1. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Physics I 95.141 LECTURE 11 10/13/10

2. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Exam Prep Problem It is the year 2030, and we have colonized the moon. In order to set-up lunar GPS, satellites must be launched to orbit the moon. Two different satellites are launched, to orbit at altitudes of 8x10 5 m and 1x10 6 m, respectively. –A) (5pts) What is the acceleration due to the Force of Gravity on the surface of the moon? –B) (10pts) What are the speeds of the two satellites? –C) (10pts) What are the periods and frequencies of the satellites orbits?

3. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Exam Prep Problem It is the year 2030, and we have colonized the moon. In order to set-up lunar GPS, satellites must be launched to orbit the moon. Two different satellites are launched, to orbit at altitudes of 8x10 5 m and 1x10 6 m, respectively. –A) (5pts) What is the acceleration due to the Force of Gravity on the surface of the moon?

4. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Exam Prep Problem It is the year 2030, and we have colonized the moon. In order to set-up lunar GPS, satellites must be launched to orbit the moon. Two different satellites are launched, to orbit at altitudes of 8x10 5 m and 1x10 6 m, respectively. –B) (10pts) What are the speeds of the two satellites?

5. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Exam Prep Problem It is the year 2030, and we have colonized the moon. In order to set-up lunar GPS, satellites must be launched to orbit the moon. Two different satellites are launched, to orbit at altitudes of 8x10 5 m and 1x10 6 m, respectively. –C) (10pts) What are the periods and frequencies of the satellites orbits?

6. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Outline Work by Constant Force Scalar Product of Vectors Work done by varying Force What do we know? –Units –Kinematic equations –Freely falling objects –Vectors –Kinematics + Vectors = Vector Kinematics –Relative motion –Projectile motion –Uniform circular motion –Newton’s Laws –Force of Gravity/Normal Force –Free Body Diagrams –Problem solving –Uniform Circular Motion –Newton’s Law of Universal Gravitation –Weightlessness –Kepler’s Laws

7. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Work and Energy Up until this point, we have discussed motion of objects using the idea of Force, and Newton’s Laws We are going to start looking at describing physical situations using the concepts of Work/Energy and momentum. –Another way of approaching problems –Can often be an extremely powerful method, allowing us to solve problems that Newton’s Laws make very complicated.

8. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 What is Work? Obviously in the vernacular, Work can have many different meanings In Physics, there is one meaning for work Work done on an object is given by the product of the physical displacement of that object and the component of the Force parallel to the displacement. Work has units of N-m, or Joules (J), and is a scalar

9. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Example Say I pull on a crate, as show below, with a Force of 10N across a distance of 10m. How much work have I done?

10. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Example Say I pull on a crate, as show below, with a Force of 10N across a distance of 10m. How much work have I done? What about other Forces?

11. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Example Problem II Sisyphus was condemned to Hades and forced to continually push a large boulder (1000kg) up a hill, only to have it roll down every time he neared the top. How much work does Sisyphus do each time he pushes the boulder up the hill, assuming he pushes the block with a constant speed? h Free body diagram F II-Sisyphus

12. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Example Problem II Sisyphus was condemned to Hades and forced to continually push a large boulder up a hill, only to have it roll down every time he neared the top. How much work does Sisyphus do each time he pushes the boulder up the hill, assuming he pushes the block with a constant speed? h

13. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Example Problem II Sisyphus was condemned to Hades and forced to continually push a large boulder up a hill, only to have it roll down every time he neared the top. How much work does gravity do? How much does the Normal Force do? How much Net Work is done on the boulder? h

14. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Scalar Product of 2 Vectors Both Force and Displacement are vectors. So Work, which is a scalar, comes from the product of two vectors. Three ways to multiply vectors –Multiplication by a scalar –Scalar (or dot) product –Vector (or cross) product

15. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Scalar Product of Two Vectors The scalar product of two vectors is written as: And gives a result of:

16. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Work as a Scalar Product If we look at the definition of the scalar product of two vectors: We can see that this is exactly what we found when we determined the work done by a force over a distance:

17. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Scalar Products (parallel and perpendicular) For the case that: Or, if

18. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Properties of Scalar Product Commutative Distributive

19. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Scalar Product in Component Form

20. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Equivalence of two methods For two vectors given by:

21. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Equivalence of two methods For two vectors given by:

22. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Example A constant Force F acts on an object as it moves from position x 1 to x 2. What is the work done by this Force?

23. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Example A constant Force F acts on an object as it moves from position x 1 to x 2. What is the work done by this Force?

24. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Work Done By a Varying Force If Force is constant, then finding work simply entails knowing change of position, and magnitude and direction of constant Force However, in many situations, the Force acting on an object is NOT constant! –Rocket leaving Earth –Springs –Electrostatic Forces

25. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Work Done by a Varying Force

26. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Work Done by a Varying Force

27. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Work Done by a Varying Force

28. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Work Done by a Spring The force exerted by a spring is given by: Hooke’s Law

29. Department of Physics and Applied Physics 95.141, F2010, Lecture 11 Example Problem How much work must I do to compress a spring with k=20N/m 20cm?