Static and Kinetic Friction

Friction opposes motion (or tendency to motion) along a contact surface. There are two regimes:

• Static friction holds an object stationary even as you push on it, up to a maximum value $f_s^{\max} = \mu_s\,N$.
• Kinetic friction acts on a moving object with a slightly smaller (usually) constant value $f_k = \mu_k\,N$.

In both cases, $N$ is the normal force between the object and the surface.

Step-by-Step Solution

Given: $m = 20\;\text{kg}$, $\mu = 0.3$ (assume the same for static and kinetic for simplicity), $g = 9.81\;\text{m/s}^{2}$.

Find: The minimum horizontal force needed to start the box sliding, and the friction force when it's already moving.

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Step 1 — Compute the normal force.

On a flat floor, the only vertical forces are gravity (down) and the normal force (up). They must balance, so:

$N = mg = 20 \times 9.81 = 196.2\;\text{N}$

Step 2 — Compute the maximum static friction.

The box doesn't move until you exceed the maximum static friction:

$f_s^{\max} = \mu_s\,N = 0.3 \times 196.2 = 58.86\;\text{N}$

So if you apply less than 58.86 N horizontally, the box stays put — static friction adjusts to exactly cancel your push. The moment you exceed 58.86 N, the box breaks free and starts to slide.

Step 3 — Compute the kinetic friction (once moving).

Once the box is sliding, the friction force is constant:

$f_k = \mu_k\,N = 0.3 \times 196.2 = 58.86\;\text{N}$

(With $\mu_s = \mu_k = 0.3$, they're equal in this simplified problem.)

Step 4 — Compute the acceleration if you apply, say, 80 N.

If you push with $F = 80\;\text{N}$ on a moving box, the net force is:

$F_{\text{net}} = F - f_k = 80 - 58.86 = 21.14\;\text{N}$

The acceleration is:

$a = \dfrac{F_{\text{net}}}{m} = \dfrac{21.14}{20} \approx 1.057\;\text{m/s}^{2}$

Step 5 — Sanity check the formula.

Notice that the friction force depends only on $N$, not on the area of contact, the speed of motion, or how hard the surface looks. This is the surprising and famous Amontons-Coulomb law of friction, an empirical observation that has held up remarkably well for ordinary surfaces.

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Answer:

• Normal force: $N = mg = 196.2\;\text{N}$
• Maximum static friction: $f_s^{\max} = \mu N = 58.86\;\text{N}$
• Minimum applied force to move the box: $\boxed{\;58.86\;\text{N}\;}$
• Once moving, kinetic friction is also $\approx 58.86\;\text{N}$ (with $\mu_k = \mu_s$).

If you apply exactly 58.86 N, the box is on the verge of sliding. Anything more, and it accelerates.

Try It

• Slide the applied force widget — see whether it's enough to overcome static friction.
• Adjust the mass and friction coefficient to see how the threshold changes.
• The HUD lights up "STATIC" (still) or "KINETIC" (moving) and shows the net force on the box.