Friction, Tension, and Normal Force
Analyze friction, tension, and normal force in horizontal, vertical, and inclined-plane problems using free-body diagrams and equations.
A lesson of Physics, in Forces and Newton Laws, about 10 minutes.
By the end of this lesson, you will be able to calculate frictional forces, apply the coefficients of friction, solve tension problems, and analyze objects on inclined planes with friction.
Friction
Friction is the force that opposes relative motion (or tendency of motion) between surfaces in contact.
Static friction ($f_s$): prevents objects from starting to slide; can be less than its maximum value.
$$f_s \leq \mu_s N$$
$\mu_s$ = coefficient of static friction (dimensionless; $\mu_s > \mu_k$ always).
Kinetic friction ($f_k$): acts on sliding objects; constant magnitude.
$$f_k = \mu_k N$$
$\mu_k$ = coefficient of kinetic friction.
Friction always opposes the direction of motion (or intended motion).
Example 1 — Pulling a Block
A 20 kg block on a horizontal surface ($\mu_k = 0.30$) is pulled by a 70 N horizontal force. Find acceleration.
$N = mg = 20 \times 9.8 = 196$ N
$f_k = 0.30 \times 196 = 58.8$ N
$\sum F_x = 70 - 58.8 = 11.2$ N
$a = 11.2/20 = 0.56$ m/s²
Tension
Tension is the pulling force transmitted through a string, rope, or cable.
For a massless string (ideal): tension is the same throughout.
For a string with mass: tension varies along its length.
Example 2 — Box on Inclined Plane with Friction
A 10 kg box slides down a 30° incline with $\mu_k = 0.20$. Find acceleration.
Along incline ($x$): $mg\sin 30° - f_k = ma$
Perpendicular to incline ($y$): $N - mg\cos 30° = 0 \implies N = mg\cos 30°$
$f_k = \mu_k N = 0.20 \times 10(9.8)\cos 30° = 16.97$ N
$a = g\sin 30° - f_k/m = 9.8(0.5) - 16.97/10 = 4.9 - 1.70 = 3.2$ m/s²
Example 3 — Maximum Angle Before Sliding
For what angle does a block just start to slide on a surface with $\mu_s = 0.50$?
At the critical angle: static friction = maximum.
$\tan\theta = \mu_s \implies \theta = \tan^{-1}(0.50) = 26.6°$
Apparent Weight
Normal force is not always equal to weight.
In an elevator accelerating upward: $N = m(g + a)$ → "heavier"
In an elevator accelerating downward: $N = m(g - a)$ → "lighter"
In free fall: $N = 0$ → weightlessness (astronauts).
Summary
Kinetic friction: $f_k = \mu_k N$ (always opposite to motion).