5.4 Inclined Planes

Friction and Inclined Planes

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Static vs. Kinetic Friction

Static friction acts between two surfaces that aren't moving relative to each other. It opposes the start of motion, and it's variable: it matches whatever force is trying to get the object moving, up to a maximum value.

$f_{s,\max} = \mu_s N$

• $\mu_s$ is the coefficient of static friction, determined by the materials in contact (rubber on concrete has a high $\mu_s$; ice on steel has a very low one)
• $N$ is the normal force, perpendicular to the surface

A common point of confusion: static friction isn't always at its maximum. If a 20 N force pushes a box and the box doesn't move, static friction is exactly 20 N, even if $f_{s,\max}$ is 50 N. It only reaches $f_{s,\max}$ right at the threshold of motion.

Kinetic friction acts between two surfaces that are sliding relative to each other. It opposes the direction of motion and has a fixed magnitude for a given pair of surfaces:

$f_k = \mu_k N$

• $\mu_k$ is the coefficient of kinetic friction, also determined by the materials in contact
• For any pair of surfaces, $\mu_k < \mu_s$. That's why it takes more force to start sliding a heavy box than to keep it sliding.

Once an applied force exceeds $f_{s,\max}$, the object begins to move and friction drops to $f_k$. From there, kinetic friction decelerates the object if no other net force is present.

Forces on Inclined Planes

The key skill here is resolving the weight vector into components aligned with the incline. Gravity pulls straight down with magnitude $mg$, but on a tilted surface, that force does two things at once: it pushes the object into the surface and pulls it along the slope.

To break $mg$ into components along and perpendicular to the ramp:

1. Define your coordinate axes so that one axis runs parallel to the incline surface and the other runs perpendicular to it.
2. The component pulling the object down the slope (parallel) is $mg\sin\theta$.
3. The component pushing the object into the surface (perpendicular) is $mg\cos\theta$.
4. Since the object doesn't accelerate through the surface, the normal force balances the perpendicular component: $N = mg\cos\theta$.

Why sine for parallel and cosine for perpendicular? Think about the extremes. When $\theta = 0°$ (flat ground), the parallel component should be zero and the perpendicular component should equal $mg$. Since $\sin 0° = 0$ and $\cos 0° = 1$, the assignments check out.

Applying Newton's Second Law

For an object sliding down an incline with kinetic friction:

$F_{\text{net}} = mg\sin\theta - f_k = mg\sin\theta - \mu_k mg\cos\theta$

Dividing both sides by $m$:

$a = g\sin\theta - \mu_k g\cos\theta$

Notice that mass cancels entirely. The acceleration depends only on the angle, the friction coefficient, and $g$.

If the object is at rest, it stays at rest as long as static friction can balance the parallel component:

$mg\sin\theta \leq \mu_s mg\cos\theta$

This simplifies to $\tan\theta \leq \mu_s$. So the steepest angle at which an object can sit without sliding depends only on $\mu_s$, not on mass. This critical angle is sometimes called the angle of repose.

Variables Affecting Inclined Motion

Mass does not affect acceleration on an incline (with or without friction), because both the gravitational force component and the friction force scale with $m$. However, mass does affect the magnitudes of the normal force and friction force themselves. A 20 kg box on a ramp has twice the friction force of a 10 kg box on the same ramp, but both boxes accelerate at the same rate.

Angle has a double effect. As $\theta$ increases:

• The parallel component $mg\sin\theta$ increases, pulling the object down the slope more strongly.
• The perpendicular component $mg\cos\theta$ decreases, which reduces the normal force and therefore reduces friction.

Both effects work together to increase acceleration at steeper angles.

Surface properties determine $\mu_s$ and $\mu_k$. Rougher surfaces (like sandpaper on wood) have higher coefficients and produce more friction. Smoother surfaces (like polished metal) have lower coefficients and allow faster sliding.

Equilibrium and Mechanical Advantage

Equilibrium on an incline means the net force along the slope is zero. This can happen when friction alone balances the parallel weight component, or when an applied force (like someone pushing a crate up a ramp) combines with friction to produce zero net force.

Mechanical advantage is the ratio of the output force to the input force. An inclined plane lets you raise an object to a height $h$ by applying a smaller force over a longer distance $d$ along the ramp. The ideal mechanical advantage of an incline is:

$\text{IMA} = \frac{d}{h} = \frac{1}{\sin\theta}$

A longer, shallower ramp means a greater mechanical advantage: you push with less force, but over a longer distance. The trade-off is that work (force times distance) stays the same in an ideal case, consistent with energy conservation.