Momentum

1. Momentum Conservation of Force

2. Impulse and Momentum • Impulse and Momentum

3. Impulse and Momentum • Impulse and Momentum • The right side of the equation FΔt = mΔv, mΔv involves the change in velocity: Δv = vf − vi • Therefore, mΔv = mvf − mvi • The product of the object’s mass, m, and the object’s velocity, v, is defined as the momentum of the object. Momentum is measured in kg·m/s. An object’s momentum, also known as linear momentum, is represented by the following equation: p = mv

4. Impulse and Momentum • Impulse and Momentum • Because mvf = pf and mvi = pi: • pf − pi, describes change in momentum of an object • The right side of this equation, pf − pi, 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: FΔt = mΔv = pf − pi • FΔt = pf − pi

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

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 N·s. The direction of the impulse is in the direction of the force. Therefore, the change in momentum of the ball is also 13.1 N·s

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 baseball’s final velocity?

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 Newton’s law for rotating motion, τ = IΔω/Δt • This equation can be rearranged in the same way as Newton’s second law of motion was, to produce τΔt = IΔω • The left side of this equation is the angular impulse of the rotating object and the right side can be rewritten as Δω = ωf− ωi

9. Impulse and Momentum • Angular Momentum • The angular momentum of an object is equal to the product of a rotating object’s moment of inertia and angular velocity. Angular momentum is measured in kg·m2/s L = Iω

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 object’s angular momentum, which is called the angular impulse-angular momentum theorem • The angular impulse-angular momentum theorem is represented by the following equation: τΔt = Lf− Li

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 object’s mass cannot be changed, if its momentum is constant, then its velocity is also constant

12. Impulse and Momentum • Angular Momentum • In the case of angular momentum, however, the object’s angular velocity does not remain constant. • This is because the moment of inertia depends on the object’s 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

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, Δt, 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

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

15. Impulse and Momentum • Angular Momentum

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?

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 107 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 103 kg, is heading east at 20.0 m/s, and car 2, with a mass of 1.45 x 103 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?

18. Conservation of Momentum • Two-Particle Collisions

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

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

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?

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

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

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

25. Conservation of Momentum Before After pCi+pDi=pCf+pDf 0 = pCf+pDf pCf= −pDf mCvCf= −mDvDf Recoil

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

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 Newton’s third law of motion

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 person’s 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

29. Conservation of Momentum • Two-Dimensional Collisions • Two billiard balls system • Original momentum of moving ball pCi and momentum of the stationary ball zero • Momentum of system before collision equal to pCi • After collision, both billiard balls moving and have momenta • If friction ignored, system closed and isolated

30. Conservation of Momentum pCi = pCf+pDf pCf, y + pDf, y = 0 pCi= pCf, x + pDf, x • 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

31. Conservation of Momentum Lf = Li • 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 • Earth’s angular momentum constant and conserved. So, length of a day does not change

32. Conservation of Momentum Li = Lf so, Iiωi = Ifωf • Conservation of Angular Momentum • When ice skater pulls in arms, he begins spinning faster • Without external torque, angular momentum does not change; L = Iω 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

33. Conservation of Momentum • Conservation of Angular Momentum • Frequency is f = ω/2π, so:

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

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

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

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

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?

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?