1. 1 Honors Physics 1 Class 07 Fall 2013 Center of mass – multivariable integration Impulse – Integration of force Integrating equations of motion

2. 2 Calculating the Center of Mass This approach is used for calculating the average value of a quantity for any distribution.

3. 3 Center of Mass Integration

4. 4 Computing Center of mass for a Shaped Object 1) Use symmetry to reduce the amount of direct calculation. 2) Set up the multivariable integral relation for calculating the average value along the axis of interest. 3) Do the integration in one of the variables (e.g.- y), putting the limitations on the shape (in terms of the other variables) into the limits of the integral. 4) Repeat for as many directions as needed until you get to the last one, where the upper and lower bounds are then the global max and min of the shape.

5. 5 A Math Interlude: Differentials

6. 6 Changing the variable of integration

7. 7 Impulse

8. 8 Impulse example 1 (KK131) 0.2 kg ball bounces from floor in time 10 -3 s. 8 m/s. Find average force.

9. 9 Propulsion example Is it possible to build a jetpack using downward firing machine guns? (Yes!) We need to know: –mass of a machine gun, (AK 47 = 4.8 kg) –mass of a bullet, (0.008 kg) –rate at which bullets can be fired. (10/s) –speed of exiting bullet wrt gun (715 m/s) http://what-if.xkcd.com/21/

10. 10 Propulsion example: AK47 jet pack My drawing Randall Munroe’s drawing Only 3 seconds of lift We need more ammunition and ammunition has mass.

11. 11 But what if the mass changes with time?

12. 12 Filling in some details on rocket discussion