OBJETIVOS ESPECIFÍCOS
1. GENERALIDADES DEL CAMPO CHUIRA
1.1 HISTORIA DEL CAMPO
(a) is an inertial frame by definition
(b) cannot be an inertial frame because the earth is revolving around the sun
(c) is an inertial frame because Newton's laws are applicable in this frame
(d) cannot be an inertial frame because the earth is rotating about its axis.
2. A particle stays at rest as seen in a frame. We can conclude that
(a) the frame is inertial
(b) resultant force on the particle is zero
(c) the frame may be inertial but the resultant force on the particle is zero
(d) the frame may be noninertial but there is a nonzero resultant force.
3. A particle is found to be at rest when seen from a frame and moving with a constant velocity when seen from another frame S2. Mark out the possible options.
(a) Both the frames are inertial. (b) Both the frames are noninertial. (c) 5, is inertial and S2 is noninertial.
(d) S, is noninertial and St is inertial.
4. Figure (5-Q3) shows the displacement of a particle going along the X-axis as a function of time. The force acting on the particle is zero in the region
Figure 5-Q3
Figure (5-Q4) shows a heavy block kept on a frictionless surface and being pulled by two ropes of equal mass m. At £ = 0, the force on the left rope is withdrawn but the force on the right end continues to act. Let F, and F2 be
the magnitudes of the forces by the right rope and the left rope on the block respectively.
mmzt Figure 5-Q4 (a) AB (b) BC (c) CD (d) DE. (a) F = F2 = F for £<0 (b) F = F + mg for £ < 0 (c) F - F, F2 = F for £>0 (d) <F, F2- F for £>0.
tf. The force exerted by the floor of an elevator on the foot of a person standing there is more than the weight of the person if the elevator is
(a) going up and slowing down (b) going up and speeding up (c) going down and slowing down (d) going down and speeding up.
7. If the tension in the cable supporting an elevator is equal to the weight of the elevator, the elevator may be (a) going up with increasing speed
(b) going down with increasing speed (c) going up with uniform speed (d) going down with uniform speed.
8. A particle is observed from two frames S, and S2 . The
frame S2 moves with respect to S, with an acceleration
a. Let Fl and F., be the pseudo forces on the particle
when seen from S', and S'2 respectively. Which of the
following are not possible ?
(a) F, = 0 , F2 * 0 (b F, * 0, F2 - 0
(c) F, * 0, F, * 0 (d) Fl - 0, F2 - 0.
9. A person says that he measured the acceleration of a particle to be nonzero while no force was acting on the particle.
(a) He is a liar.
(b) His clock might have run slow.
(c) His meter scale might have been longer than the standard.
(d) He might have used noninertial frame.
EXERCISES 1. A block of mass 2 kg placed on a long frictionless
horizontal table is pulled horizontally by a constant force
F. It is found to move 10 m in the first two seconds. Find the magnitude of F.
:. A car moving at 40 knyh is to be stopped by applying brakes in the next 4"0 m. If the car weighs 2000 kg, what average force must be applied on it ?
. In a TV picture tube electrons are ejected from the cathode with negligible speed and reach a velocity of 5 x 100 m/s in travelling one centimeter. Assuming
straight line motion, find the constant force exerted on the electron. The mass of the electron is 9-1 x 10 kg. •1. A block of mass 0'2 kg is suspended from the ceiling by
a light string. A second block of mass 0'3 kg is suspended from the first block through another string. Find the tensions in the two strings. Take g - 10 m/s 2.
5. Tw"o blocks of equal mass m are tied to each other through a light string. One of the blocks is pulled along the line joining them with a constant force F. Find the tension in the string joining the blocks.
6. A particle of mass 50 g moves on a straight line. The variation of speed with time is shown in figure (5-E1). Find the force acting on the particle at t = 2, 4 and 6 seconds.
8 t(s)
Figure 5-E1
A and B of mass m. and mB respectively
Two blocks
are kept in contact on a frictionless table. The experimenter pushes the block A from behind so that
9.
10.
11.
the blocks accelerate. If the block A exerts a force F on the block B,what is the force exerted by the experimenter on A ?
Raindrops of radius 1 mm and mass 4 mg are falling with a speed of 30 m/s on the head of a bald person. The drops splash on the head and come to rest. Assuming equivalently that the drops cover a distance equal to their radii on the head, estimate the force exerted by each drop on the head.
A particle of mass 0'3 kg is subjected to a force
F = - k x with k - 15 N/m. What will be its initial acceleration if it is released from a point x = 20 cm ? Both the springs shown in figure (5-E2) are unstretched. If the block is displaced by a distance x and released, what will be the initial acceleration?
Figure 5-E2
A small block B is placed on another block A of mass 5 kg and length 20 cm. Initially the block B is near the right end of block A (figure 5-E3). A constant horizontal force of 10 N is applied to the block A. All the surfaces are assumed frictionless. Find the time elapsed before the block B separates from .4.
W///W/WMW///////7/M
Figure 5-E3
12. A man has fallen into a ditch of width d and two of his friends are slowly pulling him out using a light rope and two fixed pulleys as shown in figure (5-E4). Show that
the force (assumed equal for both the friends) exerted by each friend on the rope increases as the man moves up. Find the force when the man is at a depth h.
Figure 5-E4
13. The elevator shown in figure (5-E5) is descending with an acceleration of 2 m/s 2. The mass of the block A is
0"5 kg. What force is exerted by the block A on the block B ?
2 m / s
Figure 5-E5
14. A pendulum bob of mass 50 g is suspended from the ceiling of an elevator. Find the tension in the string if the elevator (a) goes up with acceleration 1'2 m/s 2,
(b) goes up with deceleration 1'2 m/s 2, (c) goes up with
uniform velocity, (d) goes down with acceleration 1'2 m/s , (e) goes down with deceleration 1"2 try's 2 and
(0 goes down with uniform velocity.
15. A person is standing on a weighing machine placed on the floor of an elevator. The elevator starts going up with some acceleration, moves with uniform velocity for a while and finally decelerates to stop. The maximum and the minimum weights recorded are 72 kg. and 60 kg. Assuming that the magnitudes of the acceleration and the deceleration are the same, find (a) the true weight of the person and (b) the magnitude of the acceleration. Take g - 9*9 m/s 2.
E
1.5kg £ 1 J 3.0kg g '1016. Find the reading of the spring balance shown in figure (5-E6). The elevator is going up with an acceleration of g/10, the pulley and the string are light and the pulley is smooth.
17. A block of 2 kg is suspended from the ceiling through a massless spring of spring constant k - 100 N/m. What is the elongation of the spring ? If another 1 kg is added to the block, what would be the further elongation ? 18. Suppose the ceiling in the previous problem is that of
an elevator which is going up with an acceleration of 2'0 m/s 2. Find the elongations.
19. The force of buoyancy exerted by the atmosphere on a balloon is B in the upward direction and remains constant. The force of air resistance on the balloon acts opposite to the direction of velocity and is proportional to it. The balloon carries a mass M and is found to fall down near the earth's surface with a constant velocity v. How much mass should be removed from the balloon so that it may rise with a constant velocity v ?
20. An empty plastic box of mass m is found to accelerate up at the rate of g/6 when placed deep inside water. How much sand should be put inside the box so that it may accelerate down at the rate of g/6 ?
21. A force F = v * A is exerted on a particle in addition to the force of gravity, where v is the velocity of the particle y and A is a constant vector in the horizontal direction.
With what minimum speed a particle of mass m be projected so that it continues to move undeflected with a constant velocity ?
22. In a simple Atwood machine, two unequal masses m{ and m2 are connected by a string going over a clamped
light smooth pulley. In a typical arrangement (figure 5-E7) ml = 300 g and m2 = 600 g. The system is
released from rest, (a) Find the distance travelled by the first block in the first two seconds. (b) Find the tension in the string, (c) Find the force exerted by the clamp on the pulley.
I
Figure 5-E7
Consider the Atwood machine of the previous problem. The larger mass is stopped for a moment 2'0 s after the system is set into motion. Find the time elapsed before the string is tight again.
Figure (5-E8) shows a uniform rod of length 30 cm having a mass of 3'0 kg. The strings shown in the figure are pulled by constant forces of 20 N and 32 N. Find the force exerted by the 20 cm part of the rod on the 10 cm part. All the surfaces are smooth and the strings and the pulleys are light.
23.
24.
10 c m 20 c m
ft
20 N
A
32 XFigure 5-E8
5. Consider the situation shown in figure (5-E9). All the surfaces are frictionless and the string and the pulley are light. Find the magnitude of the acceleration of the two blocks.
1.0 kg
• •
Figure 5-E12
29. In the previous problem, suppose m2 = 2'0 kg and
m3 = 3"0 kg. What should be the mass m so that it
remains at rest ?
30. Calculate the tension in the string shown in figure (5-E13). The pulley and the string are light and all the surfaces are frictionless. Take g = 10 m/s 2.
Figure 5-E9
6 A constant force F=m^g/^ is applied on the block of mass 7ft, as shown in figure (5-E10). The string and the pulley are light and the surface of the table is smooth. Find the acceleration of m, .
1 kg
rmmmmsmmmmmm. m
1 kg
mmzmmmTmmm.
Figure 5-E10
In figure (5-E11) m, = 5 kg, m2 = 2 kg and F = 1 N. Find
the acceleration of either block. Describe the motion of m, if the string breaks but F continues to act.
'///////////WW/. v/m
0
F ..
[b
V F Figure 5-E11Let /ft, = 1 kg, /n, = 2 kg and m3 = 3 kg in figure (5-E12).
Find the accelerations of ml , m2 and m3. The string
from the upper pulley to ml is 20 cm when the system
is released from rest. How long will it take before m, strikes the pulley ?
3 1 .
Figure 5-E13
Consider the situation shown in figure (5-E14). Both the pulleys and the string are light and all the surfaces are frictionless. (a) Find the acceleration of the mass M. (b) Find the tension in the string, (c) Calculate the force exerted by the clamp on the pulley A in the figure.
Figure 5-E14
32. Find the acceleration of the block of mass M in the situation shown in figure (5-E15). All the surfaces are frictionless and the pulleys and the string are light.
33. Find the mass M of the hanging block in figure (5-E16) which will prevent the smaller block from slipping over the triangular block. All the surfaces are frictionless and the strings and the pulleys are light.
777777777,
Figure 5-E16
34. Find the acceleration of the blocks A and B in the three situations shown in figure (5-El7).
2 kg
• 5kg
1 kg [ jCb) (c)
Figure 5-E17
35. Find the acceleration of the 500 g block in figure (5-E18).
• 5 0 g
Figure 5-E18
36. A monkey of mass 15 kg is climbing on a rope with one end fixed to the ceiling. If it wishes to go up with an acceleration of 1 m/s 2, how much force should it apply to the rope ? If the rope is 5 m long and the monkey starts from rest, how much time will it take to reach the ceiling ?
37. A monkey Is climbing on a rope that goes over a smooth light pulley and supports a block of equal mass at the other end (figure 5-E19). Show that whatever force the monkey exerts on the rope, the monkey and the block move in the same direction with equal acceleration. If initially both were at rest, their separation will not change as time passes.
3 8 .
Figure 5-E19
The monkey B shown in figure (5-E20) is holding on the tail of the monkey A which is climbing up a ro] The masses of the monkeys A and B are 5 kg and 2 respectively. If A can tolerate a tension of 30 N in tail, what force should it apply on the rope in order carry the monkey B with it ? Take g - 10 m/s 2.
V///////////////////,
Figure 5-E20
39. Figure (5-E21) shows a man of mass 60 kg standing on a light weighing machine kept in a box of mass 30 kg. The box is hanging from a pulley fixed to the ceiling through a light rope, the o.ther end of which is held by the man himself. If the man manages to keep the box at rest, what is the weight shown by the mach/ne ? What force should he exert on the rope to get his correct weight on the machine ?
Figure 5-E21
40. A block A can slide on a frictionless incline of angle t) and length /, kept inside an elevator going up with uniform velocity u (figure 5-E22). Find the time taken by the block to slide down the length of the incline if it is released from the top of the incline.
41. A car is speeding, up on a horizontal road with an acceleration a. Consider the following situations in the car. (i) A ball is suspended from the ceiling through a string and is maintaining a constant angle with the vertical. Find this angle, (ii) A block is kept on a smooth incline and does not slip on the incline. Find the angle of the incline with the horizontal.
42. A block is kept on the floor of an elevator at rest. The elevator starts descending with an acceleration of 12 m/s '. Find the displacement of the block during the first 0 2 s after the start. Take g = 10 m/s J.
Figure 5-E22 • ANSWERS O B J E C T I V E I (c) 2. (b) 3. (d) 4. (a) 5. (c) 6. (b) 10. (d) 8. (a) 9. (b) 10. (b) 11. (c) 12. (b) (a) 14. (c) 11. O B J E C T I V E II (b), (d) 2. (c), (d) 3. (a), (b) (a), (c) 5. (a) 6. (b), (c) (c), (d) 8. (d) 9. (d) E X E R C I S E S 10 N 3 1 x 10 4 N 1-1 x 10*" N •r> N and 3 N FA2
0"2f) N along the motion, zero and 0 ' 2 5 N opposite to the motion. /•'! 1 9. 10 m/s 11. 0-45 s. .T m
12. f f
4 h 13. 4 N 14. (a) 0-55 N (d) 0-43 N 4 h' (b) 0-43 N (e) 0-55 N (c) 0-49 N (f) 0-49 N 15. 66 kg and 0*9 m/s 16. 4'4 kg 17. 0-2 m, O'l m 18. 0-24 m, 0-12 m 19. 2 M - — •1-8 N 20. 2 » i / 5 21. mg/A 22. (a) 6-5 m 23. 2 / 3 s 24. 24 N 25. gA 10 (b) 3-9 N (c) 7-8 N26.
2 (m, + nu) towards right
27. 4'3 m/s 2, moves downward with acceleration
g + 0-2 m/s 2
iq 17 21 28. g g (up), g g (down), — g (down), 0'25 s 29. 4-8 kg
30. 5 N i
31. ( a ) 2 g / 3 (b) Mg/3
(c) / 2 Mg/3 at an angle of 45° with the horizontal 32. g / 3 up the plane
M' + m 33.
cote - i
2
g34. (a) — g downward, upward 10 5 (b) — g forward, — g downward (c) ~ g downward, ^ upward g 35. — g downward 36. 165 N, /ITDs 38. between 70 N and 105 N 39. 15 kg, 1800 N 40. » f ? S I g
sin 0
41. tan 1 (a/g) in each case
42. 20 cm
Solution : (a) Take the 2 kg block as the system. The forces
on this block are shown in figure (6-W16) 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.
' Mg mS
Figure 6-W16 Figure 6-W17 Taking components ± to the incline
JV=Mg cos 45° = Mg/'/2. Taking components 11 to the incline
T+f=Mg sin 45° = Mg//2 or, T=Mg//2-f. As it is a case of limiting equilibrium,
Afe Mg Mg
or, T = 7 2 "M s7 2 = / 2
Now consider the other block as the system. The forces acting on this block are shown in figure (6-W17). Taking components _L to the incline,
mg cos 45° = mg//2. Taking components 11 to the incline
(i)
T = mg sin 45° + f = + f . As it is the case of limiting equilibrium
mg f'-P.-zM'-p. / 2
Thus, T" _ mg
/ 2 (1 + M.) •
From (i) and (ii)
(ii) (iii) or, _ (1 - , , 1 - 0-28 " (1 + mJ " 1 + 0'28 = 8 kg. x 2 kg
When maximum possible value of m is supplied, the directions of friction are reversed because m has the tendency to slip down and 2 kg block to slip up. Thus, the maximum value of m can be obtained from (iii) by putting ps= - 0-28. Thus, the maximum value of m is
1 + 0-28
1 - 0 - 2 8
32. "7T kg-
x 2 kg
(b) If m = 9/8 kg and the system is gently pushed, kinetic friction will operate. Thus,
, Mg
f-Vk' 7 2 and f / 2
where = 0-25. If the acceleration is a, Newton's second law for M gives (figure 6-W16).
Mg sin 45° - T- f - Ma or, Mg /2 ' T- VkMg
/ 2 Ma.
Applying Newton's second law m (figure 6-W17),
T- mg sin 45° - / ' = ma
or, mg
/ 2 ' / 2
(iv)
(v) Adding (iv) and (v)
Mg mg 7 f ( l - ^ ) " 7 f (l + M*)-(M+m)a or, M( 1 - - m (1 + / 2 (M+m) 1 _ 2 x Q-75 - 9/8 x 1-25 / 2 (2 + 9/8) = 0-31 m/s2. g
•
QUESTIONS F O R SHORT ANSWER 1. For most of the surfaces used in daily life, the friction
coefficient is less than 1. Is it always necessary that the friction coefficient is less than 1 ?
-• 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 ?
f Why is it difficult to walk on solid ice ?
5. Can you accelerate 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 released. You want to keep the door open for a long time say for an hour. If you put a half kg stone in front of the open 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 plastic 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 coefficient 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 gently 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 goioj
on a level road than while going on an incline ? 10. You are standing with your bag in your hands on tht
ice in the middle of a pond. The ice is so slippery thai it can offer no friction. How can you come out of the ice'i 11. When two surfaces are polished, the friction coefficient between them decreases. But the friction coefficii increases and becomes very large if the surfaces i 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 frictional force between the surface and the body will
(a) increase (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 ensure (a) larger friction (b) smaller friction
(c) larger normal force (d) smaller normal force. 3. A body of mass M is kept on a rough horizontal surface
(friction coefficient = ji). A person is trying to pull the body by applying a horizontal force but the body is not moving. The force by the surface on A is F where ( a ) F - M g (b) F = /i Mg
(c) Mg <F< Mg V1 + p2 (d) Mg >F> Mg •/1 - p 2.
4. A scooter starting from rest moves with a constant acceleration for a time Atv then with a constant velocity
for the next At2 and finally with a constant deceleration
for the next At3 to come to rest. A 500 N man sitting on
the scooter behind the driver manages to stay at rest with respect to the scooter without touching any other part. The force exerted by the seat on the man is (a) 500 N throughout the journey
(b) less than 500 N throughout the journey (c) more than 500 N throughout the journey (d) > 500 N for time A«, and At3 and 500 N for At,.
5. Consider the situation shown in figure (6-Q1). The wall is smooth but the surfaces of A and B in contact are rough. The friction on B due to A in equilibrium
I
(a) is upward (b) is downward (c) is ze| (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 zefi (d) depends on the masses of A and B.
7. l\vo cars of unequal masses use similar tyres. If thej are moving at the same initial speed, the minimu stopping distance
(a) is smaller for the heavier car (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