1. Engineering Mechanics
L22: Relative Velocity and System of Particles
Indian Institute of Technology Jodhpur

2. Relative Motion
All measurement were with reference to a fixed frame of reference attached on Earth.
Is earth really stationary?
It has rotation about its own axis and also revolve around sun
Acceleration of centre of earth around sun is m/s2
Acceleration of a point on equator at sea level with reference to centre of earth considered fixed is m/s2
These accelerations are small in comparison to g and hence generally ignored on earth.

3. RELATIVE MOTION Motion of particle A is observed from axes x-y-z
Frame x-y-z translates with respect to Fixed frame X-Y-Z
 F Z X z x
m A rA Y rB B O
r 𝐴/𝐵 = r 𝑟𝑒𝑙 y
𝐚 𝐴 = 𝐚 𝐵 + 𝐚 rel
Σ𝐅=𝑚 𝐚 𝐴
Σ𝐅=𝑚 𝐚 𝐵 + 𝐚 rel
Σ𝐅≠𝑚 𝐚 rel
We can conclude immediately that Newton’s second law does
not hold with respect to an accelerating system

4. D’Alembert’s Principle
When acceleration of particle is observed from fixed set of axes X-Y
Σ𝐅=𝑚𝐚
When motion of particle is observed from moving set of axes x-y the particle appear to be at rest in x-y
Thus, the observer who is accelerating with x-y-z concludes that a force −ma acts on the particle to balance Σ F.
This point of view, allows the treatment of a dynamics problem by the methods of statics by rewriting equation as
Σ𝐅−𝑚𝐚=0
This is D’Alembert’s Principle. −𝑚𝐚 is fictitious force is known as the inertia force

5. Example: Conical Pendulum
Newton’s Second Law
Σ 𝐅 𝑛 =𝑚 𝐚 𝑛
𝑇 sin 𝜃 =𝑚𝑟 𝜔 2
Σ 𝐅 𝑦 =0
𝑇 cos 𝜃−𝑚𝑔 =0
D’Alembert’s Principle
Σ𝐅−𝑚 𝐚 𝑛 =0
𝑇 sin 𝜃 −𝑚𝑟 𝜔 2 =0

6. Constant-Velocity, Nonrotating Systems
𝐚 𝐴 = 𝐚 𝐵 + 𝐚 rel
Newton’s Second Law
𝐚 𝐵 =0
Σ𝐅=𝑚 𝐚 rel
𝐚 𝐴 = 𝐚 rel
Newton’s second law holds for measurements made in a system moving with a constant velocity. z y m
Path relative
to x-y-z x Z Y O B X
 F
a 𝐴 = a 𝑟𝑒𝑙
V 𝑟𝑒𝑙
dr 𝑟𝑒𝑙
r 𝑟𝑒𝑙
Such a system is known as an inertial system or as a Newtonian frame of reference.

7. Constant-Velocity, Nonrotating Systems
Work-Energy
𝑑 𝑼 𝑟𝑒𝑙 = Σ𝐅. 𝑑 𝐫 𝑟𝑒𝑙
Σ𝐅=𝑚 𝐚 rel
𝑑 𝑼 𝑟𝑒𝑙 = 𝑚 𝐚 rel . 𝑑 𝐫 𝑟𝑒𝑙
𝐚 𝑟𝑒𝑙 .𝑑 𝐫 𝑟𝑒𝑙 = 𝐯 𝑟𝑒𝑙 .𝑑 𝐯 𝑟𝑒𝑙
𝑑 𝑼 𝑟𝑒𝑙 = 𝑣 𝑟𝑒𝑙 𝑑 𝑣 𝑟𝑒𝑙
( 𝑎 𝑡 𝑑𝑠=𝑣𝑑𝑣)
=𝑑 1 2 𝑚 v 𝑟𝑒𝑙 2 z y m
Path relative
to x-y-z x Z Y O B X
 F
a 𝐴 = a 𝑟𝑒𝑙
V 𝑟𝑒𝑙
dr 𝑟𝑒𝑙
r 𝑟𝑒𝑙
𝑇 𝑟𝑒𝑙 = 1 2 𝑚 v 𝑟𝑒𝑙 2
𝑈 𝑟𝑒𝑙 = 𝑑𝑇 𝑟𝑒𝑙
or 𝑈 𝑟𝑒𝑙 = ∆𝑇 𝑟𝑒𝑙
work-energy equation holds for measurements made relative to a constant-velocity, nonrotating system.

8. Constant-Velocity, Nonrotating Systems
Linear Impulse-Momentum
Σ𝐅=𝑚 𝐚 rel
Σ𝐅𝑑𝑡=𝑚 𝐚 rel 𝑑𝑡
Σ𝐅𝑑𝑡= 𝑑(𝑚 𝐯 𝑟𝑒𝑙
𝑚 𝐚 rel 𝑑𝑡=𝑚 𝑑 𝐯 𝑟𝑒𝑙
Σ𝐅𝑑𝑡=𝑑 𝐆 𝑟𝑒𝑙
𝐆 𝑟𝑒𝑙 = 𝑚 𝐯 𝑟𝑒𝑙 z y m
Path relative
to x-y-z x Z Y O B X
 F
a 𝐴 = a 𝑟𝑒𝑙
V 𝑟𝑒𝑙
dr 𝑟𝑒𝑙
r 𝑟𝑒𝑙
Σ𝐅= 𝐆 𝑟𝑒𝑙
Σ𝐅 𝑑𝑡=∆ 𝑮 𝑟𝑒𝑙
Linear Impulse-Momentum equation holds for measurements made relative to a constant-velocity, nonrotating system.

9. Constant-Velocity, Nonrotating Systems
Angular Impulse-Momentum
𝐇 B 𝑟𝑒𝑙 = 𝐫 𝑟𝑒𝑙 × 𝐆 𝑟𝑒𝑙
𝐇 B 𝑟𝑒𝑙 = 𝐫 𝑟𝑒𝑙 × 𝐆 𝑟𝑒𝑙 + 𝐫 𝑟𝑒𝑙 × 𝐆 𝑟𝑒𝑙
𝐇 B 𝑟𝑒𝑙 = 𝐫 𝑟𝑒𝑙 × 𝐆 𝑟𝑒𝑙
𝐆 𝑟𝑒𝑙 =Σ𝐅 z y m
Path relative
to x-y-z x Z Y O B X
 F
a 𝐴 = a 𝑟𝑒𝑙
V 𝑟𝑒𝑙
dr 𝑟𝑒𝑙
r 𝑟𝑒𝑙
𝐇 B 𝑟𝑒𝑙 = 𝐫 𝑟𝑒𝑙 ×Σ𝐅
𝐇 B 𝑟𝑒𝑙 =Σ 𝐌 𝐵
Angular Impulse-Momentum equation holds for measurements made relative to a constant-velocity, nonrotating system.

10. Caution
Although the work-energy and impulse-momentum equations hold relative to a system translating with a constant velocity, the individual expressions for work, kinetic energy, and momentum differ between the fixed and the moving systems, i.e,
𝑑𝑈=Σ𝐅. 𝑑 𝐫 𝐴 )≠(𝑑 𝑼 𝑟𝑒𝑙 =Σ𝐅. 𝑑 𝐫 𝑟𝑒𝑙
𝑇= 1 2 𝑚 𝑣 𝐴 2 )≠(𝑇 𝑟𝑒𝑙 = 1 2 𝑚 𝑣 𝑟𝑒𝑙 2
𝐆=𝑚 𝐯 𝐴 )≠( 𝐺 𝑟𝑒𝑙 = 𝑚 𝐯 𝑟𝑒𝑙

11. Example 1
A simple pendulum of mass m and length r is mounted on the flatcar, which has a constant horizontal acceleration a0 as shown. If the pendulum is released from rest relative to the flatcar at the position 𝜃 = 0, determine the expression for the tension T in the supporting light rod for any value of 𝜃. Also find T for 𝜃=𝜋/2 and 𝜃=𝜋 .

12. Example 1

13. System of Particles
We focused primarily on the kinetics of a single particle in previous chapter
Next major step in the development of dynamics is to extend these principles, to describe the motion of a general system of particles.

14. NEWTON’S SECOND LAW
Forces F1, F2, F3, acting on mi from sources external to
the envelope
Forces f1, f2, f3, acting on mi from sources internal to the system boundary
For ith particle
For all particles

15. WORK-ENERGY

16. Linear Impulse Momentum
Linear Momentum

17. Angular Impulse Momentum @ O

18. Angular Impulse Momentum @ G

19. Angular Impulse Momentum @ P

20. CONSERVATION OF ENERGY AND MOMENTUM
A mass system is said to be conservative if it does not lose energy by virtue of internal friction forces which do negative work or by virtue of inelastic members which dissipate energy upon cycling. or
Conservation of Linear Momentum
Total Linear Momentum is conserved If the resultant external force F acting on a conservative or nonconservative mass system is zero
Conservation of Angular Momentum
Total Angular Momentum is conserved, if the resultant moment about a fixed point O all external forces on any mass system is zero

21. Example 2

22. Example 2

23. Example 2

24. References :
Thanking you