ (flat surface): $\cos 0° = 1$, so $N = mg$. The full weight presses into the surface.
• At $\theta = 90°$ (vertical surface): $\cos 90° = 0$, so $N = 0$. The object is in free fall along the "surface."

As the angle increases, the normal force decreases and the parallel component increases. This is why steeper ramps are harder to stand on: more of your weight pulls you downhill, and less of it presses you into the surface (which also means less friction available to keep you in place, since friction depends on $N$).