Newtonian mechanics

The inclined plane

Resolving weight parallel and perpendicular to a slope explains motion and friction.

A ramp is a simple machine: raising an object along it requires less pulling force than lifting vertically, but over a longer distance. To understand sliding, resolve weight into directions tied to the slope.

P_{\parallel}=mg\sin\theta,\qquad N=mg\cos\theta

Definition: Components of weight

For a slope at angle $\theta$ above horizontal, weight $mg$ has a downslope component $mg\sin\theta$ and a perpendicular component $mg\cos\theta$ . If there is no acceleration normal to the plane, the normal force is $N=mg\cos\theta$ . Friction, when present, acts along the plane.

Adjust slope angle, mass, and friction; track weight components, normal force, and downslope acceleration.

Motion along the slope

With no friction, Newton's second law along the plane gives downslope acceleration $a=g\sin\theta$ . With kinetic coefficient $\mu_k$ , a block sliding downhill has $a=g(\sin\theta-\mu_k\cos\theta)$ ; this expression assumes downhill sliding. Static friction can hold the block if the downslope component does not exceed its maximum.

Example: Acceleration without friction

A block slides without friction down a $30^\circ$ slope. Take $g=9.8\,m/s^2$ . Find its downslope acceleration.

Solution

$a=g\sin30^\circ=9.8\times0.5=4.9\,m/s^2$ downhill.

On a 30° slope, a 4 kg crate has a downslope weight component $mgsin30°=20 N$ using $g=10 m/s²$ ; the normal force is $mgcos30°≈34.6 N$ . With no friction, its acceleration is 5 m/s², independent of mass. A ramp trades a smaller lifting force for a longer distance; it does not eliminate the work needed to raise the load.

Example: Forces on a ramp

A block slides without friction down a 30° slope. Take g=10 m/s². Find its acceleration.

Solution

a=g sin30°=10×0.5=5 m/s² down the slope.

To choose the direction of friction, first determine the actual or impending relative motion along the slope. The normal component of gravity affects contact pressure, while its parallel component tends to slide the object downhill. These roles appear on loading ramps and playground slides.

Quick check

What is the component of weight parallel to a slope at angle θ above horizontal?

With no acceleration normal to the slope, what is the normal force N?

References

Young, Freedman (2019). University Physics