# EXERCISE 3

(A)       Theoretical Questions

(a) Friction

1.  Define the terms: Friction, limiting force of friction, co-efficient of friction and angle of friction.

2.   Explain the difference between co-efficient of friction of friction and angle of friction

(AMIE Winter, 1981)

3.     (a) State the laws of static and dynamic friction.

(AMIE (S) 1982, (S) 1986)

(b) State the laws of solid friction.

(AMIE Summer 1983, (S) 1989)

4.   Prove that the angle of friction is equal to the angle of the inclined plane, when a solid body or weight W placed on the inclined plane, is about to slide down.

5.           What do you mean by ‘angle of repose’? Prove that angle of repose is equal to the angle of friction?

6.           A body of weight W is placed on an inclined rough plane. The inclination of the plane with the horizontal is less than the angle of friction. The body willbe

1. in equilibrium
2. move downwards
3. move upwards
4. none of the above

[Ans. (a)]

7.   If in the above question, the inclination of the plane with the horizontal is more than the angle of friction, the body willbe in

1. equilibrium
2. move downwards
3. move upwards
4. none of the above

[Ans. (b)]

8.    A body of weight W is placed on a rough inclined plane having inclination α to the horizontal. The force P is applied horizontally to drag the body. If the body is on the point of motion up this plane, prove that P is given by

P = W tan (α+Φ)

where Φ = Angle of friction.

9.  In the above question. If the body is on the point of motion down the plane, prove that the force P is given by

P = W tan (α+Φ)

10.  A body of weight is placed on a rough inclined plane having inclination α to the horizontal. A force P is applied to the body in such a way that it makes an angle θ to the inclined plane. Prove that the force required to drag the body when the body is on the point of motion up the plane is given by

12. Define statics and kinetic friction and state the laws of solid-friction.

(AMIE Summer, 1983)

13.   Derive an expression for the least force required to drag a body on a rough horizontal plane.

(b)        Belt Friction

14. Derive an expression for the ratio of tensions for a flat belt passing over a pulley, when it is just on the point of slipping.

(AMIE Winter, 1981)

15. Distinguish between initial tension and centrifugal tension in a belt.

(AMIE Winter, 1987)

16. Distinguish between slip and creep in a belt drive. Derive an expression for the ratio of tensions in the tight and slack sides in terms of µ and θ, when the belt is just on the point of slipping.

(AMIE Summer, 1984)

17. Derive the expression for optimum speed of flat for the transmission of maximum power considering the effect of centrifugal tension.

(AMIE Summer, 1986)

20.  Derive an expressions for the length of

1. an open belt, and
2. a crossed belt.

21. Prove that the angle of contact (θ) is equal to (180-2α) for an open belt whereas it is equal to (180+2α) for a crossed-belt.

(c)       Wedge and Screw-Jack

22. What is a wedge? Explain how a wedge is used to raise heavy loads.

(AMIE Winter, 1982 ; Summer, 1986)

23.  What is a screw-jack? Show that for an ideal screw-jack, the efficiency is independent of the weight lifted.

(AMIE Winter, 1982)

24.  Prove that the effort applied at the end of the handle of a screw-jack to lift a load W is given by

(B)       Numerical Questions

(a)        Friction

1.           A body of weight 90 N is placed on a rough horizontal plane. Determine the co-efficient of friction if a horizontal force of 63 N causes the body to slide over the horizontal plane.

[Ans. 0.7]

2.           A body of weight 150 N is placed on a rough horizontal plane. If the co-efficient of friction between the body and the horizontal plane is 0.4, determine the horizontal force required to just slide the body on the plane.

[Ans. 60 N]

3.           The force required to pull a body of weight N on a rough horizontal plane is 15 N. Determine the co-efficient of friction if the force is applied at an angle of 200 with the horizontal.

[Ans. 0.404]

4.           A body of weight of 60 N is placed on a rough horizontal plane. To just move the body on the horizontal plane, a push of 18 N inclined at 200 to the horizontal plane is required. Find the co-efficient of friction.

[Ans. 0.255]

5.           A pull of 60 N inclined at 250 to the horizontal plane, is required just to move a body placed on a rough horizontal plane. But the push required to move the body is 75 N. If the push is inclined at 250 to the horizontal, find the weight of the body and co-efficient of friction.

[Ans. 253.83 N, 238]

6.           Find the least force required to pull a body of weight W placed on a rough horizontal plane, which the force is applied at an angle θ wi the horizontal.

[Ans. W sin θ]

7.           A body of weight 450 N is pulled up an inclined plane, by a force of 300 N. The inclination of the plane is 300 to the horizontal and the force is applied parallel to the plane. Determine the co-efficient of friction.

[Ans. .192]

8.           A body of weight 400 N is pulled up along an inclined plane having inclination 300 to the horizontal at a steady speed. If the co-efficient of friction between the body and the plane is 0.3 and force is applied parallel to the inclined plane, find the force required. Find also the work done on the body if the distance travelled by the body is 10 m along the plane.

[Ans. 303.92 N, 3039.2 Nm]

9.           An effort of 180 N is required just to move a certain body up an inclined plane of angle 150, the force being parallel to the plane. If the angle of inclination of the plane is made 200, the effort required, again applied parallel to the plane, is found to be 210 N. Find the weight of the body and co-efficient of friction.

[Ans. 386.68 N, 0.214]

10.           A cord connects two bodies of weights 500 N and 1000 N. The two bodies are placed on an inclined plane and cord is parallel to inclined plane. The co-efficient of friction for the weight of 500 N is 0.20 and that of 1000 N is 0.4. Determine the inclination of the plane to the horizontal and tension in the cord when the motion is about to take place, down the inclined plane. The body weight 500 N is below the body weighing 1000 N.

[Ans. 18.4170, 63.086 N]

11.           A uniform ladder of length 13 m and weighing 30 N is placed against a smooth vertical wall with its lower end 10 m from the wall. In this position the ladder is just to slip.the co-efficient of friction between the ladder and the floor, andfrictional force acting on the ladder at the point of contact between ladder and floor.

[Ans. (a) .602 (b) 18.06N]

12.           A uniform ladder of length 10 m and weighing 20 N is placed against a smooth vertical wall with its lower end 6 m from the wall. The co-efficient of friction between the ladder and floor is 0.45. Show that the ladder will remain in equilibrium in this position. What is the frictional force acting on the ladder at the point of contact between the ladder and floor.

[Ans. 7.5 N]

13.           A uniform ladder of weight 800 N and of length 7 m rests on a horizontal ground and leans against a smooth vertical wall. The angle made by the ladder with the horizontal is 600. When a man of weight 600 N stands on the ladder at a distance 4 m from the top of the ladder, the ladder is at the point of sliding. Determine the co-efficient of friction between the ladder and the floor.

[Ans. 0.237]

14.           A uniform ladder of weight 250 N and of length 5 m rests on a horizontal ground and leans against a rough vertical wall. The co-efficient of friction between the ladder and floor is 0.3 and between the ladder and vertical wall is 0.2. When a weight of 900 N is placed on the ladder at a distance of 2 m from the top of the ladder; the ladder is at the point of sliding.
Find

(ii) Reaction at the foot of the ladder, and
(iii) Reaction at the top of the ladder.

[Ans. (i) 61.520 (ii) 1084.9 N (iii) 325.47 N]

15.           A uniform ladder of length 15 m rests against a vertical wall making an angle of 600 with the horizontal. Co-efficient of friction between the wall and ladder, and ground and ladder are 0.3 and 0.25. A mad weighing 500 N ascends the ladder. How high will he be able to go before the ladder slips? Find the weight necessary to be put at the bottom of this ladder so as to be just sufficient to permit the man to go to top. Assume weight of ladder as 850 N.

(AMIE Winter, 1974, Old Scheme)
[Ans. 6.39 m, 607.6 N]

16.           A uniform ladder 3 m long weights 18 N. is placed against a wall making an angle 600with the floor as shown in Fig. 3.53. The co-efficient of friction between the floor and ladder is 0.35. The ladder, in addition to its own weight, has to support a weighing 90 N at its top at A. Calculate:The horizontal force F to be applied to the ladder at the floor level to prevent slipping.
If the force F is not applied what would be the minimum inclination of the ladder with the horizontal so that there is no slipping of it with the man at the top?

(AMIE Winter, 1977)
[Ans. (a) 16.6 N, (b) 690]

17.           A ladder 5 m long rests on a horizontal ground and leans against a smooth vertical wall at an angle of 700 with the horizontal. The weight of the ladder is 90 N and acts at its middle. The ladder is at the point of sliding, when a man weighing 75 N stands on a rung 3.5 m from the top of the ladder. Calculate the co-efficient of the friction between the ladder and the floor.

[Ans. 0.15]

18.           A uniform ladder of 7 m rests against a vertical wall with which it makes an angle of 450. The co-efficient of friction between the ladder and the wall is 1/3 and that between ladder and floor is 0.5. If a man whose weight is one-half of that of the ladder, ascends its, how high will it be when the ladder slips.

[Ans. 5.25 m]

19.           A body of weight of 100 N is at rest on a horizontal plane. A horizontal force of 70 N just causes the body to slide.
Determine:

(i) limiting force of friction and
(ii) co-efficient of friction.

[Ans. (i) 70 N, (ii) 0.7]

20.           The co-efficient of friction between a body of weight 100 N and the rough horizontal plane on which the body rests is 0.3 calculate the horizontal force required just to cause the body to slide over the horizontal plane.
If the body is loaded with an additional weight of 40 N, find the least horizontal force which will cause the body to side.

[Ans. (i) 30 N, (ii) 45 N]

(b)        Belt Friction

21.           With the help of a belt, an engine running at 150 r.p.m., drives a line shaft. The diameter of the pulley on the engine is 70 cm and the diameter of the pulley on the line shaft is 35 cm. A 80 cm diameter pulley on the line shaft drives a 20 cm diameter pulley keyed to a dynamo shaft. Find the speed of the dynamo shaft whenthere is no slip there is a slip of 3% at each drive.

[Ans. (a) 1200 r.p.m. (b) 1129 r.p.m]

22.           Two parallel shafts 12 metres apart are to be connected by a belt running over pulleys of diameters 480 cm and 80 cm respectively. Determine the length of the belt requiredif the belt is open, and belt is crossed.

[Ans. (a) 33.13 m, (b) 33.45 m]

23.           A shaft which rotates at a constant speed of 160 r.p.m. is connected by belling to a parallel shaft 72 cm apart which has to run at 60, 80 and 100 r.p.m. The smallest pulley on the driver shaft is 4 cm in radius Determine the remaining radii of the two stepped pulleys for:a crossed belt, and
an open belt.

[Ans. (a) r2 = 10.67 cm, r3 = 4.89 cm, r4 = 9.78 cm, r5 = 5.64 and r6 = 9.02 cm;
(b) r2 = 10.69, r3 = 5,r4 = 10, r5 = 6 and r6 = 9.6 cm]

24.           A shaft rotating at 200 r.p.m. drives another shaft at 300 r.p.m., and transmits 8 H.P. through a belt. The belt is 10 cm wide and 1 cm thick. The distance between the shafts is 4 m. The smaller pulley is 50 cm in diameter. Calculate the stress inopen-belt, and crossed-belt.
Take µ = 0.3. Neglect centrifugal tension.

(AMIE Summer, 1985)
[Ans. (a) 12.68 kgf/cm2 (b) 11.847 kgf/cm2]

25.           An open-belt drive connects two pulleys 120 cm and the 50 cm diameter on parallel shafts 4 m apart. The belt has a mass 0.9 kg/m length and a maximum tension in it is not to exceed 2000 N. The co-efficient of friction is 0.3. The 120 cm pulley, which is the driver runs at 200 r.p.m. Due to belt ship on one of the pulleys, the velocity of the driven shaft is only 450 r.p.m. Calculate the torque on each of the two shafts, the power transmitted and power lost in friction. What is the efficiency of the drive.

(AMIE Summer, 1979)
[Ans. 656.1 Nm ; 273.4 Nm ; 13.73 kW ; 1.0725 kW, 93.7%]

26.           The maximum allowable tension in a flat belt is 1500 N. The angle of lap is 1700 and co-efficient of friction between the belt and material of the pulley is 0.27. Neglecting the effect of centrifugal tension, calculate the net driving tension and power transmitted if the belt speed is 2 m/s.

[Ans. (i) 826.7 N ; (ii) 1.6534 kW]

27.           Two pulleys on parallel shafts are connected by a crossed belt. The diameters of the pulleys are 450 mm and 200 mm. The shafts are 1.95 apart. Find the length of the belt required and angle of contact between the belt and each pulley.

[Ans. (i) L=4.975 m, (ii) θ=199.20]

28.           In question 12, what power can be transmitted by the above belt when the larger pulley rotates at 100 rev/min, if the maximum permissible tension the belt is 1000 N and the co-efficient of friction between the belt and pulley is 0.25.

[Ans. 1.37 kW]

29.           The maximum permissible stress in a belt is 1.4 N/mm2 and ratio of tensions is 2.0. Find the maximum power transmitted by a belt 150 mm × 10 mm if the density of leather is 1 Mg/m3.

[Ans. 14.75 kW]

30.           An open flat drive connects two parallel shafts 1.2 m apart. The driving and driven shafts rotate at 350 r.p.m. and 140 r.p.m. respectively and the driven pulley is 40 cm in diameter. The belt is 5 mm thick and 8 cm wide. Co-efficient of friction between belt and pulley is 0.3 and maximum permissible tension the belting is 140 N/cm2. Determine:

1. diameter of driving pulley,
2. maximum power that may be transmitted by the belting, and
3. required initial tension in the belt. Neglect centrifugal tension.

(AMIE Winter, 1977)

[Ans. (a) 16 cm, (b) 0.96 kW, (c) 396 N]

31.           A cross-belt drive is to transmit 7.5 kW at 1000 r.p.m. of the smaller pulley. The diameter of the smallest pulley is 25 cm and velocity ratio is 2. The centre distance between the pulley is 125 cm. A flat belt of thickness 6 mm and of co-efficient friction is 0.3 is used over the pulleys. Determine the necessary width of the belt if the maximum allowable stress in the belt is 175 N/cm2 and density of the belt is 1 gm/cm3.

(a)        Wedge and Screw-Jack

32.           A load of 4000 N is to be lifted by a screw-jack, having threads of 10 mm pitch. The efficiency of the jack at this load is 40%. Determine the effort applied at the end of a handle of 60 cm length.

[Ans. 26.5 N]

33.           Find the effort required to apply at the end of a handle, fitted to the screw hand of a screw-jack to lift a load of 2000 N. The length of the handle is 50 cm. The mean diameter and the pitch of the screw-jack are 6 cm and 10 mm respectively. The co-efficient of friction is given as 0.09.

[Ans. 17.245 N]

34.           If in question 10, instead of raising the load of 2000 N, the same load is lowered, determine the effort required to apply at the end of the handle.

[Ans. 4.4 N]

35.        A screw-jack is used to lift a load of 4 kN. The screw of the screw-jack is square threaded with two threads to 1 cm. If the co-efficient of friction between the nut and screw is 0.095 and outer diameter of screw is 6 cm, find the force required at the end of the handle of length 50 cm to lift the load.

[Ans. 28.29 N]

36.        A screw-jack has a square thread, 6 cm mean diameter and 1 cm pitch. The load on the jack revolves with the screw. The co-efficient of friction at the screw threads is 0.045:Find the tangential force to be applied to the jack at 40 cm radius so as to lift a load of 700 N.
State whether the jack is self-locking. If it is, find the torque necessary to lower the load. If not, find the torque, which must be applied to keep the load from descending.

[Ans. (a) 5.16 N, (b) No, .1675 Nm]

37.        A screw-jack has a square thread of mean diameter 6 cm and pitch 9 mm. The co-efficient of friction at the screw-thread is 0.09. A load of 2 N is to be lifted through 10 cm. Determine the torque required and work done in lifting the load through 10 cm. Find the efficiency of the jack also.

[Ans.  .00826 kN ; 5792 kN ; 34.53%]

38.        In a differential screw-jack, the screw-threads have pitch of 12 mm and 9 mm. If the efficiency of the machine is 30%, find the effort required at the end of the arm 40 cm long to lift a load of 4 kN.

[Ans. 15.9 kN]