6Friction

📚
FRICTION
▢ Introduction:

Table 1: Friction Basics

Point
Description
Friction
Opposing force acting when one body moves / tries to move over another surface
Acts during
Sliding, rolling or tendency of motion
Old view
Due to interlocking of surface irregularities
Modern view
Due to atomic / molecular attraction at actual contact points
Nature
Contact force
Type of force
Non-conservative force
Origin
Electrical / molecular interaction origin
▢ Types of Friction:

Table 1: Types of Frictional Forces

Type
Meaning
Special point
Static friction
Opposing force when body tends to move but actual motion has not started
Self-adjusting force
Limiting friction
Maximum value of static friction
Acts when body is just about to move
Kinetic / dynamic friction
Opposing force when body is actually moving over surface
Almost constant for given surfaces
Sliding friction
Friction when body slides over surface
Greater than rolling friction
Rolling friction
Friction when body rolls over surface
Much smaller than sliding friction
❖ Order: \(\mu_r < \mu_k < \mu_s\)
❖ Meaning of Order: Rolling friction < kinetic friction < limiting/static friction
❖ Applied Force vs Friction:
  • •
    Applied force increases → static friction increases gradually
  • •
    Static friction increases up to limiting friction
  • •
    After motion starts → kinetic friction remains nearly constant
▢ Laws of Limiting Friction:
  1. Limiting friction \(F\) is directly proportional to normal reaction \(R\): \(F\propto R\)
  2. Direction of limiting friction is opposite to direction in which body is about to move
  3. Limiting friction is independent of apparent area of contact if normal reaction is constant
  4. Limiting friction depends on nature of surfaces in contact
▢ Coefficient of Friction:

Table 1: Coefficient of Friction

Coefficient
Formula
Meaning
Coefficient of limiting/static friction
\(\mu_s=\frac{F}{R}\)
Limiting friction / normal reaction
Coefficient of kinetic friction
\(\mu_k=\frac{F_k}{R}\)
Kinetic friction / normal reaction
Coefficient of rolling friction
\(\mu_r=\frac{F_r}{R}\)
Rolling friction / normal reaction
❖ Depends On:
  • •
    Nature of materials
  • •
    State of polish
  • •
    Condition of surfaces in contact
▢ Angle of Friction:
❖ Definition: Angle made by resultant of frictional force and normal reaction with direction of normal reaction

Table 1: Angle of Friction

Quantity
Formula
Angle of friction
\(\theta\)
Coefficient of friction
\(\mu=\tan\theta\)
Friction-normal relation
\(\tan\theta=\frac{F}{R}\)
▢ Angle of Repose:
❖ Definition: Maximum angle of inclination of plane with horizontal at which body is just in limiting equilibrium

Table 1: Angle of Repose

Quantity
Formula / Point
Angle of repose
\(\alpha\)
At limiting equilibrium
\(F=mg\sin\alpha\)
Normal reaction
\(R=mg\cos\alpha\)
Coefficient of friction
\(\mu=\frac{F}{R}=\tan\alpha\)
Relation
\(\mu=\tan\alpha=\tan\theta\)
Result
\(\alpha=\theta\)
❖ Conclusion: Angle of repose = Angle of friction
▢ Pulling a Block on Rough Horizontal Surface:
❖ Condition: Block of mass \(M\), force \(F\) pulled at angle \(\alpha\), coefficient of friction \(\mu\), angle of friction \(\theta\)
❖ _*table:
    ❖ Important Point: Pulling decreases normal reaction, so pulling is easier than pushing
    ▢ Pushing a Block on Rough Horizontal Surface:
    ❖ Condition: Block pushed by force \(F\) at angle \(\alpha\) with horizontal

    Table 1: Pushing Formulae

    Quantity
    Formula
    Normal reaction
    \(R=Mg+F\sin\alpha\)
    Condition for motion
    \(F\cos\alpha \geq \mu R\)
    Required pushing force
    \(F\geq \frac{\mu Mg}{\cos\alpha-\mu\sin\alpha}\)
    In terms of angle of friction
    \(F\geq \frac{Mg\sin\theta}{\cos(\alpha+\theta)}\)
    Possible pushing condition
    \(\alpha \leq \theta\)
    ❖ Do You Know:
    • •
      Block cannot be pushed, however large force is applied, if \(\alpha>\theta\)
    • •
      For minimum force, block should be pulled at angle of friction \(\theta\)
    ▢ Stopping Time and Stopping Distance:
    ❖ Definition: When driving force is removed, moving body stops due to friction after time \(t\) and distance \(s\)

    Table 1: Stopping Formulae

    Quantity
    Formula
    Retarding force
    \(f=ma\)
    Retardation due to friction
    \(a=\mu g\)
    Stopping distance
    \(s=\frac{v^2}{2\mu g}\)
    Stopping time
    \(t=\frac{v}{\mu g}\)
    Mass dependence
    Stopping distance and time are independent of mass
    ▢ Minimum Force to Move Block:

    Table 1: Minimum Force Formulae

    Case
    Condition / Formula
    Minimum force at angle \(\theta\)
    \(\tan\theta=\mu\)
    Minimum force
    \(F=mg\sin\theta=\frac{mg\mu}{\sqrt{1+\mu^2}}\)
    Minimum horizontal force
    \(F=mg\tan\theta=\mu mg\)
    Least force along rough horizontal plane
    \(W\sin\theta=mg\sin\theta\)
    Least horizontal force along rough horizontal plane
    \(W\tan\theta=mg\tan\theta\)
    ▢ Block Against Vertical Wall:
    ❖ Condition: Block of mass \(m\) is held against vertical wall by horizontal force \(F\)

    Table 1: Vertical Wall Formula

    Step
    Formula
    Condition to prevent sliding
    \(f\geq mg\)
    Friction
    \(f=\mu R\)
    Normal reaction
    \(R=F\)
    Minimum force
    \(F=\frac{mg}{\mu}=mg\cot\theta\)
    ▢ Hanging Chain Over Table:
    ❖ Condition: Uniform chain of mass \(M\), length \(L\), length \(x\) hanging from edge of horizontal table

    Table 1: Chain on Rough Table

    Quantity
    Formula
    Frictional force
    \(\mu\frac{M}{L}(L-x)g\)
    Weight of hanging part
    \(\frac{M}{L}xg\)
    Equilibrium condition
    \(\mu\frac{M}{L}(L-x)g=\frac{M}{L}xg\)
    Coefficient of friction
    \(\mu=\frac{x}{L-x}=\frac{Lower\ part}{Upper\ part}\)
    Maximum overhanging length
    \(x=\frac{\mu}{1+\mu}L\)
    Fraction overhung
    \(\frac{x}{L}=\frac{\mu}{1+\mu}\)
    Condition for no sliding
    \(\mu\geq\frac{x}{L-x}\)
    ▢ Friction on Inclined Plane:

    Table 1: Inclined Plane with Friction

    Case
    Acceleration / Formula
    Body sliding downward
    \(a=g(\sin\theta-\mu\cos\theta)\)
    Body moving upward
    \(a=g(\sin\theta+\mu\cos\theta)\)
    Block slides down rough plane in \(n\) times of climbing-up time
    \(\mu=\frac{n^2-1}{n^2+1}\tan\theta\)
    Smooth distance \(x\), rough distance \(x/n\), equal time
    \(\mu=\left(1-\frac{1}{n}\right)\tan\theta\)
    Rough plane velocity \(v/n\), smooth plane velocity \(v\)
    \(\mu=\left(1-\frac{1}{n^2}\right)\tan\theta\)
    Rough plane time \(n\) times smooth plane time
    \(\mu=\left(1-\frac{1}{n^2}\right)\tan\theta\)
    Lower \(1/n\) part rough, upper smooth; final velocity zero
    \(\mu=n\tan\theta\)
    Horizontal force to move block up an incline
    \(F=\frac{\mu\cos\theta+\sin\theta}{\cos\theta-\mu\sin\theta}Mg\)
    ❖ Special Example: If lower \(\frac{1}{3}\) is rough, upper \(\frac{2}{3}\) smooth and \(\theta=45^\circ\), then \(\mu=3\tan45^\circ=3\)
    ▢ Read and Digest:

    Table 1: Important Friction Points

    Fact
    Point
    Friction origin
    Molecular interaction; electrical origin
    Friction depends on
    Nature of material and surfaces
    Friction type
    Non-conservative force
    Coefficient order
    \(\mu_r<\mu_k<\mu_s\)
    Limiting friction
    Greater than kinetic and rolling friction
    Sliding ball moving east
    Friction acts west
    Limiting friction relation
    \(F\propto R\)
    If normal reaction triples
    Limiting friction becomes 3 times
    Work against static friction
    No work in pure static contact
    Work against kinetic friction
    Work is done
    Body begins to slide
    Applied force = limiting friction
    Angle between friction and instantaneous velocity
    \(\pi\)
    Walking on ground
    Friction by ground acts forward
    Smooth surface
    Friction and coefficient of sliding friction are zero
    Fast vehicles
    Streamlined to reduce friction
    Lubricant
    Allows surfaces to slide easily
    Pulling lawn roller
    Easier than pushing because pulling decreases normal reaction
    Walking on ice
    Take smaller steps to avoid slipping
    Block slides down plane with constant speed
    \(\theta=\) angle of repose and \(\mu=\tan\theta\)
    \(1/n\) part of chain overhangs
    \(\mu=\frac{1}{n-1}\)
    Work to pull hanging chain back
    \(W=\frac{mgl}{2n^2}\)
    Body in freely falling lift pulled horizontally
    Frictional force = 0
    ▢ Bicycle and Friction:

    Table 1: Friction in Bicycle

    Case
    Direction / Explanation
    Cycle with brakes on
    Difficult to move because sliding friction > rolling friction
    Pedalled bicycle: front wheel
    Friction acts backward
    Pedalled bicycle: rear wheel
    Friction acts forward
    Bicycle not pedalled
    Friction on both wheels acts backward
    ▢ High-Yield Recall:

    Table 1: Friction One-Liners

    Fact
    Answer
    Friction
    Opposing contact force
    Static friction
    Self-adjusting force
    Limiting friction
    Maximum static friction
    Kinetic friction
    Friction during actual motion
    Rolling vs sliding friction
    Rolling friction is smaller
    Order of coefficients
    \(\mu_r<\mu_k<\mu_s\)
    Limiting friction law
    \(F\propto R\)
    Coefficient of friction
    \(\mu=\frac{F}{R}\)
    Angle of friction
    \(\mu=\tan\theta\)
    Angle of repose
    \(\mu=\tan\alpha\)
    Angle of repose vs angle of friction
    \(\alpha=\theta\)
    Pulling normal reaction
    \(R=Mg-F\sin\alpha\)
    Pushing normal reaction
    \(R=Mg+F\sin\alpha\)
    Minimum pulling force
    \(F*{min}=Mg\sin\theta\)
    Minimum horizontal force
    \(F=Mg\tan\theta\)
    Vertical wall minimum force
    \(F=\frac{mg}{\mu}=mg\cot\theta\)
    Retardation due to friction
    \(a=\mu g\)
    Stopping distance
    \(s=\frac{v^2}{2\mu g}\)
    Stopping time
    \(t=\frac{v}{\mu g}\)
    Chain over table coefficient
    \(\mu=\frac{x}{L-x}\)
    Maximum chain overhang
    \(x=\frac{\mu L}{1+\mu}\)
    Down rough incline acceleration
    \(g(\sin\theta-\mu\cos\theta)\)
    Up rough incline acceleration
    \(g(\sin\theta+\mu\cos\theta)\)
    Walking friction direction
    Forward
    Pedalled bicycle rear wheel friction
    Forward
    Pedalled bicycle front wheel friction
    Backward
    Freely falling lift friction
    Zero
    Q1.
    A body is placed on an inclined plane. At a certain angle, the body just starts to slide down the plane. The particular angle is:
    📅CEE MBBS 2024a
    Q2.
    A horse of mass \(200\,\mathrm{kg}\) pulls a cart of mass \(1000\,\mathrm{kg}\). If the resistive force on the combined system is \(\mu (M+m)g\), where \(\mu=0.2\), find the driving force needed for acceleration \(2\,\mathrm{m/s^2}\). Take \(g=10\,\mathrm{m/s^2}\).
    📅BP 2011
    Q3.
    Blocks of masses \(1\,\mathrm{kg}\) and \(2\,\mathrm{kg}\) are joined by an initially unstretched spring on a horizontal surface with \(\mu_s=0.4\). What minimum horizontal force is needed to initiate motion of both blocks quasi-statically? Take \(g=10\,\mathrm{m/s^2}\).
    📅BP 2009
    Q4.
    A block of mass \(m\) starts from rest and accelerates at \(a\) on a rough horizontal surface with kinetic friction coefficient \(\mu\). Find the instantaneous power of the horizontal pulling force after time \(t\).
    📅BP 2009
    Q5.
    A uniform rectangular block has base width \(b\) and height \(h\). A horizontal force is applied at its top. What condition allows toppling before sliding if the static friction coefficient is \(\mu_s\)?
    📅BP 2014
    Q6.
    A body starts from rest and slides down a \(45^\circ\) incline. It takes twice as long to cover a given distance as it would without friction. Find the coefficient of kinetic friction.
    📅MOE 2012
    Q7.
    A \(30\,\mathrm{kg}\) box is pushed horizontally at constant speed on a floor with \(\mu_k=0.2\). Find the applied force. Take \(g=10\,\mathrm{m/s^2}\).
    Q8.
    A car is stopped by a constant frictional force over distance \(x\). If its initial kinetic energy is doubled, its stopping distance becomes:
    📅IE 2010
    Q9.
    A body of mass \(M\) and momentum \(p\) slides on a rough horizontal surface with kinetic friction coefficient \(\mu_k\). How far does it travel before stopping?
    📅IE 2013
    Q10.
    A \(2\,\mathrm{kg}\) block just begins to move when pulled at \(30^\circ\) above the horizontal by \(10\,\mathrm N\). Find the coefficient of limiting friction. Take \(g=10\,\mathrm{m/s^2}\).
    Q11.
    A \(2\,\mathrm{kg}\) block accelerates at \(2\,\mathrm{m/s^2}\) when pulled horizontally with \(10\,\mathrm N\). If kinetic friction remains unchanged, find its acceleration under a \(20\,\mathrm N\) pull.
    Q12.
    A \(2\,\mathrm{kg}\) block moves at constant velocity when pulled horizontally by \(8\,\mathrm N\). Find the coefficient of kinetic friction. Take \(g=10\,\mathrm{m/s^2}\).
    Q13.
    A block slides down a smooth \(30^\circ\) inclined plane. What is the coefficient of friction between the block and the plane?
    Q14.
    A \(10\,\mathrm{kg}\) block rests on a \(40\,\mathrm{kg}\) slab on a frictionless floor. The coefficients of friction between them are \(\mu_s=0.6\) and \(\mu_k=0.4\). A horizontal force of \(100\,\mathrm N\) pulls the top block. Find the acceleration of the slab, taking \(g=10\,\mathrm{m/s^2}\).
    Q15.
    A \(4\,\mathrm{kg}\) block rests on a horizontal surface with \(\mu_s=0.4\). What static friction acts when a horizontal force of \(7\,\mathrm N\) is applied?
    Q16.
    A \(2\,\mathrm{kg}\) block rests on a \(30^\circ\) rough incline with \(\mu_s=0.7\). Find the actual static friction acting on it, taking \(g=9.8\,\mathrm{m/s^2}\).
    Q17.
    A \(0.1\,\mathrm{kg}\) block is held against a vertical wall by a horizontal \(5\,\mathrm N\) force. If \(\mu_s=0.5\), find the actual frictional force, taking \(g=9.8\,\mathrm{m/s^2}\).
    Q18.
    A block takes twice as long to descend a fixed distance down a rough \(45^\circ\) incline as it would on a frictionless incline. Find the coefficient of kinetic friction.
    📅MOE 2012
    Q19.
    A uniform chain lies on a horizontal table with coefficient of static friction \(\mu_s=0.25\). What maximum percentage of its length can hang over the edge without sliding?
    Q20.
    A truck starts from rest with acceleration \(2\,\mathrm{m/s^2}\). A box is initially \(4\,\mathrm m\) from its open rear edge, and \(\mu=0.15\) between the box and truck. How far does the truck travel before the box falls? Take \(g=10\,\mathrm{m/s^2}\).
    Q21.
    A block initially moving at \(10\,\mathrm{m/s}\) stops after sliding \(50\,\mathrm m\) on a horizontal surface. Find the coefficient of kinetic friction if \(g=10\,\mathrm{m/s^2}\).
    Q22.
    A \(1\,\mathrm{kg}\) block rests on a horizontal table with \(\mu_s=0.5\). What force applied at \(60^\circ\) above the horizontal just starts motion? Take \(g=10\,\mathrm{m/s^2}\).
    Q23.
    A block of mass \(m\) experiences a horizontal force \(P\) and another force \(Q\) directed downward at angle \(\theta\) to the vertical, with its horizontal component acting in the same direction as \(P\). What minimum coefficient of static friction can prevent motion?
    Q24.
    The upper half of a plane inclined at \(\theta\) to the horizontal is smooth, and the lower half is rough. A block released from rest at the top stops exactly at the bottom. Find the coefficient of kinetic friction on the lower half.
    Q25.
    A block of mass \(m\) is pulled by a force of magnitude \(mg\), making angle \(\theta\) with the upward vertical. What condition on \(\mu_s\) and \(\theta\) allows motion?
    Q26.
    A coin lies on a horizontal rotating gramophone record at radius \(r\). The angular speed is \(\omega\) and the coefficient of static friction is \(\mu\). What condition prevents slipping?
    Q27.
    The maximum available traction coefficient for a car starting from rest increases linearly with distance \(x\), so \(\mu=kx\). If the engine can supply the maximum traction, how does its kinetic energy \(E\) vary with \(x\)?
    Q28.
    A \(4\,\mathrm{kg}\) block A rests on an \(8\,\mathrm{kg}\) block B on a smooth table. When \(12\,\mathrm N\) is applied to A, they are just about to slip. What maximum force may instead be applied to B without slipping?
    Q29.
    A \(0.25\,\mathrm{kg}\) body falls vertically with downward acceleration \(9.2\,\mathrm{m/s^2}\). Find the upward air-resistance force, taking \(g=9.8\,\mathrm{m/s^2}\).
    📅IIT-JEE 2015