Inclined plane

What is it about?

On an inclined plane with the angle of inclination $\alpha$, the weight $F_G = m \cdot g$ of a body acts vertically downwards. It is resolved into two components:

• the downhill force $F_H$ parallel to the slope (pointing down the slope) and
• the normal force $F_N$ perpendicular to the slope, which presses the body onto the surface.

The triangle of forces $F_G$, $F_H$, $F_N$ is similar to the triangle of length $l$, height $h$ and base of the plane. Therefore

The steeper the plane, the larger $F_H$ and the smaller $F_N$. At $\alpha = 30^\circ$ the downhill force is exactly half the weight.

Gradient in per cent: Road signs give the gradient as the change in height per horizontal distance, i.e. $\tan\alpha$. “12 %” means $\tan\alpha = 0.12$, so $\alpha \approx 6.8^\circ$.

Friction: A block stays at rest as long as static friction can balance the downhill force. Static friction is at most

($\mu_s$: coefficient of static friction, written $\mu_H$ in the simulation). If $F_H$ is larger, the block slides. While sliding, the smaller kinetic friction $F_R = \mu_k \cdot F_N$ (in the simulation $\mu_G$) slows it down. The block then slides down with constant acceleration

Limiting angle: Without any other force, the block starts to slide when $F_H = F_{R,\max}$, i.e. at $\tan\alpha_G = \mu_s$. The mass does not matter. Measuring the limiting angle gives the coefficient of static friction.

Force converter: A cart rolls almost without friction. To hold it on the slope or to pull it up steadily, you only need the force $F_H$ instead of $F_G$ – but the distance $l$ is longer than the height $h$. This is the golden rule of mechanics: $F_H \cdot l = F_G \cdot h$.

Try it

1. Increase the angle by dragging the top end of the plane. Watch $F_H$ and $F_N$ in the scene and in the graph.
2. Press Release at $20^\circ$ and at $35^\circ$. Wooden block at 35°
3. Find the limiting angle for each pair of materials, from which the block slides. Compare with the intersection of $F_H$ and $F_{R,\max}$ in the graph. Rubber on asphalt · Steel on ice
4. Double the mass. Do the limiting angle and the acceleration change? Mass 2 kg
5. Hook the spring balance to the cart and set the pulling force so that the cart stays put. Record measurements for several angles. Cart with spring balance
6. Pull the wooden block up. From which pulling force does it start to move? Pulling up

Tasks

A block with $m = 2\,\text{kg}$ lies on a plane with $\alpha = 30^\circ$. Calculate $F_G$, $F_H$ and $F_N$.

A ramp is $l = 1\,\text{m}$ long and $h = 34\,\text{cm}$ high. What force do you need at least to hold a $1\,\text{kg}$ cart on it (ignore friction)?

A wooden block ($\mu_s = 0.5$) lies on a plane with $\alpha = 25^\circ$. Does it start to slide by itself?

How strongly does the wooden block ($\mu_k = 0.3$) accelerate at $\alpha = 35^\circ$? How long does it take for $40\,\text{cm}$?

Task 5: Measuring the coefficient of friction

A block of an unknown material starts to slide at $\alpha = 31^\circ$. How large is the coefficient of static friction? Check with your own values. Check

With what force must you pull the wooden block ($m = 1\,\text{kg}$, $\mu_k = 0.3$) at $\alpha = 20^\circ$ so that it slides up steadily? Why do you need more force to get it moving?

Task 7: Gradient in per cent

A road sign warns of a $12\,\%$ gradient. How large is the angle of inclination? What fraction of a car’s weight acts as the downhill force?

Ideas for the classroom

• Use: resolving forces on the inclined plane (graphically with the triangle of forces, with $h/l$ or with trigonometry), friction and limiting angle, the inclined plane as a force converter (grades 8–10); the equation of motion $a = g(\sin\alpha - \mu_k\cos\alpha)$ rather from grade 10.
• Terms: Here the component of the weight perpendicular to the plane is called the normal force $F_N$. Some textbooks distinguish it from the equally large force that the surface exerts on the body.
• Model: plane $1\,\text{m}$ long, $g = 9.81\,\tfrac{\text{N}}{\text{kg}}$. The coefficients of friction are typical values; real values depend strongly on surface, moisture and cleanliness. The cart has a very small rolling friction ($0.005$). Before release, a hand holds the body; that is why the static friction shown is at most $F_{R,\max}$. At the bottom end a stop holds the body.
• Series of measurements: With the spring balance attached, a measurement can only be recorded if the body stays at rest with this pulling force. For the cart, the points lie on the curve $F_H(\alpha)$; for the block there is a whole range of holding forces because of static friction – a good reason for discussion.
• Common stumbling blocks:
  • “The normal force always equals the weight.” – On the inclined plane it is smaller: $F_N = F_G \cdot \cos\alpha$.
  • “Heavy bodies start sliding earlier.” – The limiting angle depends only on $\mu_s$, not on the mass.
  • “Friction is always $\mu \cdot F_N$.” – Static friction is only as large as needed, at most $\mu_s \cdot F_N$.
  • “The inclined plane saves work.” – It saves force, but you have to cover a longer distance.