T1 + U1-2 = T2

Conservation of Energy

Principle of Work and Energy

CHAPTER 15

Kinetics of a Particle: Impulse and Momentum

Objectives

1. To develop the principle of linear impulse and momentum for a particle

2. To study the conservation of linear momentum for a particle

3. To analyze the mechanics of impact

T1 + U1-2 = T2

Principle of Work and Energy

v = ds/dta = dv/dt

dt = ds/vdt = dv/a

ads = vdv

a = dv/dtads = vdv

Principle of Linear Impulse and Momentum

Linear Impulse

Variable Force Constant Force

+ =

Principle of Linear Impulse and Momentum

a particle

system of particles

100 kg stone is originally at rest on the smooth horizontal space.If a towing force of 200 N, acting at an angle of 45o, is appliedto the stone for 10 s, determine the final velocity and the normal forcewhich the surface exerts on the stone during the time interval.

ex1

Principle of Impulse and momentum

+

Principle of Impulse and momentum

+

The 250 N crate is acted upon by a force having a variable magnitude P = 100t, where t is in seconds. Determine the crate’s velocity 2 s afterP has been applied. The initial velocity is v1 = 1 m/s down the plane,and k = 0.3.

Principle of Linear Impulse and Momentum

a particle

system of particles

Conservation of Linear Momentum for a System of Particles

When the sum of the external impulses acting on a system of a particleis zero,

Nonimpulsive force : - impulse that may be neglected - considered equal to zero

Impulsive : - forces that are very large and act for a very short period of time

produce significant change in momentum

The force of the rackets on the ball is impulsiveThe ball’s weight will have the negligible effect, nonimpulsive

Boxcar A : 15 MgBoxcar B : 12 Mg

Ex1

If the car meets and couple together, determine(a) the speed of both cars just after the coupling(b) the average force between them if the coupling takes place in 0.8 s

Conservation of Linear Momentum

(a) the speed of both cars just after the coupling

Principles of Impulse and Momentum

(b) the average force between them if the coupling takes place in 0.8 s

The 600 kg cannon shown fires a 4 kg projectile with a muzzle velocity450 m/s relative to the ground. If firing takes place in 0.03 s, determine

(a) the recoil velocity of the cannon just after firing(b) the average impulsive force acting on the projectile

The cannon support is fixed to the ground, and the horizontal recoil of the cannon is absorbed by two springs.

Ex 2

Conservation of Linear Momentum

Mc = 600 [kg]Mp = 4 [kg](Vp )1 = 450 [m/s]t = 0.03 [s]

Priciples of Impulse and Momentum

Lab test 2 : 12/04/07 (Khamis)

Mid-Term 2 : 16/04/07 (Isnin)

Objectives

1. To develop the principle of linear impulse and momentum for a particle

2. To study the conservation of linear momentum for a particle

3. To analyze the mechanics of impact

Impact

Impact occurs when two bodies collide with each other during a veryshort period of time, causing relatively large (impulsive ) forces to beexerted between the bodies.

Central Impact

Coefficient of Restitution e

e = 1 : Perfectly elastic impact

e = 0 : Plastic impact / inelastic impact

Ex

The bag A, having a mass of 6 kg, is released from rest at the position = 0o. After falling to = 90o, it strikes an 18 kg box B. If the coefficientof restitution between the bag and box is e = 0.5, determine

(i) the velocities of the bag and box just after impact(ii) the loss of energy during collision

Conservation of Energy

6 kg

Datum at = 0o

(i) the velocities of the bag and box just after impact

Conservation of Momentum

(i)

6 kg

Coefficient of restitution

(ii)

Solving (i) and (ii), (vA)2 = - 0.554 m/s = 0.554 m/s (vB)2 = 1.66 m/s

Loss of Energy

(ii) the loss of energy during collision

Oblique Impact

The particles move away withvelocities having unknown directionsand unknown magnitudes.

(vA)2,

(vB)2

2 2

If v1 velocities are known;

Ex

The smooth disks A and B, having a mass of 1 kg and 2 kg, respectively,collide with the velocities shown in the figure. If e = 0.75, determine the x and y components of the final velocity of each disk just after collision.

x components

y components

Conservation of “x” Momentum

(i) for x components of the final velocity;

(i)

Coefficient of Restitution (x)

Both disk are assumed to have components of velocity in the +x direction after collision

(ii)

Solving (i) and (ii),

Ans.

Ans.

(i) for x components of the final velocity;

(ii) for y components of the final velocity;

Conservation of “y” Momentum

A man kicks the 200g ball such that it leaves the ground at an angle of 30oWith the horizontal and strikes the ground at the same elevation a distance of 15m away. Determine the impulse of hiss foot F on the ball. Neglect theImpulse caused by the ball’s weight while its being kicked.