1. Wednesday, Mar. 3, 2004 1 PHYS 1443-501, Spring 2004 Dr. Andrew Brandt PHYS 1443 – Section 501 Lecture #12 Newton’s Law of Gravitation and Kepler’s Laws Work Done by Constant Force Work Done by Varying Force Kinetic Energy Wednesday, Mar 3, 2004 Dr. Andrew Brandt

2. Wednesday, Mar. 3, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 2 Announcements HW#5 on Ch. 6 is due Mon. March 8 at midnight Test Solutions available Interim grades posted on website: If you don’t like your grade there is still time to change it; it starts with the HW’s, spend more time on them and everything else will follow

3. Wednesday, Mar. 3, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 3 Newton’s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. How would you write this principle mathematically? G is the universal gravitational constant, and its value is This constant is not given by the theory but must be measured by experiment. With G Unit? This form of force is known as an inverse-square law, because the magnitude of the force is inversely proportional to the square of the distance between the objects.

4. Wednesday, Mar. 3, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 4 Free Fall Acceleration & Gravitational Force Weight of an object with mass m is mg. Using the force exerting on a particle of mass m on the surface of the Earth, one can get The gravitational acceleration is independent of the mass of the object The gravitational acceleration decreases as the altitude increases If the distance from the surface of the Earth gets infinitely large, the weight of the object approaches 0. What would the gravitational acceleration be if the object is at an altitude h above the surface of the Earth? What does this tell us about the gravitational acceleration?

5. Wednesday, Mar. 3, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 5 Kepler’s Laws & Ellipse Kepler’s laws can be derived from Newton’s laws. Kepler’s third law is the direct consequence of the inverse square nature of the law of gravitation. 1.All planets move in elliptical orbits with the Sun at one focal point. 2.The radius vector drawn from the Sun to a planet sweeps out equal area in equal time intervals. (Angular momentum conservation) 3.The square of the orbital period of any planet is proportional to the cube of the semi-major axis of the elliptical orbit. F1F1 F2F2 b c a Ellipses have two different axis, major (long) and minor (short) axis, and two focal points, F 1 & F 2 a is the length of a semi-major axis b is the length of a semi-minor axis quiz&example

6. Wednesday, Mar. 3, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 6 x y Work Done by a Constant Force Work in physics is done only when a sum of forces exerted on an object causes motion of the object. M F  Free Body Diagram M d  Which force did the work?Force How much work did it do? What does this mean? Physical work is done only by the component of the force along the movement of the object. Unit? Work is energy transfer!!

7. Wednesday, Mar. 3, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 7 Example of Work w/ Constant Force A man cleaning a floor pulls a vacuum cleaner with a force of magnitude F=50.0N at an angle of 30.0 o with East. Calculate the work done by the force on the vacuum cleaner as it is displaced by 3.00m to East. Does work depend on mass of the object being worked on? M F   M d Yes Why don’t I see the mass term in the work at all then? It is reflected in the force. If the object has smaller mass, its would take less force to move it the same distance as the heavier object. So it would take less work.

8. Wednesday, Mar. 3, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 8 Scalar (Dot) Product of Two Vectors Product of magnitude of the two vectors and the cosine of the angle between them Operation is commutative Operation follows distribution law of multiplication How does scalar product look in terms of components? Scalar products of Unit Vectors Is this a vector or a scalar?

9. Wednesday, Mar. 3, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 9 Example of Work by Scalar Product A particle moving in the xy plane undergoes a displacement d =(2.0 i +3.0 j )m as a constant force F =(5.0 i +2.0 j ) N acts on the particle. a) Calculate the magnitude of the displacement and that of the force. b) Calculate the work done by the force F. Y X d F Can you do this using the magnitudes and the angle between d and F ?

10. Wednesday, Mar. 3, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 10 Work Done by a Varying Force If the force depends on the position of a particle during the motion –one must consider the work increment in a small segment of the position where the force can be considered constant –Then add them all up throughout the entire motion (x i  x f ) –If more than one force is acting, the net work is done by the net force In the limit where  x  0 An example of a force that depends on position is the spring force The work done by the spring force is

11. Wednesday, Mar. 3, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 11 Kinetic Energy and Work-Kinetic Energy Theorem Some problems are hard to solve using Newton’s second law –If forces exerted on the object during the motion are complicated –Relate the work done on the object by the net force to the change of the speed of the object M FF M d vivi vfvf Suppose a net force  F was exerted on an object over a displacement d to increase its speed from v i to v f. The work on the object by the net force  F is Acceleration Work Kinetic Energy Work The work done by the net force causes a change of object’s kinetic energy.