1. Exam 2 Review 8.02 W08D1

2. Announcements Test Two Next Week Thursday Oct 27 7:30-9:30 Section Room Assignments on Announcements Page Test Two Topics: Circular Motion, Energy, Momentum, and Collisions

3. Circular Motion: Vector Description Position Angular Speed Velocity Speed Angular Acceleration Acceleration

4. Modeling the Motion: Newton’s Second Law Define system, choose coordinate system. Draw free body force diagrams. Newton’s Second Law for each direction. Example: x -direction Example: Circular motion

5. Strategy: Applying Newton’s Second Law for Circular Motion Always has a component of acceleration pointing radially inward May or may not have tangential component of acceleration Draw Free Body Diagram for all forces mv 2 /r is not a force but mass times acceleration and does not appear on force diagram Choose a sign convention for radial direction and check that signs for forces and acceleration are consistent

6. Concept Question: Tension and Circular Motion A stone attached to a string is whirled in a vertical plane. Let T 1, T 2, T 3, and T 4 be the tensions at locations 1, 2, 3, and 4 required for the stone to have the same speed v 0 at these four locations. Then 1.T 3 > T 2 > T 1 = T 4 2. T 1 = T 2 = T 3 = T 4 3. T 1 > T 2 = T 4 > T 3 4.none of the above

7. Dot Product A scalar quantity Magnitude: The dot product can be positive, zero, or negative Two types of projections: the dot product is the parallel component of one vector with respect to the second vector times the magnitude of the second vector

8. Review: Potential Energy Difference Definition: Potential Energy Difference between the points A and B associated with a conservative force is the negative of the work done by the conservative force in moving the body along any path connecting the points A and B.

9. Review: Examples of Potential Energy with Choice of Zero Point (1) Constant Gravity: (2) Inverse Square Gravity (3) Spring Force

10. Work-Energy Theorem: Conservative Forces The work done by the total force in moving an object from A to B is equal to the change in kinetic energy When the only forces acting on the object are conservative forces then the change in potential energy is Therefore

11. Change in Energy for Conservative and Non-conservative Forces Force decomposition: Work done is change in kinetic energy: Mechanical energy change:

12. Strategy: Using Multiple Ideas Energy principle: No non-conservative work For circular motion, you will also need to Newton’s Second Law in the radial direction because no work is done in that direction hence the energy law does not completely reproduce the equations you would get from Newton’s Second Law Constraint Condition : Conservation

13. Modeling the Motion Energy Concepts Change in Mechanical Energy: Identify non-conservative forces. Calculate non-conservative work Choose initial and final states and draw energy diagrams. Choose zero point P for potential energy for each interaction in which potential energy difference is well- defined. Identify initial and final mechanical energy Apply Energy Law.

14. Bead on Track A small bead of mass m is constrained to move along a frictionless track. At the top of the circular portion of the track of radius R, the bead is pushed with an unknown speed v 0. The bead comes momentarily to rest after compressing a spring (spring constant k) a distance x f. What is the direction and magnitude of the normal force of the track on the bead at the point A, at a height R from the base of the track? Express your answer in terms of m, k, R, g, and x f.

15. Block Sliding off Hemisphere A small point like object of mass m rests on top of a sphere of radius R. The object is released from the top of the sphere with a negligible speed and it slowly starts to slide. Find an expression for the angle θ f with respect to the vertical at which the object just loses contact with the sphere. There is a non-uniform friction force with magnitude f=f 0 sinθ acting on the object.

16. Table Problem: Potential Energy Diagram A body of mass m is moving along the x- axis. Its potential energy is given by the function U(x) = b(x 2 -a 2 ) 2 where b = 2 J/m 4 and a = 1 m. a) On the graph directly underneath a graph of U vs. x, sketch the force F vs. x. b) What is an analytic expression for F(x)?

17. Momentum and Impulse: Single Particle Momentum SI units Change in momentum Impulse SI units

18. External Force and Momentum Change The momentum of a system of N particles is defined as the sum of the individual momenta of the particles Force changes the momentum of the system Force equals external force, internal forces cancel in pairs Newton’s Second and Third Laws for a system of particles: The external force is equal to the change in momentum of the system

19. Strategy: Momentum of a System 1. Choose system 2. Identify initial and final states 3. Identify any external forces in order to determine whether any component of the momentum of the system is constant or not i) If there is a non-zero total external force: ii) If the total external force is zero then momentum is constant

20. Problem Solving Strategies: Momentum Flow Diagram Identify the objects that comprise the system Identify your choice if reference frame with an appropriate choice of positive directions and unit vectors Identify your initial and final states of the system Construct a momentum flow diagram as follow: Draw two pictures; one for the initial state and the other for the final state. In each picture: choose symbols for the mass and velocity of each object in your system, for both the initial and final states. Draw an arrow representing the momentum. (Decide whether you are using components or magnitudes for your velocity symbols.)

21. Modeling: Instantaneous Interactions Decide whether or not an interaction is instantaneous. External impulse changes the momentum of the system. If the collision time is approximately zero, then the change in momentum is approximately zero.

22. Collision Theory: Energy Types of Collisions Elastic: Inelastic: Completely Inelastic: Only one body emerges. Superelastic:

23. Elastic Collision: 1-Dim Conservation of Momentum and Relative Velocity Momentum Relative Velocity

24. Table Problem: One Dimensional Elastic Collision: Relative Velocity Consider the elastic collision of two carts; cart 1 has mass m 1 and moves with initial speed v 0. Cart 2 has mass m 2 = 4 m 1 and is moving in the opposite direction with initial speed v 0. Immediately after the collision, cart 1 has final speed v 1,f and cart 2 has final speed v 2,f. Find the final velocities of the carts as a function of the initial speed v 0.