Describe the method of measuring the angle of repose.

III. Long Answer Questions · Q9

Q. Describe the method of measuring the angle of repose.

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Concept understanding — Friction on an Inclined Plane

Friction on an Inclined Plane

Imagine a block placed on a ramp. If the ramp is flat, the block stays put. As you tilt the ramp, something changes. At a small tilt, the block still doesn't move — something is holding it in place. That something is static friction, acting up the slope, exactly balancing the component of gravity that tries to pull the block down.

Now tilt the ramp a little more. At some critical angle, the block just begins to slide. At that instant, static friction has reached its maximum possible value. Tilt it further, and the block accelerates downward — now kinetic friction acts up the slope, but it is weaker than the maximum static friction was.

That is the entire story in a nutshell. Let's make it precise.

The forces on an inclined plane

Take a block of mass $m$ on a plane inclined at angle $\theta$ to the horizontal. The weight $mg$ points straight down. Resolve it into two components:

Perpendicular to the plane: $mg\cos\theta$ — this pushes the block into the surface. The normal reaction $N$ balances it: $N = mg\cos\theta$ .
Parallel to the plane (down the slope): $mg\sin\theta$ — this is what tries to make the block slide.

Friction $f$ acts parallel to the plane, opposing the relative motion (or the tendency to move). So if the block is at rest or sliding down, friction acts up the slope. If you push the block up the slope, friction acts down the slope.

Three cases

1. Block at rest (static friction)

The block does not move. So the net force along the slope is zero:

f_s = mg\sin\theta

Here $f_s$ is static friction. It is not a fixed value — it adjusts itself to whatever is needed, up to a maximum. The maximum possible static friction is:

f_{s,\text{max}} = \mu_s N = \mu_s mg\cos\theta

The block stays at rest as long as:

mg\sin\theta \le \mu_s mg\cos\theta \quad\Rightarrow\quad \tan\theta \le \mu_s

The angle $\theta_s = \tan^{-1}(\mu_s)$ is called the angle of repose. At this angle, the block is just about to slide.

Important

The angle of repose $\theta_s$ satisfies $\tan\theta_s = \mu_s$ . This is a direct link between friction and geometry — you can measure $\mu_s$ by finding the tilt at which sliding begins.

2. Block sliding down (kinetic friction)

Once $\theta > \theta_s$ , the block accelerates down. Kinetic friction acts up the slope, with magnitude:

f_k = \mu_k N = \mu_k mg\cos\theta

The net force down the slope is:

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

So the acceleration down the plane is:

a = g(\sin\theta - \mu_k\cos\theta)

Note

| Condition | What happens |

|-----------|--------------|

| $\tan\theta < \mu_s$ | Block stays at rest |

| $\tan\theta = \mu_s$ | Block just about to slide |

| $\tan\theta > \mu_s$ | Block slides down with acceleration $g(\sin\theta - \mu_k\cos\theta)$ |

3. Block pushed up the slope …

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Step 1. Take a hard-bound notebook lying flat, and place a small object (a coin or eraser) on its stiff cover.

Step 2. Slowly tilt the cover (about the spine, keeping the rest of the notebook flat) so the angle of inclination with the horizontal steadily increases, while the coin stays on the cover.

Step 3. At small angles, the component of gravity along the incline ( $mg\sin\theta$ ) is smaller than the maximum available static friction ( $\mu_smg\cos\theta$ ), so the coin stays put.

Step 4. Continue increasing the angle until, at some particular angle, the coin just begins to slide down — at exactly this angle, $mg\sin\theta=\mu_smg\cos\theta$ , i.e. $\tan\theta=\mu_s$ .

Step 5. Measure this critical angle θ (e.g. with a protractor against the horizontal); its tangent directly gives the coefficient of static friction between the coin and the cover — this angle IS the angle of repose (equivalently the angle of friction) for that pair of surfaces. …

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Q.Describe the method of measuring the angle of repose.

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