t 1 + u 1-2 = t 2 conservation of energy principle of work and energy
TRANSCRIPT
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T1 + U1-2 = T2
Conservation of Energy
Principle of Work and Energy
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CHAPTER 15
Kinetics of a Particle: Impulse and Momentum
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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
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T1 + U1-2 = T2
Principle of Work and Energy
v = ds/dta = dv/dt
dt = ds/vdt = dv/a
ads = vdv
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a = dv/dtads = vdv
Principle of Linear Impulse and Momentum
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Linear Impulse
Variable Force Constant Force
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+ =
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Principle of Linear Impulse and Momentum
a particle
system of particles
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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
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Principle of Impulse and momentum
+
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Principle of Impulse and momentum
+
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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.
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Principle of Linear Impulse and Momentum
a particle
system of particles
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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
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The force of the rackets on the ball is impulsiveThe ball’s weight will have the negligible effect, nonimpulsive
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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
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Conservation of Linear Momentum
(a) the speed of both cars just after the coupling
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Principles of Impulse and Momentum
(b) the average force between them if the coupling takes place in 0.8 s
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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
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Conservation of Linear Momentum
Mc = 600 [kg]Mp = 4 [kg](Vp )1 = 450 [m/s]t = 0.03 [s]
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Priciples of Impulse and Momentum
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Lab test 2 : 12/04/07 (Khamis)
Mid-Term 2 : 16/04/07 (Isnin)
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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
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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.
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Central Impact
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Coefficient of Restitution e
e = 1 : Perfectly elastic impact
e = 0 : Plastic impact / inelastic impact
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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
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Conservation of Energy
6 kg
Datum at = 0o
(i) the velocities of the bag and box just after impact
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Conservation of Momentum
(i)
6 kg
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Coefficient of restitution
(ii)
Solving (i) and (ii), (vA)2 = - 0.554 m/s = 0.554 m/s (vB)2 = 1.66 m/s
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Loss of Energy
(ii) the loss of energy during collision
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Oblique Impact
The particles move away withvelocities having unknown directionsand unknown magnitudes.
(vA)2,
(vB)2
2 2
If v1 velocities are known;
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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.
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x components
y components
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Conservation of “x” Momentum
(i) for x components of the final velocity;
(i)
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Coefficient of Restitution (x)
Both disk are assumed to have components of velocity in the +x direction after collision
(ii)
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Solving (i) and (ii),
Ans.
Ans.
(i) for x components of the final velocity;
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(ii) for y components of the final velocity;
Conservation of “y” Momentum
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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.