Linear Momentum

  • Momentum is a vector describing how difficult it is to stop a moving object

  • Total momentum is the sum of individual momenta

  • C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image118.png

  • Units are kg·m/s or N·s

Angular Momentum

  • C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image252.png

  • Total angular momentum is the sum of individual angular momenta

  • C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image253.png

  • Units are kg·m2/s

Calculating Angular Momentum

  • C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image254.png

    C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image255.png

Spin Angular Momentum

  • For an object rotating about its center of mass

    • C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image256.png

  • This is known as an object's spin angular momentum

  • Spin angular momentum is constant regardless of your reference point

Example 1: Object in Circular Orbit

  • Find the angular momentum of a planet orbiting the sun. Assume a perfectly circular orbit

    C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image257.png

Example 2: Angular Momentum of a Point Particle

C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image258.png

  • Find the angular momentum for a 5-kg point particle located at (2,2) with a velocity of 2 m/s east

  • About point O at (0,0)

    C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image259.png

  • About point P at (2,0)

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  • About point Q at (0,2)

    Ita /:

Angular Momentum and Net Torque

C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image262.png

Conservation of Angular Momentum

  • Spin angular momentum, the product of an object's moment of inertia and its angular velocity about the center of mass, is conserved in a closed system with no external net toques applied

    • C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image263.png

Example 3: Ice Skater Problem

  • An ice skater spins with a specific angular velocity. She brings her arms and legs closer to her body, reducing her moment of inertia to half its original value. What happens to her angular velocity? What happens to her rotational kinetic energy?

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Example 4: Combining Spinning Discs

  • A disc with moment of inertia 1 km·m2</sup spins about an axle through its center of mass with angular velocity 10 rad/s. An identical disc which is not rotating is slide along the axle until it makes contact with the first disc.

  • If the two discs stick together, what is their combined angular velocity?

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Example 5: Catching While Rotating

  • Angelina spins on a rotating pedestal with an angular velocity of 8 radians per second.

  • Bob throws her an exercise ball, which increases her moment of inertia from 2 kg·m2</sup to 2.5 kg·m2

  • What is Angelina's angular velocity after catching the exercise ball? (Neglect any external torque from the ball)

    C:\25225E85\B09A51C6-0574-4A0C-A2C1-496768C10C63_files\image266.png

2005 Free Response Question 3

Before Collision After Collision TOP VIEWS Mech. 3. A system consists of a ball of mass M2 and a uniform rod of mass Ml and length d. The rod is attached to a horizontal frictionless table by a pivot at point P and initially rotates at an angular speed , as shown above left. The rotational inertia of the rod about point P is —Mld2. The rod strikes the ball, which is initially at rest. As a result of this collision, the rod is stopped and the ball moves in the direction shown above right. Express all answers in terms of Ml , M2, o , d, and fundamental constants. (a) Derive an expression for the angular momentum of the rod about point P before the collision. (b) Derive an expression for the speed v of the ball after the collision. (c) Assuming that this collision is elastic, calculate the numerical value of the ratio Ml/M2 . Before Collision (d) A new ball with the same mass Ml as the rod is now placed a distance x from the pivot, as shown above. Again assuming the collision is elastic, for what value of x will the rod stop moving after hitting the ball?

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2014 Free Response Question 3

m/20 Top View Mech. 3. Side View A large circular disk of mass m and radius R is initially stationary on a horizontal icy surface. A person of mass m 2 stands on the edge of the disk. Without slipping on the disk, the person throws a large stone of mass m/ 20 horizontally at initial speed vo from a height h above the ice in a radial direction, as shown in the figures above. The coefficient of friction between the disk and the ice is g . All velocities are measured relative to the ground. The time it takes to throw the stone is negligible. Express all algebraic answers in terms of m, R, Do , h, g , and fundamental constants, as appropriate. (a) (b) (c) Derive an expression for the length of time it will take the stone to strike the ice. Assuming that the disk is free to slide on the ice, derive an expression for the speed of the disk and person immediately after the stone is thrown. Derive an expression for the time it will take the disk to stop sliding.

o Top View The person now stands on a similar disk of mass m and radius R that has a fixed pole through its center so that it can only rotate on the ice. The person throws the same stone horizontally in a tangential direction at initial speed vo , as shown in the figure above. The rotational inertia of the disk is mR 2. (d) Derive an expression for the angular speed (D of the disk immediately after the stone is thrown. (e) The person now stands on the disk at rest R/2 from the center of the disk. The person now throws the stone horizontally with a speed Do in the same direction as in part (d). Is the angular speed of the disk immediately after throwing the stone from this new position greater than, less than, or equal to the angular speed found in part (d) ? Greater than Justify your answer. Less than Equal to

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