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KINETICS OF PARTICLES: ENERGY AND MOMENTUM METHODS. Consider a force F acting on a particle A . The work of F corresponding to the small displacement d r is defined as. A 2. ds. s 2. a. A. d r. F. A 1. dU = F d r. s. s 1.

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KINETICS OF PARTICLES: ENERGY AND MOMENTUM METHODS

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KINETICS OF PARTICLES: ENERGY AND MOMENTUM METHODS

Consider a force F acting on a particle A. The work of Fcorresponding to the small displacement dr is defined as

A2

ds

s2

a

A

dr

F

A1

dU = Fdr

s

s1

Recalling the definition of scalar product of two vectors,

dU = Fds cosa

where a is the angle between F and dr.

dU = Fdr = Fds cosa

A2

ds

s2

a

The work of F during a finite displacement from A1 to A2 , denoted by U1 2 , is obtained

A

dr

F

A1

s

s1

by integrating along the path described by the particle.

A2

ٍ

U1 2 = Fdr

A1

For a force defined by its rectangular components, we write

A2

ٍ

U1 2 = (Fxdx + Fydy + Fzdz)

A1

A2

The work of the weight W of a body as its center of gravity moves from an elevation y1to y2 is obtained by setting Fx = Fz = 0 and

Fy = - W.

W

dy

y2

A

A1

y

y1

y2

ٍ

U1 2 = - Wdy = Wy1 - Wy2

y1

The work is negative when the elevation increases, and

positive when the elevation decreases.

A

O

x

2

U

= -

k x dx

A

x

1

1 2

1

x

1

1

1

=

kx

- kx

1

2

2

2

A

x

2

2

The work of the force F exerted by a spring on a body A during a finite displacement of the body from A1 (x = x1) to A2 (x = x2) is obtained by writing

spring undeformed

B

dU = -Fdx = -kx dx

ٍ

B

F

2

2

A

x

The work is positive when the spring is returning to its undeformed position.

.

B

A2

The work of the gravitational force F exerted by a particle of mass M located at O on a particle of mass m as the latter moves from A1 to A2 is obtained from

A’

dr

m

r2

A

r2

r

ٍ

GMm

r2

dr

F

U1 2 =

dq

r1

A1

-F

GMm

r2

GMm

r1

r1

q

M

= -

O

The kinetic energy of a particle of mass m moving with a velocity v is defined as the scalar quantity

1

2

T = mv2

From Newton’s second law the principle of work and energy is derived. This principle states that the kinetic energy of a particle atA2can be obtained by adding to its kinetic energy at A1 the work done during the displacement from A1 to A2 by the force F exerted on the particle:

T1 + U1 2 = T2

The power developed by a machine is defined as the time rate at which work is done:

dU

dt

Power = = F v

where F is the force exerted on the particle and v is the velocity of the particle. The mechanical efficiency, denoted by h, is expressed as

power output

power input

h =

When the work of a force F is independent of the path followed, the force F is said to be a conservative force, and its work is equal to minus the change in the potential energy V associated with F :

U1 2 = V1 - V2

The potential energy associated with each force considered earlier is

Vg = Wy

Force of gravity (weight):

GMm

r

Vg = -

Gravitational force:

1

2

Ve = kx2

Elastic force exerted by a spring:

U1 2 = V1 - V2

This relationship between work and potential energy, when combined with the relationship between work and kinetic energy (T1 + U1 2 = T2) results in

T1 + V1 = T2 + V1

This is the principle of conservation of energy, which states that when a particle moves under the action of conservative forces, the sum of its kinetic and potential energies remains constant. The application of this principle facilitates the solution of problems involving only conservative forces.

1

2

1

2

v

When a particle moves under a central force F, its angular momentum about the center of force O remains constant. If the central force F is also conservative, the principles of conservation of angular momentum and conservation of energy can be used jointly to analyze the motion of the particle. For the case of oblique launching, we have

f

P

v0

r

f0

r0

O

P0

(HO)0 = HO : r0mv0 sin f0 = rmv sin f

GMm

r

GMm

r0

T0 + V0 = T + V :

mv2 - = mv2 -

0

where m is the mass of the vehicle and M is the mass of the earth.

The linear momentum of a particle is defined as the product mv of the mass m of the particle and its velocity v. From Newton’s second law, F = ma, we derive the relation

t2

ٍ

mv1 + Fdt = mv2

t1

where mv1 and mv2 represent the momentum of the particle at a time t1 and a time t2 , respectively, and where the integral defines the linear impulse of the force F during the corresponding time interval. Therefore,

mv1 + Imp1 2 = mv2

which expresses the principle of impulse and momentum for a particle.

When the particle considered is subjected to several forces, the sum of the impulses of these forces should be used;

mv1 + SImp1 2 = mv2

Since vector quantities are involved, it is necessary to consider their x and y components separately.

The method of impulse and momentum is effective in the study of impulsive motion of a particle, when very large forces, called impulsive forces, are applied for a very short interval of timeDt, since this method involves impulses FDt of the forces, rather than the forces themselves. Neglecting the impulse of any nonimpulsive force, we write

mv1 + SFDt= mv2

In the case of the impulsive motion of several particles, we write

Smv1 + SFDt= Smv2

where the second term involves only impulsive, external forces.

In the particular case when the sum of the impulses of the external forces is zero, the equation above reduces to

Smv1 = Smv2

that is, the total momentum of the particles is conserved.

Line of

Impact

vB

B

A

Before Impact

vA

v’B

B

A

v’A

After Impact

In the case of direct central impact, two colliding bodies A and B move along the line of impact with velocities vA and vB , respectively. Two equations can be used to determine their velocities v’A and v’B after the impact. The first represents the conservation of the total momentum of the two bodies,

mAvA + mBvB = mAv’A + mBv’B

Line of

Impact

vB

B

A

Before Impact

vA

v’B

B

A

v’A

After Impact

mAvA + mBvB = mAv’A + mBv’B

The second equation relates the relative velocities of the two bodies before and after impact,

v’B - v’A = e (vA - vB )

The constant e is known as the coefficient of restitution; its value lies between 0 and 1 and depends on the material involved. When e = 0, the impact is termed perfectly plastic; when e = 1 , the impact is termed perfectly elastic.

Line of

Impact

In the case of oblique central impact, the velocities of the two colliding bodies before and after impact are resolved into n components along the line of impact and t components along the common tangent to the surfaces in contact. In the t direction,

n

t

vB

B

A

Before Impact

(vA)t = (v’A)t (vB)t = (v’B)t

vA

v’B

while in the n direction

n

mA (vA)n + mB (vB)n =

mA (v’A)n + mB (v’B)n

t

v’A

B

vB

(v’B)n - (v’A)n = e [(vA)n - (vB)n]

A

After Impact

vA

Line of

Impact

n

(vA)t = (v’A)t (vB)t = (v’B)t

t

mA (vA)n + mB (vB)n =

mA (v’A)n + mB (v’B)n

vB

B

A

Before Impact

(v’B)n - (v’A)n = e [(vA)n - (vB)n]

vA

v’B

n

Although this method was developed for bodies moving freely before and after impact, it could be extended to the case when one or both of the colliding bodies is constrained in its motion.

t

v’A

B

vB

A

After Impact

vA