Physics I 95.141 LECTURE 11 10/13/10

1. Physics I95.141LECTURE 1110/13/10

2. 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 8x105m and 1x106m, 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. 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 8x105m and 1x106m, respectively. • A) (5pts) What is the acceleration due to the Force of Gravity on the surface of the moon?

4. 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 8x105m and 1x106m, respectively. • B) (10pts) What are the speeds of the two satellites?

5. 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 8x105m and 1x106m, respectively. • C) (10pts) What are the periods and frequencies of the satellites orbits?

6. 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. 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. 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. 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. 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. 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? • Free body diagram • FII-Sisyphus h

12. 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. 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. 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. Scalar Product of Two Vectors • The scalar product of two vectors • is written as: • And gives a result of:

16. 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. Scalar Products (parallel and perpendicular) • For the case that: • Or, if

18. Properties of Scalar Product • Commutative • Distributive

19. Scalar Product in Component Form

20. Equivalence of two methods • For two vectors given by:

21. Equivalence of two methods • For two vectors given by:

22. Example • A constant Force F acts on an object as it moves from position x1 to x2. What is the work done by this Force?

23. Example • A constant Force F acts on an object as it moves from position x1 to x2. What is the work done by this Force?

24. 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. Work Done by a Varying Force

26. Work Done by a Varying Force

27. Work Done by a Varying Force

28. Work Done by a Spring • The force exerted by a spring is given by: Hooke’s Law

29. Example Problem • How much work must I do to compress a spring with k=20N/m 20cm?