6Friction

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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 cart of mass 1000 kg is pulled by the\nhorse of 200 kg. The coefficient of friction\nbetween them and ground is 0.2 Calculate\nthe force required\nthe\nacceleration of 2 m/s in the cart [BP 201 1]
    📅BP 2011
    Q2.
    Two blocks of masses m₁ = 1 kg and m₂ = 2 kg are connected by a non - deformed light spring. They are lying on a rough horizontal surface. The coefficient of friction between the block and the surface\n\nis 0.4. What min. const force F has to be\napplied in horizontal direction to the block\nof mass m₁ in order to shift the other\nblock?\n[BP 2009]
    📅BP 2009
    Q3.
    A block of mass 'm' is moving with const.\nacceleration on a rough horizontal plane.\nIf the coefficient of friction between the\n\nblock and ground is , the power delivered\nby the external agent after a time t from\nthe beginning is equal to\n[BP 2009]
    📅BP 2009
    Q4.
    A rectangular block is moving on a\nhorizontal surface (side 'a' height 'h'). It\nwill topple down when (u = coefficient of\nfriction of surface)\n[BP 2014]
    📅BP 2014
    Q5.
    .\nStarting from rest , a body slides down a\n45 inclined plane in twice the time it takes\nto slide down the same distance in the\n\nabsence of friction. The coefficient of\nfriction between the body and the inclined\nplane is:\n[MOE 2012]
    📅MOE 2012
    Q6.
    A box weighting 30 kg is pushed along\nfloor at a constant speed by applying\nhorizontal force. If the coefficient\nfriction is 0.2, then force applied is\nIMOE 20
    Q7.
    A car of mass 'm' moving with speed 'v'\nstopped at a distance 'x' by the friction\n\nbetween the tyres and the road. If K.E\nthe car is doubled, stopping distance w\nbe\n[LE 2010
    Q8.
    A body of mass "M" is moving on a rou\nhorizontal surface with kinetic friction\n"Hx" and momentum "p". Find out th\ndistance covered by body before coining\nrest\n[L.E 2013)
    Q9.
    A block 2kg on a horizontal surface begins\nto move when it is pulled at 30 with\nhorizontal by 10N force. Then coefficient\nof limiting friction for the block and\nsurface is
    Q10.
    A 2 kg block moves at constant\nacceleration of 2ms when it is pulled\nhorizontally by 10N. If it is pulled by 20\nforce on the same surface then acceleration\nwill be:
    Q11.
    A block of 2kg slides at constant velocity of\n20m/s on a horizontal surface if it is pulled\nhorizontally by 8N. Then coefficient of\nsliding friction will be
    Q12.
    A block is sliding down a 30 smooth\ninclined plane. Then coefficient of s\nfriction will be
    Q13.
    A 40 kg slab rests on a frictionless floor. A\nblock rests on top of the slab. The\nstatic coefficient of friction between the\nslab is 0.6 while the kinetic\n\ncoefficient is 0.4. The 10kg block is acted\nupon by a horizontal force of 100N, what\nwill be the resulting acceleration of the\nslab?
    Q14.
    A block of mass 4 kg is placed on a\nhorizontal surface. The coefficient of static\nfriction is 0.4. If a force of 7N is applied on\nthe block, then frictional force is
    Q15.
    A body of mass 2kg rests on a rough\ninclined plane making an angle of 30 with\nthe horizontal. The coefficient of static\nfriction between the block and the plane is\n0.7. The frictional force on the block is
    Q16.
    A block of mass 0. 1kg is held against a wall\nby applying a horizontal force of 5N on the\n\nblock. If the coefficient of friction between\nthe block and the mass is 0.5, the\nmagnitude of the frictional force acting on\nthe block is
    Q17.
    Starting from rest a body slides on an 45\ninclined plane through certain distance in\ntwice the time it takes to slide down the\nsame distance in the absence of friction.\nThe coefficient of friction between the\nbody and the inclined plane is [MOE 2012]
    📅MOE 2012
    Q18.
    A heavy uniform chain lies on a horizontal\n\ntable top. If the coefficient of friction\nbetween the chain and the table surface is\n0.25, then the maximum fraction of the\nlength of the chain, that can hung over the\none edge of the table is
    Q19.
    The rear side of a truck is open and a box\nof mass 20kg is placed on the truck 4n\naway from the open end. The coefficient of\nfriction between the truck and box is 0.15\n\nand g = 10m/s'. The truck starts from rest\nwith an acceleration of 2m/s' on a straight\nroad. The box will fall off the truck when it\nis at a distance from the starting point\nequal to
    Q20.
    A block moving initially with velocity of\n10m/s on a rough horizontal surface,\n\ncomes to rest a\n50m. If g = 10m/s, the coefficient of\ndynamic friction between the block and\nthe surface is
    Q21.
    A block of mass 1 kg is at rest on a horizontal table. The coefficient of static friction between the block and the table is 0.5 (g = 10 m/s²). The magnitude of the force acting upward at an angle 60° with the horizontal that will just start the block moving is:
    Q22.
    A block of mass m, lying on a rough horizontal plane is acted upon by a horizontal force P and another force Q, inclined at an angle θ to vertical. The block will remain in equilibrium if coefficient of friction between it and surface is:
    Q23.
    The lower half of an inclined plane of inclination θ with horizontal is rough and its upper half is frictionless. If a block released at the top of the plane comes to rest at the bottom again, then coefficient of sliding friction between the block and rough part of the plane will be:
    Q24.
    A block of mass m resting on a horizontal surface is pulled at an angle θ with vertical by a force mg. The coefficient of friction for the block and the surface is μ. The block can move if:
    Q25.
    A gramophone is revolving at angular speed ω with a coin placed on its surface at a distance r from the centre of record and coefficient of friction is μ. The coin will revolve with record without sliding if:
    Q26.
    A car starts from rest and moves on a surface in which coefficient of friction between the road and tyres increases linearly with distance (x). The car moves with maximum possible acceleration. The K.E. of the car (E) will depend on x as:
    Q27.
    A 4 kg block A is placed on the top of a block B of mass 8 kg which rests on a smooth table. A just slips on B when a force of 12 N is applied on A. Then the maximum horizontal force required to make both A and B move together is:
    Q28.
    Two masses A and B of 10 kg and 15 kg respectively are connected with a spring passing over a frictionless pulley fixed at the corner of a table as shown in figure. The coefficient of friction of A with the table is 0.2. The minimum mass of C that may be placed on A to prevent it from moving is equal to:
    Q29.
    The frictional force exerted by air on a body of mass 0.25 kg moving with acceleration 9.2 m/s² is:
    📅IIT-JEE 2015