Friction

Friction

A force acting on the point of contact of the objects, which opposes the relative motion is called friction.

It acts parallel to the contact surfaces.

Frictional forces are produced due to intermolecular interactions acting between the molecules of the bodies in contact.

Friction is of three types:

1. Static Friction

It is an opposing force which comes into play when one body tends to move over the surface of the other body but actual motion is not taking place.

Static friction is a self adjusting force which increases as the applied force is increases. Static friction oppose impending motion.

Impending Motion

The motion that would take place under the applied force, if friction were absent.

2. Limiting Friction

It is the maximum value of static friction when body is at the verge of starting motion.

$Limitingfriction\left(f_s\right)={\mu }_{s}R$

$where{\mu }_s=coefficientof\lim itingfrictionandR=normalreaction.$

Limiting friction do not depend on area of contact surfaces but depends on their nature, i.e., smoothness or roughness.

If angle of friction is $\theta ,$ then coefficient of limiting friction

${\mu }_s=\tan \theta$

3. Kinetic Friction

If the body begins to slide on the surface, the magnitude of the frictional force rapidly decreases to a constant value f_k kinetic friction.

$Kineticfriction,f_k={\mu }_{k}N$

Where ${\mu }_k=$ coefficient of kinetic friction and N = normal force.

Kinetic friction is of two types:

(a) Sliding frictin

(b) Rolling friction

As, rolling friction < sliding friction, therefore it is easier to roll a body than to slide.

Where ${\mu }_k=$ coefficient of kinetic friction and R = normal reaction.

Angle of repose of angle of sliding it is the minimum angle of inclination of a plane with the horizontal, such that a body placed on it, just begins to slide down.

If angle of repose is $\alpha$ and coefficient of limiting friction is $\mu ,$ then

${\mu }_s=\tan \alpha$

Motion on an Inclined Plane

When an object moves along an inclined plane then different forces act on it like normal reaction of plane, friction force acting in opposite direction of motion etc. Different relations for the motion are given below.

Normal reaction of plane

$R=mg\cos \theta$

and net force acting downward on the block.

$F=mg\sin \theta -f$

Acceleration on inclined plane $a=g\left(\sin \theta -\mu \cos \theta \right)$

When angle of inclination of the plane from horizontal is less than the angel of repose $\left(\alpha \right),$ then

(i) minimum force required to move the body up the inclined plane

$f_1=mg\left(\sin \theta +\mu \cos \theta \right)$

(ii) minimum force required to push the body down the inclined plane

$f_2=mg\left(\mu \cos \theta +\sin \theta \right)$

Tension

Tension force always pulls a body.

Tension is a reactive force. It is not an active force.

Tension across a massless pulley or frictionless pulley remain constant.

Rope becomes slack when tension force becomes zero.

Motion of Bodies in Contact

(i) Two Bodies in Contact If F force is applied on object of mass m₁, then acceleration of the bodies

$a=\frac{F}{\left(m_1+m_2\right)}$

$contactforceon{m}_1=m_{1}a=\frac{m_{1}F}{\left(m_1+m_2\right)}$

$contactforceon{m}_2=m_{2}a=\frac{m_{2}F}{\left(m_1+m_2\right)}$

(ii) mass m₁, then acceleration of the bodies $=\frac{F}{\left(m_1+m_2+m_3\right)}$

Contact force between m1 and m2

$F_1=\frac{\left(m_2+m_3\right)F}{\left(m_1+m_2+m_3\right)}$

Contact force between m₂ and m₃

$F_2=\frac{m_{3}F}{\left(m_1+m_2+m_3\right)}$

(iii) Motion of Two Bodies, One Resting on the Other

(a) The coefficient of friction between surface of A and B be $\mu .$ If a force F is applied on the lower body A, then common acceleration of two bodies

$a=\frac{F}{\left(M+m\right)}$

Pseudo force acting on block B due to the accelerated motion

$f^{\prime }=ma$

The pseudo force tends to produce a relative motion between bodies A and B consequently a frictional force

$f=\mu N=\mu mg$ is developed. For equilibrium

$ma⩽\mu mgora⩽\mu g$

(b) Let friction is also present between the ground surface and body A. Let the coefficient of friction between the given surface and body A is ${\mu }_1$ and the coefficient of friction between the surfaces of bodies A and B is ${\mu }_2.$ if a force F is applied on the lower body A.

Net accelerating force $F-f_A=F-{\mu }_1\left(M+m\right)g$

$\therefore$ Net acceleration

$a=\frac{F-{\mu }_1\left(M+m\right)g}{\left(M+m\right)}$

$=\frac{F}{\left(M+m\right)}-{\mu }_{1}g$

Pseudo force acting on the block B

$f^{\prime }=ma$

The pseudo force tends to produce a relative motion between the bodies A and B are consequently a frictional force $f_B={\mu }_2$ mg is developed. For equilibrium

$ma⩽{\mu }_{2}mg$

Or $a⩽{\mu }_{2}g$

If acceleration produced under the the effect of force F is more than ${\mu }_{2}g,$ then bodies will not move together.