Momentum Conservation of Force. Impulse and Momentum Impulse and Momentum.

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Transcript of Momentum Conservation of Force. Impulse and Momentum Impulse and Momentum.

  • Slide 1
  • Momentum Conservation of Force
  • Slide 2
  • Impulse and Momentum Impulse and Momentum
  • Slide 3
  • Impulse and Momentum The right side of the equation Ft = mv, mv involves the change in velocity: v = v f v i Therefore, mv = mv f mv i The product of the objects mass, m, and the objects velocity, v, is defined as the momentum of the object. Momentum is measured in kgm/s. An objects momentum, also known as linear momentum, is represented by the following equation: Impulse and Momentum p = mv
  • Slide 4
  • Impulse and Momentum Because mv f = p f and mv i = p i : p f p i, describes change in momentum of an object The right side of this equation, p f p i, describes the change in momentum of an object. Thus, the impulse on an object is equal to the change in its momentum, which is called the impulse-momentum theorem The impulse-momentum theorem is represented by the following equation: Impulse and Momentum Ft = mv = p f p i Ft = p f p i
  • Slide 5
  • Impulse and Momentum Impulse and Momentum If the force on an object is constant, the impulse is the product of the force multiplied by the time interval over which it acts Because velocity is a vector, momentum also is a vector Similarly, impulse is a vector because force is a vector This means that signs will be important for motion in one dimension
  • Slide 6
  • Impulse and Momentum Impulse-Momentum Theorem Look at the change in momentum of a baseball. The impulse, that is the area under the curve, is approximately 13.1 Ns. The direction of the impulse is in the direction of the force. Therefore, the change in momentum of the ball is also 13.1 Ns
  • Slide 7
  • Impulse and Momentum Impulse-Momentum Theorem What is the final momentum of ball after collision if initial momentum is -5.5 kg m/s? What is the baseballs final velocity?
  • Slide 8
  • Impulse and Momentum Angular Momentum The angular velocity of a rotating object changes only if torque is applied to it This is a statement of Newtons law for rotating motion, = I/t This equation can be rearranged in the same way as Newtons second law of motion was, to produce t = I The left side of this equation is the angular impulse of the rotating object and the right side can be rewritten as = f i
  • Slide 9
  • Impulse and Momentum Angular Momentum The angular momentum of an object is equal to the product of a rotating objects moment of inertia and angular velocity. Angular momentum is measured in kgm 2 /s L = I
  • Slide 10
  • Impulse and Momentum Angular Momentum Just as the linear momentum of an object changes when an impulse acts on it, the angular momentum of an object changes when an angular impulse acts on it Thus, the angular impulse on the object is equal to the change in the objects angular momentum, which is called the angular impulse- angular momentum theorem The angular impulse-angular momentum theorem is represented by the following equation: t = L f L i
  • Slide 11
  • Impulse and Momentum Angular Momentum If there are no forces acting on an object, its linear momentum is constant If there are no torques acting on an object, its angular momentum is also constant Because an objects mass cannot be changed, if its momentum is constant, then its velocity is also constant
  • Slide 12
  • Impulse and Momentum Angular Momentum In the case of angular momentum, however, the objects angular velocity does not remain constant. This is because the moment of inertia depends on the objects mass and the way it is distributed about the axis of rotation or revolution Thus, the angular velocity of an object can change even if no torques are acting on it
  • Slide 13
  • Impulse and Momentum Angular Momentum The diver uses the diving board to apply an external torque to her body Then, the diver moves her center of mass in front of her feet and uses the board to give a final upward push to her feet This torque acts over time, t, and thus increases the angular momentum of the diver Before the diver reaches the water, she can change her angular velocity by changing her moment of inertia. She may go into a tuck position, grabbing her knees with her hands
  • Slide 14
  • Impulse and Momentum Angular Momentum By moving her mass closer to the axis of rotation, the diver decreases her moment of inertia and increases her angular velocity When she nears the water, she stretches her body straight, thereby increasing the moment of inertia and reducing the angular velocity As a result, she goes straight into the water
  • Slide 15
  • Impulse and Momentum Angular Momentum
  • Slide 16
  • Impulse and Momentum Impulse and Momentum A golfer uses a club to hit a 45 g golf ball resting on an elevated tee, so that the golf ball leaves the tee at a horizontal speed of 38 m/s. What is the impulse on the golf ball? What is the average force that the club exerts on the golf ball if they are in contact for 2.0 x 10 -3 s? What average force does the golf ball exert on the club during this time interval?
  • Slide 17
  • Impulse and Momentum Impulse and Momentum A single uranium atom has a mass of 3.97 x 10 -25 kg. It decays into the nucleus of a thorium atom by emitting an alpha particle at a speed of 2.10 x 10 7 m/s. The mass of an alpha particle is 6.68 x 10 -27 kg. What is the recoil speed of the thorium nucleus? Two cars enter an icy intersection. Car 1, with a mass of 2.50 x 10 3 kg, is heading east at 20.0 m/s, and car 2, with a mass of 1.45 x 10 3 kg is going north at 30.0 m/s. The two vehicles collide and stick together. What is the speed and direction of the cars as they skid away together just after colliding?
  • Slide 18
  • Conservation of Momentum Two-Particle Collisions
  • Slide 19
  • Conservation of Momentum Momentum in a Closed, Isolated System Under what conditions is momentum of system of two balls conserved? First condition: no balls lost and no balls gained. Closed system: one which does not gain or lose mass Second condition: forces internal (no forces acting on system by objects outside it) When net external force on closed system zero, system is an isolated system
  • Slide 20
  • Impulse and Momentum Momentum in a Closed, Isolated System No system on Earth absolutely isolated, because always some interactions between system and its surroundings Often interactions small enough to be ignored Systems contain any number of objects, and objects stick together or come apart in collision Law of conservation of momentum: momentum of any closed, isolated system does not change
  • Slide 21
  • Conservation of Momentum Momentum in a Closed, Isolated System A 1875-kg car going 23 m/s rear-ends a 1025-kg compact car going 17 m/s on ice in the same direction. The two cars stick together. How fast do the two cars move together immediately after the collision?
  • Slide 22
  • Conservation of Momentum Recoil The momentum of a baseball changes when the external force of a bat is exerted on it. The baseball, therefore, is not an isolated system On the other hand, the total momentum of two colliding balls within an isolated system does not change because all forces are between the objects within the system
  • Slide 23
  • Conservation of Momentum Recoil Assume that a girl and a boy are skating on a smooth surface with no external forces. They both start at rest, one behind the other. Skater C, the boy, gives skater D, the girl, a push. Find the final velocities of the two in-line skaters
  • Slide 24
  • Conservation of Momentum Recoil After clashing with each other, both skaters are moving, making this situation similar to that of an explosion. Because the push was an internal force, you can use the law of conservation of momentum to find the skaters relative velocities The total momentum of the system was zero before the push. Therefore, it must be zero after the push
  • Slide 25
  • Conservation of Momentum Recoil Before p Ci + p Di = p Cf + p Df 0 = p Cf + p Df p Cf = p Df m C v Cf = m D v Df After
  • Slide 26
  • Conservation of Momentum Recoil Are the skaters velocities equal and opposite? The last equation, for the velocity of skater C, can be rewritten as follows: Velocities depend on skaters relative masses. Less massive skater moves at greater velocity
  • Slide 27
  • Conservation of Momentum Propulsion in Space How does a rocket in space change its velocity? Rocket carries both fuel and oxidizer. When the fuel and oxidizer combine in rocket motor, resulting hot gases leave exhaust nozzle at high speed Rocket and chemicals are closed system Forces that expel gases internal forces, so system also isolated Thus, objects in space accelerate using law of conservation of momentum and Newtons third law of motion
  • Slide 28
  • Conservation of Momentum Propulsion in Space Deep Space 1 performed flyby of asteroid Braille in 1999 Had ion engine that exerted as much force as sheet of paper resting on persons hand In ion engine, xenon atoms expelled at speed of 30 km/s, produced force of only 0.092 N. Runs continuously for days, weeks, or months Impulse delivered large enough to increase momentum
  • Slide 29
  • Conservation of Momentum Two-Dimensional Collisions Two billiard balls system Original momentum of moving ball p Ci and momentum of the stationary ball zero Momentum of system before collision equal to p Ci After collision, both billiard balls moving and have momenta If friction ignored, system closed and isolated
  • Slide 30
  • Conservation of Momentum Two-Dimensional Collisions Law of conservation of momentum used Initial momentum equals vector sum of final momenta. So: Components of vectors before and after collision equal. X-axis in direction of initial momentum. Y-component of initial momentum zero Sum of final y-components also zero Sum of horizontal components equal to initial momentum p Ci = p Cf + p Df p Cf, y + p Df, y = 0 p Ci = p Cf, x + p Df, x
  • Slide 31
  • Conservation of Momentum Conservation of Angular Momentum Like linear momentum, angular momentum can be conserved The law of conservation of angular momentum states that if no net external torque acts on an object, then its angular momentum does not change Initial angular momentum equal final angular momentum Earths angular momentum constant and conserved. So, length of a day does not change L f = L i
  • Slide 32
  • Conservation of Momentum Conservation of Angular Momentum When ice skater pulls in arms, he begins spinning faster Without external torque, angular momentum does not change; L = I constant Increased angular velocity makes decreased moment of inertia By pulling arms close to body, ice-skater brings more mass closer to axis of rotation, decreasing radius of rotation and decreasing his moment of inertia L i = L f so, I i i = I f f
  • Slide 33
  • Conservation of Momentum Conservation of Angular Momentum Frequency is f = /2, so:
  • Slide 34
  • Conservation of Momentum Conservation of Angular Momentum If torque-free object starts with no angular momentum, must continue with no angular momentum Thus, if part of an object rotates in one direction, another part must rotate in the opposite direction For example, if you switch on a loosely held electric drill, the drill body will rotate in the direction opposite to the rotation of the motor and bit
  • Slide 35
  • Conservation of Momentum Tops and Gyroscopes Because of conservation of angular momentum, direction of rotation of a spinning object can be changed only by applying a torque When a top is vertical, there is no torque on it, and the direction of its rotation does not change If the top is tipped a torque tries to rotate it downward. Rather than tipping over, however, the upper end of the top revolves, or precesses slowly about the vertical axis
  • Slide 36
  • Conservation of Momentum Tops and Gyroscopes Gyroscope wheel or disk that spins rapidly around one axis while being free to rotate around one or two other axes Direction of large angular momentum changed only by applying appropriate torque Without torque, direction of axis of rotation does not change
  • Slide 37
  • Conservation of Momentum Tops and Gyroscopes Gyroscopes are used in airplanes, submarines, and spacecraft to keep an unchanging reference direction Giant gyroscopes are used in cruise ships to reduce their motion in rough water. Gyroscopic compasses, unlike magnetic compasses, maintain direction even when they are not on a level surface
  • Slide 38
  • Conservation of Momentum Momentum At 9.0 s after takeoff, a 250 kg rocket attains a vertical velocity of 120 m/s. What is the impulse on the rocket? What is the average force on the rocket? What is its altitude?
  • Slide 39
  • Conservation of Momentum Momentum In a circus act, a 18 kg dog is trained to jump onto a 3.0 kg skateboard moving with a velocity of 0.14 m/s. At what velocity does the dog jump onto the skateboard if afterward the velocity of the dog and skateboard is -10.0 m/s?