Solved Examples

1.    A body of mass 400 g slides on a rough horizontal surface . If the frictional force is 3.0 N, find (a) the angle made by the contact force on the body with the vertical and (b) the magnitude of the force. Take g = 10 m/s2.
Sol.    Let the contact force on the block by the surface be F which makes an angle  with the vertical (figure)

2.    A heavy box of mass 20 kg is pulled on a horizontal force. If the coefficient of kinetic friction between the box and the horizontal surface is 0.25, find the force of frictionless exerted by the horizontal surface on the box.
Sol.    The situation is shown in figure (6.6). In the vertical direction there is no acceleration , so

3.    A boy (30 kg) sitting on his horse whips it. The horse speeds up at an average acceleration of 2.0 m/s2.(a) If the boy does not slide back, what is the force of friction exerted by the hoerse on boy.(b) If the boy slides back during the acceleration , what can be said about the cofficientof static friction between the horse and the boy ? Take g = 10 m/s2.
Sol.    (a) The horse acting on the boy are
(i) the weight Mg1
(ii) the normal contact force  and
(iii) the static friction fs.

4.    A wooden block is kept on a polished wooden plank and the inclination of the plank is gradually incresed. It is found that starts sliping when the plank makes an angle of 18º with the horizontal . However, once  strated, the block can continue with uniform speed if the inclination is reduced to 15°. Find the coefficients of static and kinetic fraction between the block and the plank.

1.    For most of the surfaces used in daily life, the friction cofficient is less than 1. Is it always necessary that the friction cofficient is less than 1?

2.    Why is it easier to push a heavy block from behind than to press it on the top and push ?

3.    What is the average friction force when a person has a usual 1 km walk ?

4.    Why is it difficult to walk on solid ice ?

5.    Can you acceleration a car on a frictionless horizontal road by putting more petrol in the engine?Can you stop a car going on a frictionless horizontal road by applying brakes?

6.    Spring   fitted doors close by themselves when relesed.You want to keep the door open for  a long time say for

...
hour. If you put a half kg stone in front of the door, it does not help. The stone slides with the door and the door gets closed . However , if you sandwitch a 20 g piece of wood in the small gap between the door and the floor, the door stays open. Explain why a much lighter piece of wood is able to keep the door open while the heavy stone fails.

7.    A classroom demonstration of Newton’s first law is as follows : A glass is covered with a plasstic card and a coin is placed on the card. The card is given a quick strike and the coin falls  in the glass.

(a) Should the friction cofficient between the card and the coin be small or large ?
(b) Should the coin be light or heavy ?
(c) Why does the experiment fail if the card is genntly pushed ?

8.    Can a tug of war be ever won on a frictionless surface ?

9.    Why do tyres have a better grip of the road while going on a level road than while going on an incline?

10.    You are standing with your bag in your hands on the ice in the middle of a pond. The ice is so slippery that it can offer no friction. How can you come out of the ice?

11.    When two surfaces are polished, the friction cofficient between them decreases. But the friction cofficient increses and becomes very large if the surface are made highly smooth. Explain.

Objective - I

1.    In a situation the contact force by a rough horizontal surface on a body placed on it has constant magnitude. If the angle between this force and the vertical is decreased , the force and the vertical is decreased, the friction force between the surface and the body will

(a) increse             (b*) decrease      (c) remain the same    (d) may increase or decrease

2.    While walking on ice, one should take small steps to avoid slipping . This is because smaller steps to avoid sliping . This is because smaller steps ensure

(a) larger friction         (b*) smaller friction (c) larger normal force    (d) smaller normal force

5.    Consider the situation shown in figure. The wall is smooth but the surface of A and B in contact are rough. The friction on B due to A in equilibrium

(a) is upward                          (b) is downward
(c)is zero                               (d*) the system cannot remain in equilibrium.

6.    Suppose all the surfaces in the previous problem are rough. The direction of friction on B due to A

(a*) is upward                         (b) is downward
(c) is zero                                (d) depends on the masses of A and B.

7.    Two cars of unequal masses use simmilar tyres .If they are moving at the same intial speed, the minimum stopping distance

(a) is smaller for the heaviercar        (b) is smaller for the lighter car
(c*) is same for both cars            (d) depends on the volume of the car.

8.    In order to stop a car in shortest distance on a horizontal road, one should

(a) apply the brakes very hard so that the weels stop rotating
(b*) apply the brakes hard enough to just prevent slipping
(c) pump the brakes (press and release)
(d) shut the engin off and not apply brakes .

9.    A block A kept on an inclined surface just begins to slide if the inclination is 30º. The block is replaced by another block B and it is found that it just begins to slide if the inclination is 40º.
(a) mass of A > mass of B        (b) mass of A < mass of B
(c) mass of A = mass of B        (d*) all the three are possible.

Objective - II

1.    Let F, FN and f denote the magnitudes of the contact force , normal force and the friction exerted by one surface on the other kept in contact. If none of these is zero,
(a*) F > FN        (b*) F > f        (c) FN > f    (d*) FN – f < F < FN + f

2.    The contact force exerted by a body A on another body B is equal to the normal force between the bodies. We conclude that
(a) the surface must be frictionless
(b*) the force of friction between the bodies is zero
(c) the magniyude of normal force equals that of friction
(d*) the bodies may be rough but they don’t slip on each other.

3.    Mark the correct statements about the friction between two bodies.
(a) Static friction is always greater than the kinetic friction.
(b*) Cofficient of static friction is always greater than the cofficient of kinetic friction.
(c*) Limiting friction is always greater than the kinetic friction.
(d*) Limiting friction is never less than static friction.

4.    A block is placed on a rough floor and a horizontal force F is applied on it. The force of friction f by the floor on the block is measured for different values of F and a graph is plotted between them.
(a) The graph is a straight line of slope 45º
(b) The graph is straight line parallel to the F-axis.
(c*) The graph is a straight line of slope 45º for small F and a straight line parallel to the F-axis for large F.
(d*) There is a small kink on the graph.

5.    Consider a vehicle going on a horizontal road toards east. Neglect any force  by the air. The frictional forces on the vehicle by the road.
(a*) is towards east if the vehicle is accelerating
(b*) is zero if the vehicle is moving with a uniform velocity
(c) must be towards east.
(d) must be towards east.

Worked Out Examples

2.    A horizontal force of 20N is applied to a block of mass 4kg resting on a rough horizontal table. If the block does not move on the table, how much frictional force the table is applying on the block ? What can be said about the coefficient of static friction between the block and the table ?
Take g = 10 m/s2.

(a) 4kg × 10 m/s2 = 40 N downward by the earth,
(b) N upward by the table,
(c) f = 20 N and N = 40 N
(d) f towards left by the table (friction).

4.    The coefficient of static friction between the two blocks shown in figure is  and table is smooth. What maximum horizontal force F can be applied to the block of mass M so that the blocks move together ?

5.    A block slides down on incline of angle 30° with an acceleration g/4. Find the kinetic frinction coefficient.

6.    A block of mass 2.5 kg is kept on a rough horizontal surface . It is found that the block does not slide if a horizontal force less than 15 N is applied to it. Also it is found that it takes 5 seconds to slide throught the first 10 m  if a horizontal force of 15 N is applied and the block is genntly pushed to start the motion. Taking g = 10 m/s2, calculate the cofficients of static and kinetic friction between the block and the surface.
Sol.    The forces acting on the block are shown in figure. Here M  = 2.5 kg and F = 15 N.

.

10.    Figure shows two blocks connected by a light string placed on the two inclined parts of  a triangular structure. The cofficients of static and kinetic friction are 0.28 and 0.25 respectively at each of the surfaces . (a) Find the minimum and maximum values of m for which the system ramains at rest.(b) Find the acceleration of either block if m is given the minimum value calculate in the first part (a) and is gently pushed up the incline for a short while.

Sol.    (a) Take the 2 kg block as the system. The forces on this block are shown in figure with M = 2 kg. It is assumed that m has its minimum value so that the 2 kg block has a tendency to slip down. As the block is in equilibrium , the resultant force should be zero.

Exercise

1.    A body slipping on a rough horizontal plane moves with a deceleration of 4.0 m/s2 . What is the cofficient of kinetic friction between the block and the plane ?             [HCV_Chp_6_Ex_1]
[Ans : 0.4]

[Ans : 0.4]

2.    A block  is projected along a rough horizontal road with a speed of 10 m/s. If the cofficient of kinetic friction is 0.10 , how far will it travel before coming to rest ?        [HCV_Chp_6_Ex_2]

[Ans : 50 m]

3.    A block of mass m is kept on a horizontal table . If the static friction cofficient is , find the frictional force acting on the block.
Ans : zero]

4.    A block slides down an inclined surface of inclination 30º with the horizontal. Starting from rest it covers 8 m in the first two seconds. Find the cofficient of kinetic friction between two.
[Ans : 0.11]

5.    Suppose the block  of the previous problem is pushed down the incline with a force of 4N. How far will the block move in the first two seconds after starting from rest? The mass of the block is 4 kg.
[Ans : 10m]
6.    A body of mass 2 kg is lyaing on a rough inclined plane of        inclination 30º. Find the magnitude of the force parallel to the incline needed to make the block move.
(a) up the incline    (b) down the incline.
Cofficient of static friction = 0.2
[Ans : (a) 13N (b) zero]

7.    Repeat part (a) of problem 6 if the push is applied horizontally and not parallel to the incline.
Ans : 17.5 N

8.    In a children-park an inclined plane is constructed with an angle of incline 45º in the part (figure). Find the acceleration of a boy slising on it if the friction cofficient between the cloth of the boy and the incline is 0.6 and g = 10 m/s2.

9.    A body starts sliping down an incline and moves half meter in half second . How long will it taketo move the next half meter ?
Ans : 0.21 s

10.    The angle between the resultant contract force and the normal force exerted by a body on the other is called the angle of friction. Show that, if  be the angle of friction and  the cofficient of static friction, tan–1

11.    Consider the situation shown in figure. In different cases calculate (a) the acceleration of the 1.0 kg blocks, (b) the tension in the string connecting the 1.0 kg blocks and (c) the tension in string atached to 0.50 kg.

Case I : m1 = 0.2 , m2 = 0.5
Case II : m1 = 0.2 , m2 = 0.3
Case III : m1 = 0.3 , m2 = 0.2
Case IV : m1 = 0.2 , m2 = 0.2
Case V : m1 = 0.5 , m2 = 0.2        (Take g = 10 m/s2)        HCV_Ch-6_Ex._11

[Ans :     Case I : (a) 0  m/s2 , (b) 0 N , (c) 5 N ]
Case II : (a) 0  m/s2 , (b) 2 N , (c) 5 N ]
Case III : (a) 0  m/s2 , (b) 3 N , (c) 5 N ]
Case IV : (a) 0.4 m/s2 , (b) 2.4 N , (c) 4.8 N
Case V : (a) 0 m/s2 , (b) 3 N , (c) 5 N ]
12.    If the tension in the string in figure is 16N and the acceleration of each block is 0.5 m/s2, find  the friction cofficients at the two constants with the blocks.

[Ans :meu1 = 0.75 ,meu2  = 0.06]
13.    The friction cofficient between the table and the block shown in figure is 0.2 . Find the tansions in the two strings.

[Ans :  96N is the left string and 68N in the right]
14.    The friction cofficient between a road and the tyre of a vehicle is 4/3. Find the maximum incline the road may have so that once hard brakes are applied and the wheel starts skidding , the vehicle going down at a speed of 36 km/hr is stopped within 5 m.

[Ans :16º]

15.    The friction cofficient between an athelet’s shoes and the ground is 0.90. Suppose a superman wears these shoes and races for 50 m. There is no upper limit on his capacity of running at high speeds. (a) Find the minimum time that he will have to take in completing the 50 m starting from rest. (b) Suppose he takes exactly this minimum time to complete the 50 m , what minimum time will he take to stop ?

Ans :(a) s   (b)  s  ]

16.    A car is going at a speed of 21.6 km/hr when it encounters a 12.8 m long slope of angle 30º (figure). The friction cofficient between the road and tyre is 1/2. Show that no matter how hard the driver applies the brakes , the car will reach the bottom with a speed greater than 36 km/hr. Take g = 10 m/s2.

17.    A car starts from rest on a half kilometer long bridge . The cofficient of friction between the tyre and the road is 1.0. Show that one cannot drive through the bridge in less than 10s.

22.    A 2 kg block is placed over a 4 kg block and the both are placed on a smooth horizontal surface . The cofficient of friction between the blocks is 0.20. Find the acceleration of the two blocks if a horizontal force of 12 N is applied to (a) the upper block, (b) the lower block. Take g = 10 m/s2.

[Ans :(a) upper block 4 m/s2, lower block 1 m/s2,      (b) both blocks 2 m/s2]

23.    Find the accelerations a1, a2, a3 of the three blocks shown in figure. If  a horizontal force of 10N is applied on (a) 2 kg block, (b) 3 kg block, (c) 7 kg block.  (Take g = 10 m/s2)

[Ans : (a) a1 = 3 m/s2, a2 = a3 = 0.4 m/s2     , (b) a1 = a2 = a3 =  m/s2    , (c)same as (b)    ]
24.    The friction cofficient between the two blocks shown in figure is  but floor is smooth. (a) What maximum horizontal force F can be applied without disturbing the equilibrium of the system?(b) Suppose the horizontal force applied is double of that found in part (a). Find the accelerations of the two masses.

26.    Consider the situation shown in figure. Suppose a small electric field E exists in the space in the a positively charge Q on its top surface .The friction cofficient between the two blocks is  but the floor is smooth. What maximum horizontal force F can be applied without disturbing the equilibrium ?
[Hint : The force on a charge Q by the electric field E is F = QE in the direction of E.]

27.    A block of mass m slips on a rough horizontal table under the action of a horizontal force applied to it. The cofficient of friction between the block and the table is . The table does not move on the floor. Find the total friction force applied by the floor on the legs of the table. Do you need the friction coefficint between the table and the floor or the mass of the table ?

[Ans :  mg ]
28.    Find the acceleration of the block of mass M in the situation of figure. The cofficient of friction between the two blocks is m1 and that between the bigger block and the ground is m2.

[Ans :  ]

29.    A block of mass 2 kg is pushed against a rough vertical wall with a force of 40 N, cofficient of static friction being 0.5. Anotherhorizontal force of 15N is  applied on the block in a direction parallel to the wall. Will the block move ? If yes , in which direction ? If no, find the frictional force exerted by the wall on the block.

Ans : it will move at an angle of 53º with the 15N force

30.    A person (40 kg) is managing to be at rest between two vertical walls by pressing one wall B by his back (figure). Assume that the friction cofficient between his body and the walls is 0.8 and that limiting friction acts at all the contacts . (a) Show that the person pushes the two walls with equal force. (b) Find the normal force exerted by either wall on the peson. Take g = 101 m/s2.

[Ans : (b) 250N ]

31.    Figure shows a small block of mass m kept at the left end of a larger block of mass M and the length l.The system can slide on a horizontal road. The system is started towards right with an initial velocity v. The friction coefficient between the road and and the bigger block is  and that between the blocks is/ 2. Find the time elapsed before the smaller block separates from the bigger block.