FRICTION. k 9) For a body moving up a rough inclined plane under the action of a force F,

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1 FRICTION POINTS TO REMEMBER ) The force that alway oppoe the relative motion between two urface in contact and parallel to the urface, oppoite to the direction of motion i called frictional force. ) The frictional force i of three type. They are. Static friction.dynamic friction 3.Rolling friction Law of friction: i) The frictional force i independent of the area of contact. ii) The frictional force i directly proportional to the normal reaction. 3) Normal reaction i the reultant contact force acting on a body placed on a rigid urface perpendicular to the plane of contact. It i equal to mg on a horizontal urface and mg coθ on an inclined plane Coefficient of tatic friction tanφ 4)Angle of repoe i defined a the angle of inclination of a plane with repect to horizontal for which the body will be in limiting equilibrium on the inclined plane.ifα i the angle of repoe tanα When a body lide down an inclined plane of angle of inclination (θ )greater than the angle of repone (α ) i.e., θ > α,the acceleration of the body a g(inθ co θ ), f 5) The coefficient of friction i, tanθ where θ i the angle of repoe. N P mg 6) Acceleration of a body on a rough horizontal urface, a m 7) a ginθ for a mooth inclined plane. 8) a g( inθ coθ ) for a body liding down rough inclined plane. 9) For a body moving up a rough inclined plane under the action of a force F, F mg(inθ + co θ ) a m 0) When a body i to be moved up a rough inclined plane with uniform velocity the force to be applied Fmg(in θ + co θ ) W inφ W inφ ) Pulling force F and puhing force F where W i the weight of co( θ φ) co( θ + φ) body, φ i the angle of friction and θ i the angle made by F with the horizontal LONG ANSWER QUESTIONS: **. Define coefficient of tatic and dynamic friction. Mention the law of tatic and dynamic friction. Derive an expreion for the acceleration of the body liding on a horizontal plane having coefficient of friction. A. The ratio of limiting friction F l to normal reaction N i called Coefficient of tatic friction i.e. F α N f N The ratio of dynamic frictional froce to the normal reaction. Coefficient of dynamic fα N, f N friction f N

2 Law of tatic friction. The Limiting friction depend on nature of urface in contact and on their urface roughne. It i independent of area of contact. The Limiting friction i tangential to the urface in contact and oppoite to the motion of the body. 3. Magnitude of limiting friction i directly proportional to normal reaction, f α N or f N Law of dynamic or inetic friction :. The inetic friction depend on nature of urface and independent of area of contact for maller relative peed.. The inetic friction i tangential and oppoite to motion of the bodie. 3. Dynamic friction i directly proportional to normal reaction F N F N Motion of a body on rough horizontal urface: Conider a horizontal force F i applied on a bloc of ma m, which i ept on a rough horizontal urface. The coefficient of inetic friction between the bloc and the urface i. The inetic friction acting on the body i oppoite to applied force and tangential at contact urface. From law of friction: Kinetic frictional force f N and N mg Net force acting on the body F F f ma net F mg a m. Decribe with an experiment how tatic friction i explained. Ditinguih tatic friction and dynamic friction. A. Conider a body be placed on a rough horizontal urface of a table. A hown the body i connected to a weight pan by a thread through mooth pulley. When weight in the pan i mall, the bloc doe not lide, ince the tenion in the thread i balanced by frictional

3 force on bloc, thi frictional force i called tatic friction. Static friction i elf adjuting; a the weight on the pan i gradually increaed, tatic friction alo increae to a maximum limit. After the limit, the body lide with an acceleration. The maximum value of tatic friction i called limiting friction. If a graph i drawn with applied force (weight) on X-axi and frictional force on Y-axi, the graph i a hown. Starting from zero applied force, tatic friction increae up to the maximum value. After the limit, body tart liding uddenly reducing frictional force by mall amount (BC). After motion ha tarted, the inetic frictional force remain contant. Thi i repreented by line PQ. It i clear for diagram that, BE > PQ. Limiting friction > Kinetic friction I.e. f > f hence > 3. Derive an equation for the acceleration of a body liding down an inclined plane without friction. Find alo the time taen by the body to lide down the mooth inclined plane through a length l tarting from ret. A. Conider a body of ma m i placed on a mooth inclined plane of inclination θ i a hown in figure. The force acting on the body are (i) The weight of the body (mg) act vertically downward (ii) The normal reaction (N), act perpendicular to the inclined plane The weight of the body mg can be reolved in to two component. They are mg coθ,and mg inθ. The component mg coθ act perpendicular to the inclined plane and i balanced by normal reaction (Nmg coθ ) and mg inθ act parallel to the inclined plane and it i un balanced force. The net down ward force on the body along the plane i given by F mg inθ net ma mg in θ ( Q F ma) agin θ net Conider the body tart from ret from the top of the plane and acquire a velocity V after travelling through diplacement l along the plane, then From the equation of motion, v u a, a u 0, l and a ginθ ( θ ) v gin l ( Q θ ) V glin θ lin h V gh Hence the velocity acquired by the body when it reache the foot of the plane i independent of the inclination of the plane and depend only on the vertical height (h) through which the body i falling.

4 If t time taen to travel the ditance l with initial velocity u 0, at the top of the plane, fromv u+ at, We get V lginθ l t t t g inθ g inθ g inθ 4. Define angle of friction and angle of repoe. Show that angle of friction i equal to angle of repoe for a rough inclined plane. A. The angle made by the reultant of the normal reaction and the limiting frictional force With normal reaction i called angle of friction( φ ). f f From the figure tanφ and. Hence tanφ. N N The angle of repoe( α ) i defined a the angle of inclination of a rough inclined plane with horizontal for which the body will be jut ready to lide down. To how that the angle of repoe i equal to the angle of friction : When the body tend to move down the rough inclined plane, the friction on the bloc i called limiting or tatic friction f.hence f mginα and N mgcoα mg inα Coefficient of tatic friction tanα mg coα. tanα Here α i the angle of repoe. Hence the angle of repoe of a rough inclined plane i equal to the angle of friction of horizontal plane 5. Obtain an expreion for the acceleration of a body down a rough inclined plane. Find the time taen by the body to lide down the inclined plane through a length l tarting from ret. (June 003, March 00, 003, and 007) A. Conider a body of ma m i placed on a rough inclined plane of inclination θ a hown in figure.

5 The force acting on the body are (i) The weight of the body (mg) act vertically downward (ii) The normal reaction (N), act perpendicular to the inclined plane (iii) The inetic friction( f ) acting up the inclined plane The weight of the body mg can be reolved into two component.they are mg coθ and mg in θ The component mg coθ act perpendicular to the inclined plane and i balanced by normal reaction (Nmg coθ ) and the component mginθ act parallel to the inclined plane and it i unbalanced force. The net downward force acting on the body i F mginθ f net F mgin θ N ( Q f N) net F mgin θ mg coθ ( Q Nmg co θ ) net ma mg(inθ co θ) ( Q F ma) a g(inθ co θ ) net Conider body tart from ret from the top of the plane and acquire a velocity V after travelling through diplacement l along the plane, then From the equation of motion, v u a, ( θ θ ) v o g in co ( θ θ) ( Q ) V gl in co l The time taen by the body to lide down i given by Here, S l, u 0 l 0 t + g in co t t () ( θ θ ) l g θ θ ( in co ) S ut+ at Friction: It i the force which alway oppoe the relative motion between two urface in contact. It act parallel to the urface and oppoite to the direction of motion. Law of friction: i) The frictional force i independent of the area of contact. ii) The frictional force i directly proportional to the normal reaction. Static friction : Coefficient of tatic friction i defined a the ratio of limiting friction to normal reaction N. ie.., Fl N f N

6 Dynamic friction: Coefficient of dynamic friction i defined a ratio of dynamic frictional force f to the normal reaction fα N.. N Rolling friction: The rolling friction i directly proportional to the normal reaction. fr Coefficient of rolling friction, r N Angle of repoe: It i defined a the angle of inclination of a plane with the horizontal for which the body will be in limiting equilibrium on the inclined plane. If α i the angle of repoe tanα Angle of friction: The angle made by the reultant of the normal reaction and the limiting friction with normal reaction i called angle of friction. SHORT ANSWERS QUESTIONS. Explain the term dynamic limiting friction and rolling friction. (June 00, 003) A. Limiting friction: The maximum frictional force developed between the bodie at ret i called limiting friction. Dynamic or inetic friction: When applied force i equal to or greater than limiting friction then the body will move. Frictional force between moving bodie i called dynamic or inetic friction. Kinetic friction i alway le than limiting friction. Rolling friction : When one body roll on the urface of the other body, (lie a ball, a drum or a cylinder) then the exited friction in between that body and the urface of the other body i called rolling friction.. Write a note on the caue of friction. A ) Frictional force i caued due to the roughne of two urface in contact and interlocing of the irregularitie between the urface. ) Nature of urface. 3) Impuritie between urface. 4) Shape of urface. 3. Explain advantage and diadvantage of friction. (March 005, 006) A. Advantage of friction: ) Safe waling on the floor i poible due to friction. ) Nail, crew etc. are driven into wall; wooden urface etc. due to friction. 3) Writing with pen, pencil, holding object with hand etc i poible due to friction. 4) A match tic i lightened due to friction. Diadvantage of friction: ) Due to friction wear and tear of engine occur. ) Due to friction ome energy get converted into heat which goe a wate 3) The large amount of power lo in engine due to friction. 4. Mention the method ued to decreae friction? (July 00 March 00, 003, 009) A.. By polihing the urface in contact, friction can be reduced. Wheel of vehicle are provided with ball bearing to reduce the friction here ball bearing convert liding friction to rolling friction. 3. The lubricant form a thin layer between urface in contact. It reduce the friction. In light vehicle or machine, oil lie three in one are ued a lubricant. In heavy machine greae i ued. In addition to thi they guard the mechanical part from over heating. 4. Automobile and aero plane are treamlined to reduce the friction due to air. 5. Sate the law of rolling friction? A. ) The maller the area of contact, a leer will be the rolling friction. ) The larger the radiu of the rolling body, the leer will be the rolling friction. 3) The rolling friction i directly proportional to the normal reaction.

7 Fr i.e., Frα N or r N 6. Why pulling the lawn roller i preferred than puhing the lawn roller? (March 006) A. Pulling of lawn roller :In the cae of Pulling a lawn roller the vertical component of the applied force, F inθ i acting in the upward direction and reduce the normal reaction. The normal reaction become N mg Finθ The frictional force while pulling fr RN () f mg Finθ R R ( ) Hence le frictional force develop and hence le force i required to pull the lawn roller. In the cae of Puhing the lawn roller, the vertical component of the applied force, F inθ, i acting in the direction of weight, and increae the normal reaction. Then normal reaction become N mg+ Finθ The frictional force while puhing fr RN () f mg+ Finθ R A fr R ( ) < f, le force i required to pull the lawn roller than puhing it. R Hence Pulling i eaier than puhing 7. Show that velocity at the bottom of the mooth inclined plane i independent of lope of the inclined. A. Conider a mooth inclined mooth plane of inclination ' θ ', height h and length L. A bloc i releaed from the top of the inclined plane Weight of the bloc i reolved into component. They are ) Component along the incline mg inθ, and ) Component normal to the incline mg coθ, Only mg inθ create acceleration of the bloc along the incline. From the equation of motion v u a Acceleration of the bloc along the inclined plane mg inθ a ginθ m Ditance covered by the bloc on reaching the bottom L From v u a

8 v ginθ L (Since body releaed at the top, u 0) glinθ But Linθ h v gh Or v gh Velocity of the bloc at the bottom of the inclined plane depend only on height at which the bloc i releaed but not on inclination. 8. Find an expreion for the time taen by a body projected up along a rough inclined plane of length l with an initial velocity u. A. Body liding up the plane If one body i projected up on the rough urface with initial velocity u, Kinetic frictional force act downward and it uffer retardation the net force acting on body i F mginθ + f ( ) Retardation of the body F mg inθ + f mg inθ + mg coθ a m m m a g inθ + coθ ( ) When body reache the top of the plane it final velocity become 0 V u+ at u 0 u g( inθ + coθ) t t g(inθ + co θ ) In thi cae, time of accent i not equal to time of decent. 9. Define angle of friction. Obtain an expreion for the coefficient of friction in term of angle of friction? A. Angle of friction( φ ): The angle made by the reultant R of normal reaction (N) and the tatic frictional force f with normal to the urface i called angle of friction. ( ) Reolve the reultant force R into two perpendicular direction. R coφ normal to the urface and Rinφ along the urface, then the coefficient of tatic friction i defined a f Rinφ tanφ N Rcoφ then tanφ i called a the coefficient of tatic friction. 0. Find the time taen by a body projected up a mooth inclined plane with an initial velocity u to travel through a ditance l. A. Conider a body i projected up along the mooth incline with an initial velocity u.

9 Initial velocity u; Acceleration a ginθ (ince body i moving up) Final velocity at heighet point v 0 ) Ditance travelled up the plane S? From v u al u gin θ. l u or ditance travelled upwardl g inθ ) Time taen to travel upward t? Initial velocity u Final velocity v 0 Acceleration a ginθ from v u+ at 0 u gin θ. t or ginθt u Time taen to travel upward t g u in θ. The upper half of an inclined plane i mooth and the lower half rough. A bloc allowed to lide down from the top i brought to ret by friction jut a it reache the bottom. Find the ratio of the force of friction to the weight of the particle, auming friction to be independent of velocity. A. Wor done againt friction (W) P.E. at A mg ( AB) mgh l f. h P.E. at A mglinθ [ Q In ACB inθ ] l W change in P.E. l f. mglinθ f inθ mg f mginθ f mginθ h inθ mg mg l. Explain Why i) Road hould not be made very mooth ii) It i not eay to write on unglazed or over a glazed paper. iii) Tyre of motor car and cycle hould have a rough urface iv) Brae hoe are made purpoely rough

10 A. i) If road are made very mooth the frictional force increae due to force of attraction between the molecule. ii) Due to more friction it i difficult to write on unglazed and due to no friction it i difficult to write on a glazed urface. iii) To have friction and to have grip on the road, the tyre of motor car and cycle hould have a rough urface. iv) To have greater grip between the two urface in contact and to top moving vehicle quicly brae hoe are made purpoely rough. VERY SHORT ANSWER QUESTIONS. Mention the factor on which frictional force depend. A. Frictional force depend on ) Nature of urface in contact ) The force with which the urface are preed 3) Nature of motion between the bodie. Can the coefficient of friction be greater than one? A. Ye, In general, coefficient of friction i le than one. When the urface i heavily polihed the adheive force between the molecule increae and then coefficient of friction will be greater than one. 3. What happen to the frictional force if the urface i moderately polihed and heavily polihed? (June 00, 003) A. When urface i moderately polihed frictional force reduce. When the urface i heavily polihed the adheive force between the molecule increae and hence frictional force increae. 4. Why doe the car with flattened tyre top ooner than the one with inflated tyre? A. Flattened tyre deform more than that of the inflated tyre. Due to more deformation of the tyre the Area of contact between the urface increae Hence Rolling friction increae that are why the car with flattened tyre top ooner. 5. A hore ha to pull harder during the tart of the motion than later. Explain? (March 009) A. For tarting motion of the cart, the limiting friction i to be over come. Once Motion i et, the frictional force reduce therefore, the hore ha to pull harder during tarting of the cart. 6. Why the wheel of vehicle are fitted with ball bearing. A. Wheel of vehicle are provided with ball bearing to reduce the friction here ball bearing convert liding friction to rolling friction. 7. Why the coefficient of friction cannot be zero? A. Friction between the urface can be reduced by polihing. If the urface are heavily polihed due to the inter atomic force the friction will increae. Even lubricant can reduce the friction but not to zero. So, friction between the urface cannot be reduced to zero. 8. What i Normal reaction of the body when it i at ret on ) a Horizontal urface ) an inclined urface A. ) on horizontal urface Normal Reaction N mg

11 ) on an inclined urface normal reaction N mg coθ 9. What happen to the coefficient of friction? If the weight of the body i doubled? A. Coefficient of friction i Independent of ma of the body. Hence it remain ame 0. Give example of liding and rolling friction. A. Example of liding friction: ) When brae are applied uddenly during motion of a motor ) A bloc i pulled on rough urface. Example of Rolling Friction: ) Rotation of wheel of vehicle on road. ) Rotation of metallic ball in a bearing.. Why aeroplane and automobile are treamlined? A. Automobile and aeroplane are treamlined to reduce the friction due to air.. Define coefficient of tatic friction. State it unit and dimenion. A. The ratio of limiting friction to the Normal Reaction i called coefficient of tatic friction. f i. e It ha no unit and dimenion N 3. Define coefficient of dynamic friction. State it unit and dimention. A. The ratio of inetic friction to the Normal reaction i called coefficient of dynamic friction. fk i.e. K. It ha no unit and dimenion. N 4. What i the direction of dynamic friction between two bodie? A. The dynamic friction i tangential between the two bodie and oppoite to their motion. 5. What i the direction of tatic friction between two bodie? A. Static frictional force i tangential between the urface in contact oppoing them to move relative to each other. 6 What i the force required to move a body up a rough inclined plane with uniform velocity? A. When a body i to be moved up a rough inclined plane with uniform velocity the force to be applied i [ in co ] F mg θ + θ ASSES YOUR SELF QUESTIONS. When a paper boat i ept on a flowing water urface will there be friction between the paper boat and the water urface? An: Ye, if there were no friction, it cannot have been carried away by the water flow.. When a running athlete want to top immediately which force doe he ee An: Kinetic friction 3. I friction a reaction of the applied force? An: No 4. Doe friction alo occur a action-reaction pair?

12 An: Ye 5. In general, if tatic friction i the caue of rolling friction. How i it le than tatic friction? An: Rolling tart before the tatic friction reache it maximum value. 6. Gue the effect of temperature on coefficient of friction? An: The coefficient of friction may decreae with increae in temperature. 7. What i the effect of lubrication on coefficient of friction? An: Coefficient of friction decreae with lubrication. 8. What i the influence of normal reaction on the area of contact? An: Area of contact increae with normal reaction. 9. How do you mae a body lide down a mooth inclined plane with contant velocity? An: By applying an external force F mg inθ up the plane, while the body i in motion downward. 0. How can you move the body up the mooth plane without acceleration? An: By Appling an external force F mg inθ up the plane, after giving an initial velocity up the plane. I it poible to lide a body down a rough plane without retardation? An: Ye, at By applying a uitable force up the plane.. How can you move a cylinder, uch that it lide up a rough inclined plane without rolling? An: By Appling a pulling force F mg(inθ + co θ ) uch that the line of action of the force i parallel to the plane and pae through the centre of ma of the cylinder. 3. If a table i to be taen from one corner of a room to another corner which i betterpulling or puhing? An: Pulling 4. If the angle of friction iφ, what will be the minimum pulling force required? An: mg inφ where m i the ma of the body pulled on the rough horizontal urface 5. To pull a body from one place to another place, at what angle the body i to be pulled to apply leat pulling force? An: The angle made by the pulling force with the horizontal mut be equal to the angle of friction. Exercie. A box of ma 0g i pulled on a horizontal rough urface by applying a horizontal force. If the coefficient of inetic friction between the box and the horizontal urface i 0.5,find the force of friction exerted by the horizontal urface on the box. Sol: m0 g, 0.5, g 9.8m f mg N

13 . A force of 98N i required to pull a body of ma 00g over the urface of ice. Calculate the coefficient of friction between the body and the urface of ice. Sol: m 00 g, f 98 N, g 9.8m f f N mg A bloc of ma 0g i dragged along a rough horizontal urface through m with uniform peed. Calculate the wor done if the coefficient of friction i 0.5 Sol: wor done againt friction fs mgs Here 0.5, m 0 g, g 9.8 m, S m W J 4. When a car of ma 000g i moving with a velocity of 0 m on a rough horizontal road, it engine i witched off. How far doe the car move before it come to ret if the coefficient of inetic friction between the road and tyre of the car i 0.75?(g0m -- ). Sol: When the engine of a car of ma m moving with a velocity v i witched off it come to ret due to the force of friction, mg where the coefficient of inetic friction i neglecting air reitance. force of friction mg Re tardation of the car g ma of the car m -a ga ; g,int ialvelocity uv; final velocity,v0 Subtituting in v 0-v ( ) u a gs v S g Here v 0 m, 0.75, g 0m (0) 400 S 6.67m A g bloc i placed over a 5g bloc and both are placed on a mooth horizontal urface.the coefficient of friction between the bloc i 0.0.Find the acceleration of the two bloc if a horizontal force of 4N i applied to the upper bloc (g0m -- ) Sol: The force on two g bloc are (a) gravitational force F mg, g00n,vertically downward, (b) Normal reaction N by the 5g bloc, vertically upward, (c) force of friction f N to the left and (d) applied force 4N. N 0N In the vertical direction, there i no acceleration.

14 In the horizontal direction, the acceleration of the g bloc i a. 4 - N a ( ) ( Q ) a Q Reultant force ma 4 - a; a 6 m 0.0 The force on 5g bloc are (a) gravitational force 5 g N, Vertically downward, (b) Normal reaction N of the horizontal urface, vertically upward, (c) force of friction f N to the right by Newton third law of motion and (d) normal reaction, N downward by g bloc. In the vertical direction, there i no acceleration. N 50 N. 6. Find the acceleration a, a, and a 3 of the three bloc hown in figure if a horizontal force of 6 N i applied on g bloc. (g 0 m ) Sol: When the force of 6 N i applied on the g bloc, the force acting on it are (a) Gravitational normal g 0 N, (b) Reaction N by the g bloc, vertically upward, (c) force of friction f N to the left and (d) applied force 6 N. In the vertical direction, there i no acceleration. N 0 N. In the horizontal direction, the acceleration of the g bloc i a. 6 - N a a 6 - a; a 4 m. Conider the motion of g bloc. The force on it are (a) gravitational force g 0 0 N, vertically downward, (b) normal reaction N by the 3 g bloc, (c) force of friction f N to the right by Newton third law of motion and (d) normal reaction N downward by g bloc. In the vertical direction, there i no acceleration.

15 N + 0 N 0+ 0 N ( Q N 0 N) N 30 N. In the horizontal direction, force of the right i N0. 0 N. Limiting friction due to 3 g bloc i g bloc move with repect to 3 g bloc i N' N. A the force to the right N i le than the limiting force of friction due to 3. g, the g bloc doe not move eparately from 3 g bloc. Both g and 3 g bloc move together. The ytem off g and 3 g bloc move on the mooth horizontal force under the action of the force N to the right. N 0. 0 Acceleration of g and 3 g bloc, a' 0.4 m A bloc of ma 0 g puhed by a force F on a horizontal rough plane move with an - - acceleration of 5 m.when force i doubled, it acceleration become8 m. Find the coefficient of friction between the bloc and rough horizontal plane. (g 0 m.) Sol: On a rough horizontal plane, acceleration of a bloc of ma m i given by Eq. P a g.... (i) m Initially, P F; a 8 m F 5 (0).... (ii) ( Q m 0 g) 0 When force i doubled i.e., P F; a 8 m. Subtituting in Eq. (i) F 8 (0).... (iii) 0 Subtituting Eq. (ii) from Eq. (iii) F 3 ; F 30 N. 0 Subtituting in Eq. (ii) F 30 5 (0) 5 (0) A pulling force maing an angleθ to the horizontal i applied on a bloc of weight W i placed on a horizontal table. If the angle of friction iφ, find the magnitude of the force required to move the bloc. Sol: A bloc of ma m i pulled by a force F maing an angleθ to the horizontal. Reolving the force into horizontal and vertical component, the horizontal component i F coθ and vertical component i F inθ. From the free body diagram hown in the figure, in vertical direction.

16 N + F in mg...(i) ( Q W mg) N W - F in and in horizontal direction f F co θ....(ii) f But N F co θ...(iii) W - F in θ The angle of friction iφ tan...(iv) From Eq. (iii) and (iv) tanφ F co θ W - F in θ W tan φ - F inθ tanφ F coθ W tanφ - F coθ + F inθ tanφ W inφ F coθ coφ + F inθ inφ F co (θ -φ ) W inφ F co θ φ ( ) Note: For puhing force it can be hown that F W co inφ ( θ + φ ) 9. A uniform chain of length L hang partly from a rough horizontal table. The chain i ept in equilibrium by friction. If the maximum length that can hang without lipping i l find the coefficient of friction between the table and the chain. Sol: Conider M be the ma of the chain of length L. The ma of the hanging part i M. L l The weight of the haning part i M l g. The weight of the remaining part on the table i L Ml M- g M l g. The limiting force of friction on the chain then become L L N M l g. When thi i balanced by the weight of the hanging part, the L reultant force become zero and the chain i in equilibrium.

17 l l L L L-l l L L l. L-l 0. Two object of mae m and m are connected by a very light tring paing over a clamped light mooth pulley. The object of ma m i one rough horizontal table and the object of ma i hanging vertically. If the coefficient friction between the table and the object of ma m i find the acceleration of the ytem. Sol: Conider the acceleration of the ytem of two object be a. Conidering the force m we can write the equation for it acceleration a a m a m g- T.... (i) Conidering the force on m, m a T - N m a T - m g... (ii) ( Q N mg) Adding Eq. (i) and (ii), we get (m + m )a ( m m )g m m m + m a g.. A body of ma 00 g i liding down an inclined plane of inclination30 o. What i the acceleration of the body if the coefficient of friction between the body and the inclined plane i 0.5? Sol: m 00 g 0. g, θ 30 o, 0.5, g 9.8 m. Acceleration of the body i a g(inθ - coθ ) 9.8( in30 o co30 o ) 9.8( ) m.. A cube of weight 0 N ret on a rough inclined plane of lope 3 in 5. The coefficient of friction between the cube and the inclined plane i 0.6. Find the inclined plane i 0.6. Find the minimum force required to tart the cube moving up the plane? Sol: The minimum force required to tart the motion of a body up the plane i F mg (inθ + coθ ). 3 4 Here, mg 0 N; in θ, co θ, F N A body tae time a much time to lide down a rough inclined plane a it tae to lide down an identical but mooth inclined plane. If the angle of inclination of the

18 inclined plane i 45 o, what i the coefficient of friction between body and the rough plane? Sol: Conider l be the length of both the inclined plane and θ be the inclination. Then the time taen by a body liding the mooth and rough inclined plane are l l tmooth and t rough. g inθ g(in θ co θ) If t n t rough mooth l l n g(in θ co θ) g inθ inθ in θ coθ n inθ coθ inθ inθ n cotθ n cotθ n cotθ n, tanθ n o ( Q θ 45 ) n In thi problem, n 3. 4 *4. When a horizontal force of 00 N i applied on a body, the acceleration produced i.0 m. When the force i 300 N, acceleration produced in the body i.0 m. Find the ma of the body. Sol: The acceleration of the body i P a.... (i) m g Initially, P 00 N, a.0 m g m When force, P 300 N, a.0 m g. m Subtituting Eq. (ii) from Eq. (iii) 00.0 ; m 00 g. m 5. An inect i crawling inide a hemipherical bowl of radiu r. If the coefficient of friction between the bowl and inect i, find the maximum height to which the inect can crawl. Sol: When the inect crawl inide a hemipherical bowl and reache a point P, the force on the inect are (i) the gravitational force, mg, vertically downward, (ii) the normal reaction, N, normal to the urface of the bowl i.e., along the radiu toward the centre and (iii) force of friction, N along the tangent. Reolving mg into rectangular component a) parallel to the tangent, mg inθ and

19 b) perpendicular to the tangent, mg coθ. Along the tangent, there i not acceleration. N mg inθ Along the radiu alo there i no acceleration. N mg co θ Hence mg coθ mg inθ tanθ. r-h co θ and in the OPQ, co θ. + r r-h r + h h ; ; h r. r r A body lide down a mooth inclined plane of bae of length b. If the time taen by the body to reach the bottom of the plane i to be minimum, what mut be the angle of inclination? Sol: If l i the length of the inclined plane and θ i the angle of inclination of the inclined plane, the time taen by body tarting from ret to reach it bottom i l t. g inθ b b But co θ ; l. l coθ b 4b t. gin θ co θ g in θ If t i to be minimum, in θ mut be maximum i.e., inθ. θ 90 o. Henceθ 45 o. 7. A 30 g bloc i to be moved up an inclined plane at an angle30 o to the horizontal with a velocity of 5 m. If the frictional force retarding the motion i 50 N find the horizontal force required to move the bloc up the plane. (g 0 m.) Sol: The force required to body up an inclined plane i

20 F mg in θ + frictional force 30(0) in 30 o N. If P i the horizontal force, F P coθ N. F P coθ coθ 3

21 EXERCISE PROBLEMS. A wooden bloc of 00 g i about to be puhed on a floor of coefficient of friction 0.4.What i the magnitude of the force of friction on the wooden bloc when it i jut puhed? A. m 00g 0.4 ( ) ( )( )( ) f N mg N. A force of 47 N i required to jut lide a ledge weighing 500 N over a urface of ice. Calculate the coefficient of friction between the urface in contact of the ledge and the ice. A. wmg500n f f N mg When a car of ma 00 g i moving with a velocity of 5 m - on a rough horizontal road, it engine i witched off. How far doe the car travel before it come to ret if the coefficient of inetic friction between the road and tyre of the car i 0.5? (g0 m -.) A. m 00 g m00g u 5m v5m/ g 0m g0m/ a We now a g 5m (or) g Kinetic energy wor done by frictional force (Where a retardation) KEW v u 5.5m mv mg a 0 v 5 5 g m 4. A train of ma 600 metric ton i pulled by a locomotive of ma 50 metric ton. The peed of the train i 54 m hr -. The locomotive pull the train on a level trac. The force of friction on the locomotive and the train 0 N per metric ton. Calculate the power of the locomotive. 3 A. ma g peed V 54mph 5m 0 a Ng 0 m 000 force velocity ma V Power ( )( ) ( ) 3 ( )( 0 )( 5).5 KW (or) Ma of train 600metricton Ma of locomotive 50 metric ton Total ma moving metric ton

22 g Total frictional force N (0 N/metric ton) Power PF.V P KW 5. A car i driven round a curved path of radiu 8 m without the danger of idding. The coefficient of friction between the tyre of the car and the urface of the curved path i 0.. What i the maximum peed in m/ph of the car for afe driving? (g0m - ) A. radiu (r) 8m 0. for afe riding, maximum peed Q frictional force centripetal force mv mg r V m 6.6mph 4 5 V rg ( 0.)( 8)( 0) 6. A body i liding down a rough inclined plane which mae angle 30 0 with the horizontal. The coefficient of the friction between the urface of contact of the body and the plane i 0.5. Calculate the acceleration. A. 0 30, 0.5 θ Acceleration (a) g ( inθ coθ ) ( ) in co m 8 7. A body of ma 0 g i on a rough inclined plane having an inclination of 30 0 with the horizontal. If coefficient of friction between the urface of contact of the body and the plane i 0.5 find the leat force required to pull the body up. A. leat force required to pull the body up i [ in co ] F mg θ + θ ( ) + ( ) in co N 4 4 ( Q 3.73) 8. A bloc of wood of ma 0.5 g i placed on a plane maing angle 30 0 with the horizontal. If the coefficient of friction between the urface of contact of the body and the plane i 0. what force i required to eep the body liding down with uniform velocity? A. m 0.5 g 0. 0 θ 30

23 Force required for the body to lide down with uniform peed ( ) 0. 3 F mg[ inθ coθ] ( 0.5)( 9.8) ( ) N ( Q 3.73) 9. A body of ma m lide down a mooth inclined plane having an inclination of 45 0 with the horizontal. It tae to reach the bottom. If the body i placed on a imilar plane having coefficient friction 0.5 what i the time taen for it to reach the bottom? A. θ 0 45, t ec, t? 0.5 Time taen by the body to lide down on a rough inclined plane t l g θ θ ( in co )... () time taen by the body to lide down a mooth inclined plane l t... () g inθ 0 t From () and (), θ 45 t t t t t.88 in 45 in co A wooden bloc i placed on an inclined plane. The bloc jut begin to lide down when the angle of the inclination i increaed to What i the coefficient of the friction? A. mg co θ mg in θ (condition for the body jut to lide down) 0 θ 45 0 tanθ tan 45. What i the magnitude of wor done againt the friction on a body of ma M to travel through a ditance S, if i the coefficient of friction? A. Wor done (W) (force)(diplacement) f N Mg Mg ( ). Find the power of an engine which can draw a train of 400 metric ton up the inclined plane of in 98 at the rate of 0 m -. The reitance due to friction acting on the train i 0 N per tone. 3 A. Given ma g inθ 98 V peed 0m 0 Retardation 0 m 000

24 a inθ + coθ V mg inθ + coθ V g 3 0 ( )( 9.8) W 440 W. (or) 3 Given ma g inθ 98 V peed 0m Retardation force f N Power mg [ ] p? p FV. p ( mg in θ +f)v p p ( ) w 3 ( ).0 p 440w 3. If the angle of inclination of the inclined plane iin when the body jut tart 3 liding find the angle of repoe and coefficient of tatic friction between the body and the inclined plane 5 5 A. θ in inθ tanθ θ tan tan ' 0 ' ( 4 ) and angle of repoe ( ) 4. A body of ma 60 g i puhed up with jut enough force to tart it moving on a rough urface with 0.5 and 0.4 and the force continue to act afterward. What i the acceleration of the body? A. Force of tatic friction mg ( 0.5)( 60)( 9.8) N Force of inetic friction mg ( 0.4)( 60)( 9.8) N Additional force needed for the body jut to move F ( ) ma ( ) mg mg Acceleration of the body a ( ) ( )( ) 5. A body tae g m 3 time a much time to lide down a rough inclined plane a it tae to lide down an identical but mooth inclined plane. If the angle of inclination i 45 0, find the coefficient of friction.

25 A. Time to lide down on a rough inclined plane 4 [time to lide down on a mooth 3 inclined plane ] l 4 l g(inθ co θ) 3 ginθ [ ] 6 inθ coθ 9inθ 7inθ 6 coθ tan θ n (or) tan θ n 0 4 θ 45, n 3 3 tan A body i liding down an inclined plane have coefficient of friction 0.5. If the normal reaction i twice that of the reultant downward force along the incline, find the angle between the inclined plane and the horizontal. A. 0.5 Given that, normal reaction (reultant downward force) mg coθ mg inθ coθ coθ inθ 0.5 coθ ( ) ( ) inθ coθ tanθ 0 θ A body of ma 5 g ret on a rough horizontal urface of coefficient of friction 0.. The body i pulled through a ditance of 0 m by a horizontal force of 5N. Find the inetic energy acquired by the body. (g 0 m -.) 5 0 A. Reultant acceleration of the body a 3m 5 K. E. wor done ma 5( 3)( 0) 50J ( ) 8. A particle i projected up along a rough inclined plane of inclination 45 0 with the horizontal. If the coefficient of friction i 0.5, calculate the retardation. A. Given θ Retardation 0 0 (a) g[ inθ + co ] in 45 + ( 0.5) co 45 θ g

26 g + 3g g + 9. A body i moving up an inclined plane of angle θ with an initial inetic energy E. The coefficient of friction between the plane and the body i. What i the wor done againt friction before the body come to ret? Expre it in term of E. E A. Given, mu E u m v 0 a the body i coming to ret a g inθ + coθ ( ) ditance travel W f. u E E a mg in co mg in co E W ( mgcoθ) mg ( θ + θ) ( θ + θ ) E coθ ( inθ + coθ) ( inθ + coθ ) 0. A bloc of ma g ret on a rough inclined plane maing an angle of 30 0 with the horizontal. If 0.6, what i the frictional force on the bloc? 0 A. m g θ when bloc i at ret frictional force mg inθ ( ) 0 mg inθ 9.8 in N. A 40 g lab ret on a frictionle floor. A 0 g bloc ret on top of the lab. The coefficient inetic friction between the bloc and the lab i A horizontal force of 00 N i applied on the 0 g bloc. Find the reulting acceleration of the lab. (g 0 m -.) A. Given m 0 g M 40 g ; 0.4 ( 0.4)( 0)( 0) mg mg Ma a m M 40. A bloc of ma m lying on a rough horizontal plane i acted upon by a horizontal force P and another force Q inclined at an angle θ to the vertical. If the bloc remain in equilibrium, find the coefficient of friction between the bloc and the horizontal plane. A. From the figure, Normal reaction R mg+ Qcoθ Force of friction f R ( mg+ Qcoθ )

27 Applied force along horizontal direction F P+ Qinθ If the body i in equilibrium, ( co ) mg + Q θ P + Qinθ P+ Qinθ mg + Q coθ 3. A bloc of ma g lie on a horizontal urface in a truc. The coefficient of friction between the bloc and the urface i 0.5. If the acceleration of the truc i 6 m - find the acceleration of the bloc. A. ma of the bloc (m) g 0.5 Acceleration of the truc 6m a a If bloc truc g g a m 0m, then bloc ( )( ) 4. A bloc of ma g lying on ice when given a velocity of 6 m - i topped by friction in 5. Find the coefficient of friction between the bloc and ice. (g 0 m -.) A. ma (m) g u 6m v 0 t 5 6 a. Retardation (a). m 0. 5 g 0 5. A bloc A of ma 3 g and another bloc B of ma g are connected by a light inextenible tring a hown in figure. If the coefficient of friction between the urface of the table and A i 0.5. What maximum ma C i to be placed on A o that the ytem i to be in equilibrium? A. For the ytem to be in equilibrium, force of friction of A and C mut be equal to the weight of B. ( M + M ) g M g 5(3 + M ) g g M 0.5M 0.5 A C B C C C MC 0.5 g 0.5