Friction, Tension, Normal Force, and Springs

12.1 Why Specific Force Models Matter

Newton’s second law, F_net = ma, is universally true, but it is only as useful as your knowledge of the forces on the right side. In Chapter 10, you learned what forces are and what Newton’s laws say about them. In Chapter 11, you learned to draw them. This chapter teaches you the quantitative rules that govern the most common mechanical forces: friction, tension, normal force, and spring force.

Each force type has its own model — a rule that specifies its magnitude and direction under given conditions. These models are empirical: they are simplifications that capture the dominant behavior of real forces to within the precision needed for most problems. They are not exact at the atomic level, but they are powerful and reliable within their domains.

12.2 Contact Forces vs. Constraint Forces

The forces in this chapter fall into two broad categories. Contact forces arise from direct physical interaction between surfaces or objects at a point of contact. They include friction and the normal force. They exist only when surfaces are in contact and disappear when contact is broken.

Constraint forces are forces that enforce a geometric or mechanical constraint on motion. Tension in a rope constrains an object to move with the rope. The normal force from a surface constrains an object to not pass through the surface. Springs constrain objects to remain connected to the spring’s equilibrium point. These forces take whatever magnitude is needed to maintain the constraint — up to the constraint’s breaking point.

12.3 The Nature of Friction

Friction is a contact force that opposes relative motion, or the tendency toward relative motion, between two surfaces. It acts parallel to the contact surface and, unlike the normal force, has a component along the direction of motion.

Friction arises from electromagnetic interactions at the atomic level. Real surfaces, no matter how polished, are microscopically rough. When two surfaces are pressed together, their irregularities interlock and form tiny adhesive bonds. Breaking these bonds requires a force. The friction we measure is the macroscopic result of billions of these microscopic bond-breaking events.

Friction depends on the types of surfaces and on how hard they are pressed together, which is the normal force. It does not depend on the area of contact — a wide box and a narrow box of the same mass on the same surface experience the same friction force. This result, which surprises many students, follows from the microscopic model.

12.4 Static Friction

Static friction acts on an object that is not sliding relative to the surface. It is a constraint force: it takes whatever magnitude is needed to prevent sliding, up to a maximum value. Static friction is variable — it adjusts to balance the applied forces and keep the object stationary.

When the applied force along the surface is small, static friction is small and exactly cancels it. As the applied force increases, static friction increases to match. When the applied force exceeds the maximum static friction, the object begins to slide and kinetic friction takes over.

f_s ≤ μ_sN

The direction of static friction opposes the tendency to slide: it points in the direction that would prevent sliding. Ask: if friction disappeared, which way would the object start to move? Static friction acts opposite to that direction.

12.5 Kinetic Friction

Kinetic friction, also called sliding friction, acts on an object that is already sliding relative to the surface. Unlike static friction, kinetic friction has a fixed magnitude, for given surfaces and normal force, that does not depend on how fast the object is moving or the applied force.

f_k = μ_kN

Because kinetic friction is constant for constant N, it produces a constant deceleration in the absence of other horizontal forces. This allows the kinematic equations from Chapter 8 to be applied directly: the deceleration from kinetic friction is a = f_k/m = μ_kg for a horizontal surface with no other horizontal forces.

12.6 Coefficients of Friction

The coefficient of friction is a dimensionless number that characterizes how rough a pair of surfaces is relative to each other. It is always defined for a surface pair — not for a single surface alone — because friction depends on both surfaces in contact.

Coefficients of friction are measured empirically. They are not exact physical constants — they depend on surface cleanliness, temperature, humidity, and surface finish. The values in Table 12.2 are representative ranges; problems in this book will always specify the coefficient to use.

12.7 Tension in Ropes and Cables

Tension is the internal force transmitted through a rope, cable, chain, or rod under pulling, or tensile, stress. When you pull one end of a rope attached to an object, the rope transmits that pulling force to the object as tension. The tension acts along the rope and pulls — never pushes.

For an ideal rope, massless and inextensible, the tension is the same at every point along the rope. This is because each infinitesimally small section of the rope has zero mass, so any net force on it would produce infinite acceleration. Newton’s second law then requires zero net force at every point, meaning the tension must be equal throughout.

When a rope is attached to multiple objects or passes over a pulley, the tension determines how forces are transmitted. Drawing an FBD of each object and a segment of the rope separately reveals how tension propagates.

12.8 Ideal Rope and Pulley Assumptions

Most rope-and-pulley problems in this book use three idealizations. Stating them explicitly at the start of each problem is good practice.

In an Atwood machine, two masses hanging from a single rope over a pulley, the inextensible rope means both masses have the same acceleration magnitude but opposite directions. The massless pulley means the tension in the rope is the same on both sides.

12.9 The Normal Force

The normal force is the contact force a surface exerts on an object perpendicular to the surface, preventing the object from passing through. “Normal” means perpendicular. It is always directed away from the surface and into the object.

The normal force is a constraint force: it takes whatever value is needed to keep the object on the surface. On a horizontal surface with only gravity and the normal force acting vertically, N = mg by Newton’s second law. In other situations, N takes a different value.

The normal force arises from the electromagnetic repulsion between the electron clouds of the surface atoms and the object’s atoms. When you push two surfaces together, their electron clouds resist interpenetration. The macroscopic normal force is the sum of billions of these repulsive interactions.

12.10 Why Normal Force Is Not Always Equal to Weight

The common shortcut “N = mg” applies only in the specific case of a horizontal surface with no other vertical forces and zero vertical acceleration. In every other case, N must be calculated from Newton’s second law applied to the perpendicular direction.

12.11 Spring Forces and Elastic Deformation

Many materials deform elastically: when deformed, they exert a restoring force that tends to return them to their original shape. Coil springs, rubber bands, and even steel rods behave this way within their elastic limits. The restoring force model for springs, derived from experiment, is one of the most useful relationships in mechanics.

Elastic deformation is reversible: remove the applied force and the material returns to its original shape. Beyond the elastic limit, or yield point, permanent plastic deformation occurs. The spring models in this chapter assume elastic deformation throughout.

Springs appear throughout physics: in mechanical devices, seismometers, shock absorbers, atomic models of crystalline solids, and as conceptual models for chemical bonds. The simple linear spring is the prototype for a vast class of oscillatory and restoring-force phenomena.

12.12 Hooke’s Law

Hooke’s Law states that for small deformations within the elastic limit, the restoring force exerted by a spring is proportional to the displacement from equilibrium and directed opposite to that displacement.

F_spring = −kx

In most mechanics problems, we specify the direction explicitly rather than relying on the sign. The rule is: a stretched spring pulls the object back toward equilibrium; a compressed spring pushes the object back toward equilibrium. Either way, the force magnitude is k|x| and the direction is toward equilibrium.

The spring constant k characterizes the stiffness of a spring: a large k means a stiff spring that requires a large force for a small deformation. A small k means a soft spring easily deformed by small forces.

12.13 Combining Multiple Forces in One System

Most real problems involve several force types simultaneously. The approach is always the same: draw the FBD for each object, identify the magnitude rule for each force (N = mg cos θ, f = μN, T from constraint, F_spring = kx), then apply Newton’s second law.

The key discipline is to derive each force from its model rather than guessing. Normal force comes from the perpendicular equilibrium condition. Friction comes from the coefficient times the normal force. Tension comes from the constraint equation. Spring force comes from Hooke’s Law.

12.14 Solving Multi-Force Newton’s Law Problems

A systematic method prevents errors when multiple force types interact.

12.15 Common Mistakes in Force Modeling

Chapter Summary

Specific force models link force magnitudes to measurable quantities. The four major contact and constraint forces in mechanics — friction, tension, normal force, and spring force — each obey a quantitative rule.

• Static friction: f_s ≤ μ_sN; equals applied force up to maximum; opposes tendency to slide.
• Kinetic friction: f_k = μ_kN; fixed magnitude for given surfaces; opposes sliding direction.
• μ_k < μ_s for the same surface pair; starting sliding requires more force than maintaining it.
• Tension in an ideal rope has the same magnitude throughout and acts along the rope, pulling toward the attachment.
• An ideal pulley redirects tension without changing its magnitude.
• Normal force is perpendicular to the surface and away from it. N = mg only on horizontal surfaces with zero vertical acceleration and no other vertical forces. Derive N from Newton’s second law.
• Spring force follows Hooke’s Law: F = kx toward equilibrium; k is in N/m; x is displacement from natural length.
• To solve multi-force problems: draw the FBD, find N first from perpendicular equilibrium, compute friction, then apply F_net = ma.

Key Terms

static friction

The friction force on a stationary object that prevents it from sliding; variable, ranging from 0 to μ_sN; direction opposes tendency to slide.

kinetic friction

The friction force on a sliding object; fixed magnitude f_k = μ_kN; direction opposes sliding.

coefficient of static friction (μ_s)

Dimensionless ratio characterizing maximum static friction relative to normal force for a surface pair; μ_s ≥ μ_k.

coefficient of kinetic friction (μ_k)

Dimensionless ratio characterizing kinetic friction relative to normal force for a surface pair; always less than μ_s.

tension

The pulling force transmitted through a rope or cable; acts along the rope toward the attachment; cannot push.

ideal rope

A massless, inextensible rope; transmits tension unchanged in magnitude throughout its length.

ideal pulley

A massless, frictionless pulley; redirects the rope without changing the tension magnitude.

normal force

The contact force perpendicular to and away from a surface; a constraint force that prevents penetration; not always equal to weight.

spring constant (k)

The stiffness of a spring in N/m; larger k means stiffer spring; determined experimentally.

Hooke’s Law

The linear relationship between spring force and displacement: F = −kx, a restoring force toward equilibrium.

elastic limit

The maximum deformation beyond which a material no longer returns to its original shape; permanent deformation occurs and Hooke’s Law fails.

Atwood machine

A simple pulley system with two masses hanging from a single rope over a pulley; used to study tension and Newton’s second law.

Concept Review Questions

1. What is the difference between static and kinetic friction? When does each apply?
2. A 5.0 kg box is pushed with a horizontal force of 10.0 N on a surface with μ_s = 0.30. The box does not move. What is the friction force on the box? Is it equal to μ_sN?
3. Explain why kinetic friction is less than maximum static friction for the same surface pair.
4. Why does the tension in a massless rope remain constant throughout its length?
5. What does a pulley do to tension in an ideal rope? What changes, and what does not?
6. Give three situations where the normal force is not equal to the weight. For each, state whether N > mg or N < mg and why.
7. What does the spring constant k tell you about a spring? What are its units?
8. A spring is stretched 5 cm by a 10 N force. What force is needed to stretch it 15 cm?
9. When setting up a multi-force Newton’s Law problem, why must you find the normal force before computing friction?
10. State Hooke’s Law in words and as an equation. Explain what the minus sign means physically.

Friction Problems

11. A 15.0 kg box on a horizontal floor has μ_s = 0.45 and μ_k = 0.32. (a) What is the minimum force to start it sliding? (b) Once sliding, what force maintains constant velocity? (c) If pushed with 80.0 N, what is the acceleration?
12. A 25.0 kg sled is pulled on packed snow (μ_k = 0.10) by a rope at 15° above horizontal. The sled moves at constant velocity. Find the tension in the rope.
13. A 6.0 kg block slides 4.0 m down a rough 30° ramp (μ_k = 0.20). Find: (a) the normal force; (b) the friction force; (c) the acceleration; (d) the speed at the bottom if it starts from rest.
14. A 10.0 kg box is on a rough surface (μ_s = 0.55). A horizontal force of 40.0 N is applied. Does it move? If not, what is the friction force?
15. A 4.0 kg block on a rough horizontal surface (μ_k = 0.25) is connected by a horizontal rope over a frictionless pulley to a hanging 3.0 kg mass. Find: (a) the acceleration; (b) the tension.

Pulley and Tension Problems

16. Two masses hang from a massless, frictionless pulley: m₁ = 5.0 kg and m₂ = 8.0 kg. Find: (a) acceleration; (b) tension.
17. A 7.0 kg mass hangs from a rope attached to a 10.0 kg block on a frictionless horizontal table. The rope passes over a pulley at the table’s edge. Find: (a) acceleration of the system; (b) tension.
18. Repeat the previous problem with μ_k = 0.20 between the block and table.
19. Three masses hang vertically: m₁ = 2.0 kg on top, m₂ = 3.0 kg in the middle, m₃ = 5.0 kg at the bottom, each connected by ideal ropes. A force of 100 N pulls upward on m₁. Find the acceleration and the tension in each rope.
20. A 60.0 kg person climbs a rope hanging from the ceiling. The rope has a breaking strength of 700 N. What is the maximum upward acceleration the person can achieve without breaking the rope?

Spring Force Problems

21. A spring stretches 6.0 cm under a 15.0 N force. (a) Find k. (b) How much force stretches it 10.0 cm? (c) How much does it stretch under 25.0 N?
22. A spring with k = 400 N/m is compressed 5.0 cm and releases a 0.20 kg ball. What is the force on the ball at the moment of release? Assume the spring is still in contact.
23. A 2.0 kg mass hangs from a spring (k = 80 N/m) at rest. By how much does the spring stretch?
24. Two springs with k₁ = 100 N/m and k₂ = 200 N/m are connected in series and stretched 15.0 cm total. Find the force on the system. Hint: in series, same force, different stretches.
25. A 0.50 kg block on a frictionless surface is attached to a spring (k = 150 N/m) and pulled 8.0 cm from equilibrium, then released. What is the acceleration at the moment of release?

Mixed Force Analysis

26. A 5.0 kg box rests on a 40° rough incline (μ_s = 0.50, μ_k = 0.38). (a) Does the box slide? (b) If it does, find the acceleration.
27. A 3.0 kg block on a frictionless 25° incline is attached to a spring (k = 120 N/m) parallel to the incline. Find the spring’s compression or stretch when the block is in equilibrium.
28. An 8.0 kg box is pulled by a rope at 20° above horizontal. The surface has μ_k = 0.30. The acceleration is 2.0 m/s². Find the tension T.
29. Challenge: Block A (6.0 kg) rests on a rough horizontal table (μ_k = 0.20). A rope connects A over a pulley at the edge to Block B (4.0 kg) hanging vertically, and another rope connects A over a second pulley to Block C (2.0 kg) also hanging vertically on the other side. Find the acceleration and both rope tensions.

Chapter 12 Checkpoint