Verification of the laws of limiting Friction and Angle of Friction

Verification of the laws of limiting Friction

Limiting friction is directly proportional to the normal reaction:

Take a block A on a horizontal table. One end of the string is attached to the block and the other end to a pan passing over a pulley as shown in the figure. Now the weight on the pan is kept increasing till the block A moves. If we keep block B over A, as shown in the figure, we will observe that the load on the pan must be doubled. That means if we double the weight of the block, double force is required to move it. As weight = normal reaction, and applied force = force of friction, force of friction α normal reaction F_a α R. Similar results will be obtained if more blocks are used in this experiment.

Limiting friction is independent of the area of contact:

Two blocks A and B are kept one over the other on the horizontal table as shown in the figure. The lower block is connected to one end of a string blocks just start to move. Now the block A and B are kept over the table connected by a slide blocks. Through the area of contact of the blocks with the table in two cases are different, the force required to slide them are equal. Thus, limiting friction is independent of area of contact.

Angle of Friction

A force F_a is applied on a block kept over a table so that the block just begins to move. As the constant applied force, F_a is equal to limiting friction F_l and normal reaction R = mg. So the resultant of F_l and R is F, which makes an angle α with R. This angle is called the angle of friction.

$$\begin{align*} \text {In} \: \Delta OYZ, \\ \tan \alpha &= \frac {XY}{OY} \\ \text {where} \\ XZ &= OX = F_l \\ \text {and} \: OY &= R \\ \therefore \tan \alpha &= \frac {F_l}{R} \\ \text {As} \: F_l &= \mu R \\ \therefore \tan \alpha &= \mu \\ \end{align*}$$

Hence the coefficient of limiting friction is equal to the tangent of the angle of friction.

Angle of Repose

Angle of repose is defined as the minimum angle of inclination of a plane structure with the horizontal such that body kept on it just begins to slide down along the plane.

As shown in the figure, an object of mass ‘m’ is kept on a plane surface AB and its inclination is slowly increased to Ï´ such that the object slides down. The component of the weight mg normal to the surface is mg cosÏ´ which balances the normal reaction R. The component mg sinÏ´ along the inclined surface acts downward along the surface while the frictional force acts upward along the surface. In equilibrium,

$$\begin{align*} mg \sin \theta &= F_s \dots (i) \\ mg\cos \theta &= R \dots (ii) \\ \text {dividing equations } (i) \text {by} (ii) \text {we get} \\ \frac {mg \sin \theta }{mg \cos \theta } &= \frac {F_s} {R} \\ \tan \theta &= \frac {F_s} {R} \\ \text {As} \frac {F_s} {R} &= \mu \\ \therefore \tan \theta &= \mu \\ \end{align*}$$

The tangent of the angle of repose is equal to the coefficient of friction between the surfaces. As tanα = µ where α is the angle of friction,

$$ \therefore \theta = \alpha $$

i.e. angle of repose is equal to angle of friction.

Methods of reducing Friction

The following methods are used to reduce friction:

1. By polishing
   Polishing makes the surfaces smoother. So friction reduces.
2. By lubrication
   Lubricants like oil, grease etc. fill up the irregularities of the surfaces makes them smoother. Hence friction decreases.
3. By proper selection of materials
   The surfaces of moving parts of machines in contact can be linked with materials having low coefficient of friction. On the same basis, tyres are made of rubber. This is because friction between rubber and concrete is much less than friction between iron and concrete.
4. By use of ball bearings
   In rotating machinery, the shafts are mounted on ball bearings to reduce friction. It is because rolling friction is very small as compared to sliding friction. The free wheels of cycles, the axles of motor cars and shafts of motors are provided with ball bearings.
5. By streaming
   Friction due to air is considerably reduced by streaming the shape of the body moving through air. For examples, jets, airplanes, fast moving cars etc. Are given streamline shape.

Importance of friction

Friction opposes the motion of objects, and causes wear and tear. However, it plays very important role in our daily life. It is due to friction that we can walk, drive vehicles and stop moving vehicles.

When we are standing on the road, no force is acting on us in the horizontal direction. As we start to walk, we must have some acceleration in forward direction. For this a net external force must act on us in forward direction in the figure. We create this force on us by pushing the road in backward direction and the friction force between the road and our shoe thus acts is forward direction. This helps us to move on the road.