Briefly explain the origin of friction. Show that on an inclined plane, the angle of friction is equal to the

III. Long Answer Questions · Q4

Q. Briefly explain the origin of friction. Show that on an inclined plane, the angle of friction is equal to 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. Origin of friction: even a well-polished surface has microscopic irregularities; when two surfaces are pressed together, real (atomic-scale) contact occurs only over a small fraction of the apparent contact area. At these true-contact points, electromagnetic (interatomic) forces between the two surfaces' atoms bond momentarily, and the making/breaking of these bonds during relative motion produces the frictional resistance.

Step 2. Angle of friction — defined as the angle θ between the normal force N and the resultant R of N and the maximum static friction $f_{s,max}$ ; from the geometry, $\tan\theta=\dfrac{f_{s,max}}{N}$ , and since $f_{s,max}=\mu_sN$ , this gives $\tan\theta=\mu_s$ .

Step 3. Angle of repose — for a block on an inclined plane, the weight resolves into $mg\sin\theta$ (along the incline, tending to slide the block down) and $mg\cos\theta$ (perpendicular, balanced by $N=mg\cos\theta$ ).

Step 4. The block is on the verge of sliding when static friction reaches its maximum: $f_{s,max}=mg\sin\theta$ . But also $f_{s,max}=\mu_sN=\mu_smg\cos\theta$ . …

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Q.Briefly explain the origin of friction. Show that on an inclined plane, the angle of friction is equal to the angle of repose.

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