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Physics 101: Lecture 15 Impulse and Momentum

Physics 101: Lecture 15 Impulse and Momentum. Today’s lecture will be a review of Chapters 7.1 - 7.2 and New material: Collisions and Center of Mass, Chapters 7.3-7.5 Rotational Motion and Angular Displacement, Chapter 8.1 . Conservation of Linear Momentum.

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Physics 101: Lecture 15 Impulse and Momentum

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  1. Physics 101: Lecture 15Impulse and Momentum • Today’s lecture will be a review of Chapters 7.1 - 7.2 and • New material: Collisions and Center of Mass, Chapters 7.3-7.5 Rotational Motion and Angular Displacement, Chapter 8.1

  2. Conservation of Linear Momentum • Consider a system of two colliding objects with masses m1 and m2 and initial velocities v01 and v02 and final velocitiesvf1 and vf2 : If the sum of the average external forces acting on the two objects is zero ( = isolated system), the total momentum of the system is conserved: SFave,ext Dt = Pf - P0 => Pf = P0 if SFave,ext = 0 Pf and Po arethetotal momenta of the system: Pf = pf1 + pf2 and P0 = p01 + p02 This is true for any number of colliding objects.

  3. Applying the Principle of Momentum Conservation • Decide which objects are included in the system. • Identify external and internal forces acting on the system. • Verify that the system is isolated. • Initial and final momenta of the isolated system can be considered to be equal. Example for an application: Determination of velocities of colliding objects after collision.

  4. Impulse and Momentum Summary Fave tJ = pf – p0 = p • For a single object…. •  Fave = 0 momentum conserved(p = 0) • For collection of objects … • Fave,ext = 0  total momentum conserved(P = 0)

  5. Collisions • If colliding objects constitute an isolated system (= no average external force), the total linear momentum is conserved. Sometimes also the kinetic energy is conserved. Elastic collision: Total kinetic energy before and after the collision is the same. Inelastic collision: Total kinetic energy is not conserved, i.e. part (or all) of the kinetic energy of the objects is converted into another form of energy. Collisions in two dimensions: SFave,ext,x Dt = Pf,x - P0,x => Pf,x = P0,x if SFave,ext,x = 0 SFave,ext,y Dt = Pf,y - P0,y => Pf,y = P0,y if S Fave,ext,y = 0 x and y components of the total linear momentum are separately conserved.

  6. Center of Mass • The center of mass of a system of objects is defined as the average location of the total mass. Consider two interacting objects (in 1-dim.) with masses m1 and m2 at the positions x1 and x2: xcm = (m1 x1 + m2 x2)/(m1+m2) Displacement of center of mass: Dxcm = (m1 Dx1 + m2Dx2)/(m1+m2) Velocity of center of mass: vcm = (m1 v1 + m2 v2)/(m1+m2) In an isolated system vcm does not change.

  7. Rotational Kinematics • The motion of a rigid body about a fixed axis is described by using the same concept as for linear motion (see C&J Chapter 2): Displacement, Velocity, Acceleration Angular Displacement: Identify the axis of rotation and choose a line perpendicular to this axis. Observe the motion of a point on this line. How can one define the change of position of this point during rotation about an axis ? Answer: Change of angle the line makes with a reference line: Dq

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