Kinetics of particles newton's 2nd law

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P
INTRODUCTION
 So far in the earlier chapter we did the motion analysis of moving particles
without taking into account the forces responsible for the motion. From this
chapter we begin our motion analysis involving the forces responsible for
the motion. This analysis is known as kinetics. Here we will analysis motion
of moving cars, elevators, blocks, airplanes, rockets etc. treating them as a
particle, since rotation of these bodies about their own Centre of gravity, if
any, is neglected.
 In this chapter we will extensively use the Newton's Second law approach to
kinetics. As stated further, this approach involves determination of
acceleration of the moving particle by knowing the forces acting on the
particle. Having determined the acceleration, the analysis is completed-
using kinematic relations which we have studied in previous chapter.
NEWTON'S SECOND LAW OF MOTION
 Newton's second law of motion states "The rate of change of momentum of a
body is directly proportional to the resultant force and takes place in the
direction of the force”.
 Consider a particle acted upon by several forces as shown in Fig. 10.1. Let
F be the resultant force. Because of the resultant force, the particle would
move in the direction of the resultant force. If u is the initial velocity, v is
final velocity and this change takes place in t sec, we have from Newton’s
second law:
 Rate of change of Momentum = Resultant Force
i.e.
Finalmomentum - Initial momentum
Time
F 
( )
mv mu
F
t
v u
F m
t

 

 
or F m a   …………10.1
Kinetics of Particles-Newton's 2nd Law

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 Equation 10.1 is the mathematical expression of Newton's Second Law and
this gives rise to another statement of Newton's Second Law, which is "If
the resultant force acting on a particle is not zero, the particle will have an
acceleration proportional to the magnitude of the resultant and in the
direction o' the resultant force'.
 Equation 10.1 is a vector relation since both the force and acceleration are
vector quantities. The scalar relations from 10.1 can be developed as,
x x
m aF   …… [10.2 (a)]
y y
m aF   …… [10.2 (b)]
z z
m aF   …… [10.2 (c)]
 Since we will normally restrict our analysis to coplanar forces, the
equations 10.2 (a) and 10.2 (b) will be mainly used.
D'ALEMBERT'S PRINCIPLE
 Referring to equation 10.1 we have
F m a  
 This is Newton's Second Law equation and relates the particle's
acceleration to the resultant of the external forces acting on the particle.
The equation is vector a equation where the resultant force vector is
equated to the ma vector.
 Transposing the R.H.S. of the equation 10.1 we get,
0F ma   …… [10.3]
 The above equation is a dynamic equilibrium equation put forth by
D'Alembert. The ma vector is treated as an inertia force and when added
with a negative sign to all other forces, results in equilibrium state of
particle.

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 Figure shows the dynamic equilibrium state of
1) A block sliding down a rough inclined plane
2) A sphere tied to a string, swinging as a pendulum to a lower position.
 Particle's acceleration can be found out by D'Alembert's principle, by
developing the figure representing a dynamic state and using the
equilibrium equations used in static viz.,
0 and 0F Fx y   
 Newton's Second Law approach to kinetics is more realistic than the
D’Alembert's principle since it does not involve inertia forces and does not
refer to an equilibrium state of a moving particle. We would therefore solve
the problems in kinetics involving forces and acceleration using Newton's
Second Law equation.

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N S L EQUATIONS APPLIED TO RECTILINEAR MOTION
 Application of Newton's equations to determine the acceleration of
rectilinear moving particles should be carried out by a systematic approach
as explained below.
step (1) Draw the FBD showing all the forces acting on the moving
particle. I more than one particle is involved, the particles may be
isolated and separate FBD may be drawn.
step (2) By the side of FBD, draw the Kinetic Diagram (KD) which shows
the particle with a ma vector acting on it. The magnitude of ma
vector is the product of particle's mass and its acceleration. ma
vector acts in the direction of particle's acceleration.
step (3) Equations of Newton's second law viz 10.2 (a) and I0.2 (b) are now
used, taking help of the FBD and KD drawn earlier. The particle's
acceleration is thus obtained.
 To understand the above outlined x steps, let us find out the
acceleration of a block of mass m which is being pulled up an
inclined plane by force P applied parallel to the plane. Let k N
be the kinetic coefficient of friction between the block and
plane. As the block moves up, the rough inclined surface
develops a frictional force k N down the plane.
 The FBD of the block showing the forces and the corresponding
Kinetic Diagram showing the ma vector is drawn as shown in
figure. Taking the x and y axes as shown and using equation
10.2 (a) of the Newton's Second Law we have,
k Substituting the value of N from equation (2) in (1)
 
 
sin cos
sin cos
k
k
P mg mg ma
P mg
a
m
  
  
  
 
 
 Thus knowing the forces acting on the particle, its acceleration
can be found out.

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EXERCISE 1
1. A 100 kg crate is projected on a horizontal floor
with an initial speed of 10 m/s. If tr1" = 0.4 and
p1 = 0.3, calculate the distance traveled and the
time taken by create to come to rest.
2. A block of mass 30 kg is placed on a plane. k Between the block and
plane is 0.3. If a force of 250 N is acting on the block, find its acceleration
for the two cases shown.
3. Find force P required to accelerate the block shown in
figure at 2.5 m/s2. Take p = 0.3.
4. Two blocks A and B are placed 15 m apart and are released simultaneously
from rest. Calculate the time taken and the distance traveled by each block
before they collide.
5. Two blocks A of weight 500 N and B of weight 300 N
are 10 m apart on an inclined plane as shown in
figure.  = O.2 for block A and 0.3 for block B. If the
blocks begin to slide down simultaneously calculate
the time and the distance travelled by each block
when block A touches block B.

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6. Two blocks A and B rest on an inclined plane as shown. Find their
acceleration on being released from rest. Also find the contact force between
the blocks.
7. Three blocks m1 and m2 and m3 of masses 1.5 kg, 2 kg and 1 kg
respectively are placed on a rough surface with  = O.2O as shown. If a
force F is applied to accelerate the blocks at 3 m/s2 what will be the force
that 1.5 kg block exerts on 2 kg block.
8. A package of total weight 4OO0 N is being pulled up a slope by a steel cable
exerting a force F = (50 t2 + 25OO) N. Knowing that the velocity of the
package was 5 m/s at t = 0, find the acceleration and velocity of the
package at t=8 sec.
9. An airplane of mass 30,000 kg starts from rest and runs on the runway for
1500 m before it takes off. If the airplane engines developed a constant
thrust of 50 kN and the air resistance force acting on the plane is F = 3 v2
(where F is in Newton and v is in m/s), find the takeoff velocity of the
airplane.

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10. A particle of mass 3 kg initially at rest at the origin is set in rectilinear
motion by a force varying with time as per the graph shown. Find
a) The velocity and position of the particle at t =1O sec.
b) The time and position at which the particle comes to rest
c) The time at which the particle comes again to its original position.
11. A locomotive train weighing 5OOO kN exerts a constant pull of 120 kN. The
train starts from rest on a level track and attains a speed of 9O kmph, after
which steam is suddenly shut off, slowly bringing it to a halt. If the tractive
resistance is 15 N/kN of the eight of the train, find the total distance
covered by the train and the total time to cover this distance. What is the
maximum power developed by the engine?
12. A vertical lift of total mass 750 kg acquires an upward velocity of 3 m/s
over a distance of 4m moving with constant acceleration starting from rest.
Calculate the tension in the cable.
13. A 400 kg lift-carrying a 60 kg person travels vertically up starting from rest
and acquires a velocity of 4m/s in a distance of 3m of motion. Find the
tension in the cable supporting the lift and the force transmitted by the
person on the lift floor. Does the person feel normal or heavy or light.
a) The lift is now brought to a halt from the speed of 4 mf s, in 2 sec time.
What is the force exerted by the man on the floor of the lift and how does
he feel now.
14. Two blocks A and B are connected by a rope and move on a rough
horizontal plane under a pull of 60 N as shown. If p is 0.2 between the
blocks and the surface, find the acceleration of the blocks and the tension
in the rope.

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15. Two masses l.4kg and 7kg connected by a flexible
inextensible cord rest on a plane inclined at 45 with the
horizontal as shown in figure. When the masses are
released what will be the tension T in the cord?
Assume the coefficient of friction between the plane
and the 14 kg mass as 0.25 and between the plane
and the 7kg mass as 0.375.
16. Two particles of masses mA = 5 kg and m= 10 kg
are supported as shown. acceleration of the
blocks
a) if the horizontal surface is smooth
b) if the horizontal surface is rough coefficient of kinetic friction k = Q.4
17. Calculate the vertical acceleration of the 10O kg block and the tension in
the connecting string. Neglect friction.
18. Determine the tension in the string and the velocity of 1500N block shown
in the figure 5 sec after starting from rest.
19. Determine the weight WA of block A required to
bring the system to stop in 6 sec, if at the instant
shown, the block C is moving down at 5 m/s.
Given WB = 2OO N, Wc = 600 N. Take the pulley to
be smooth.
a) Find weight WA to bring the system to stop in 5
sec., If at the instant shown, vc = 3 m/s  , WB
=150N and WC=500N.

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20. At a given instant 5O N block A is moving downwards with a speed of 1.8
m/sec. determine its speed 2 sec. later. Block 'B' has a weight 2O N, and
the coefficient of kinetic friction between it and the horizontal plane is k =
0.2. Neglect the mass of pulleys and chord. Use D’Alembert’s principle.
21. Two blocks A and B are connected by an inextensible string as shown.
Neglecting the effect of friction, determine the acceleration of each block
and tension in the cable.
22. Masses A (5 kg), B (10 kg), (2O kg) are connected as shown in the figure by
inextensible cord passing over massless and frictionless pulleys. The
coefficient of friction for masses A and B with ground is 0.2. If the system is
released from rest, find acceleration of the blocks and tension in the cords.
23. The two blocks starts from rest. The horizontal plane and the pulley are
frictionless. Determine the acceleration of each blocks and tension in the
cord.

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24. Block A (7 kg), B (12 kg), - (30 kg) are connected by an inextensible string
as shown. If the system is released from rest find the accelerations of each
block and tension in the cord. Assume smooth surfaces.

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N S L EQUATIONS APPLIED TO CURVILINEAR MOTION
 For curvilinear moving particles, the kinetic diagram shows the components
of ma vector viz, man, and mat. The man component acts towards the
centre of path while the mat component acts tangent to the path. Both the
components lie in the plane of curvilinear motion.
EXERCISE 2
1. A steel bob of mass 5 kg tied to a string of 3 m length is whirled with a
constant speed, such that the bob moves in a circle in the horizontal plane.
If the string makes an angle of 30" with the vertical, find the speed of the
bob and tension in the string.
2. A car weighing 12 kN goes round a flat curve of 90 m radius. Determine the
uniform limiting speed of the car in order to avoid outward skidding. Take µ
= 0.35
3. Figure shows two vehicles moving at 90 kmph on a road. Knowing that µ=
0.5 between road and the tyres. Determine total acceleration of each vehicle
when the brakes are suddenly applied and the wheels skid.
4. A vehicle of weight 8OO0 N travels with a constant speed of 72 kmph over a
vertical parabolic curve as shown. Find the pressure exerted by the tyres on
the road at the peak B.

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5. In a roller coaster ride, the coaster traveling around the loop ABC needs to
have a certain minimum velocity at the peak B. If the loop diameter is 10
m, find this velocity.
6. A pendulum of mass 600 gms, has a speed of 3.6 m/s at the position
shown. Find the tension in the string and the total acceleration at this
instant.
EXERCISE 3
Theory Questions
Q.1 State Newton's Second Law of Motion.
Q.2 Derive the mathematical expression of Newton's Second Law. Explain
D'Alembert's Principle.
UNIVERSITY QUESTIONS
1. A ball thrown down the incline strike the incline at a distance of 75 m
along. If the ball rises 20 m above the point of projection. Complete the
initial velocity and angle of projection with horizontal. (10 Marks)

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2. The system shown in figure is
released from rest. What is the height
lost by bodies A, B and C in 2
seconds? Take coefficient of kinetic
friction at rubbing surfaces as o.4.
Find also the tension in wires.
(10 Marks)
3. A 50 kg block kept on a 150 inclined plane is pushed down the plane with
an initial velocity of 20 m/s. If kμ =0.4, determine the distance travelled by
the block and the time it will take as it comes to rest. (4 Marks)
4. Two block mA = 10 kg and mB = 5 kg connected with cord and pulley
system as shown in fig. Determine the velocity of each block when system
is started from rest and block B gets displacement by 2 m. Take kμ =0.2
between block A and Horizontal surface. (6 Marks)
5. A body of mass 25 kg resting on a horizontal
table is connected by string passing over a
smooth pulley at the edge of the table to another
body of mass 3.75 kg and hanging vertically as
shown. Initially, the friction between the mass A
and the table is just sufficient to prevent the motion. If an additional 1.25
kg is added to the 3.75 kg mass, find the acceleration of the masse
(4 Marks)
6. State D’ Alembert’s principle with two examples. (4 Marks)

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7. Find out min. value of P to start the motion. (8 Marks)
8. Sphere A is supported by two writes AB, AC. Find
out tensions in wire AC:-
(i) before AB is cut
(ii) just after AB is cut. (4 Marks)
9. A block of mass 5kg is released from rest along a 40 degree inclined plane.
Determine the acceleration of the block when it travels a distance of 3m
using D’ Alemberts principle. Take coefficient of friction as 0.2. (4 Marks)
10. The mass of A is 23kg and mass of B is 36kg.
The coefficient are 0.4 between A and B, and 0.2
between ground and block B. Assume smooth
drum. Determine the maximum mass of M at
impending motion. (8 Marks)

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