Motion on an inclined plane

Tilt the axes, not the slope

On a slope, the motion is along the slope. So choose the axes along the slope and across it. Only the weight is not along one of these axes, so split it:

• along the slope, downward: $mg\sin\theta$;
• across the slope, into it: $mg\cos\theta$.

Nothing moves across the slope, so the normal reaction balances that part: $N = mg\cos\theta$.

A smooth slope

With no friction, the only force along the slope is $mg\sin\theta$:

The mass cancels. At $0^\circ$ the block stays put; at $90^\circ$ it falls freely with $g$.

A rough slope

A block sliding down feels friction $\mu_k N = \mu_k mg\cos\theta$ up the slope:

A block at rest stays at rest while static friction can hold $mg\sin\theta$, that is while $mg\sin\theta \le \mu_s mg\cos\theta$, or

The angle where it just starts to slide is the angle of repose, $\tan\theta_r = \mu_s$. It does not depend on the mass. Try it.

You should find the block starts to slide when $\tan\theta$ passes $\mu_s$, whatever its mass, and that the friction arrow grows with the angle until then.

Watch out. Friction acts against the motion. A block pushed up a slope feels friction down the slope, so its deceleration is $g(\sin\theta + \mu_k\cos\theta)$, bigger than on the way down.

A worked example

An $8.0\ \text{kg}$ crate is let go on a ramp at $26^\circ$, with $\mu_s = 0.33$ and $\mu_k = 0.24$.

1. Does it slide? $\tan 26^\circ = 0.488$, more than $0.33$. Yes.
2. The forces along the ramp. Down: $mg\sin\theta = 8.0\ \text{kg} \times 9.8\ \text{m/s}^2 \times 0.438 = 34.4\ \text{N}$. Up: friction $\mu_k mg\cos\theta = 0.24 \times 8.0\ \text{kg} \times 9.8\ \text{m/s}^2 \times 0.899 = 16.9\ \text{N}$.
3. Acceleration. $a = \dfrac{34.4\ \text{N} - 16.9\ \text{N}}{8.0\ \text{kg}} = 2.18\ \text{m/s}^2$, the same as $g(\sin\theta - \mu_k\cos\theta)$.

Think first. Before using $\mu_k$, check that the body slides at all. On a gentle slope the answer may be "it stays put".

Practise with new numbers

Try it in Python

The program tilts a rough slope more and more, and reports whether the block slides.

Remember these

• Along the slope $mg\sin\theta$; across it $mg\cos\theta = N$.
• Smooth slope: $a = g\sin\theta$.
• Sliding down a rough slope: $a = g(\sin\theta - \mu_k\cos\theta)$.
• It starts to slide when $\tan\theta > \mu_s$ (the angle of repose).

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A trolley rolls down a smooth slope at $35^\circ$. What is its acceleration? ($g = 9.8\ \text{m/s}^2$.)

2. A coin on a book starts to slide when the book is tilted to $27^\circ$. What is $\mu_s$?

3. A block slides down a slope of $18^\circ$ at a steady speed. What is $\mu_k$?

4. You push a box up a rough ramp. Which way does friction act on it?