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2. NEWTON’S LAW OF UNIVERSAL GRAVITATION Before 1687, clear under-standing of the forces causing plants and moon motions was not available. In that year, Isaac Newton knew, from his first law, that a net force had to be acting on the Moon because without such a force the Moon would move in a straight-line path rather than in its almost circular orbit. Newton reasoned that this force was the gravitational attraction exerted by the Earth on the Moon.

3. Newton’s law of universal gravitation states that 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. If the particles have masses m1and m2 and are separated by a distance r , the magnitude of this gravitational force is

4. where G is a constant, called the universal gravitational constant, that has been measured experimentally. Its value in SI units is

5. MOTION WITH CONSTANT ACCELERATION The instantaneous acceleration a is defined as the limiting value of the ratio ∆v/∆t as ∆t approches zero : Velocity vector as a function of time Position vector as a function of time Velocity vector as a function of Position vector

6. FORCES OF FRICTION * As long as the book is not moving, f = F. Because the book is stationary, we call this frictional force the force of static friction fs * When the book is in motion, we call the retarding force the force of kinetic friction fk

7. • The direction of the force of static friction between any two surfaces in contact with each other is opposite the direction of relative motion and can have values where the dimensionless constant µsis called the coefficient of static friction and n is the magnitude of the normal force. When fs= fs,max = µsn. The inequality holds when the applied force is less than µsn.

8. • The direction of the force of kinetic friction acting on an object is opposite the direction of the object’s sliding motion relative to the surface applying the frictional force and is given by where µk is the coefficient of kinetic friction.

9. Work done by a constant force The work W done on an object by an agent exerting a constant force on the object is the product of the component of the force in the direction of the displacement and the magnitude of the displacement:

10. Work expressed as a dot product scalar product allows us to indicate how F and d interact in a way that depends on how close to parallel they happen to be Work is a scalar quantity, and its units are force multiplied by length. Therefore, the SI unit of work is the Newton meter (Nm). This combination of units is used so frequently that it has been given a name of its own: the joule (J).

11. KINETIC ENERGY If the particle is displaced a distance d, the net work done by the total force ΣF is when a particle undergoes constant acceleration we have , where vi is the speed at t= 0 and vf is the speed at time t. After substituting we get:

12. It is often convenient to write this equation in the form: In general, the kinetic energy K of a particle of mass m moving with a speed v is defined as This equation is an important result known as the work–kinetic energy theorem. Kinetic energy is a scalar quantity and has the same units as work.

13. Potential Energy and Conservation of Energy We introduced the concept of kinetic energy, which is the energy associated with the motion of an object. • Potential energy U • Is the energy associated with the arrangement of a system of objects that exert forces on each other. • It can be thought of as stored energy that can either do work or be converted to kinetic energy. System Is consists of two or more objects that exert forces on one another

14. Gravitational Potential Energy Ug Gravitational potential energy is the potential energy of the object–Earth system. As an object falls toward the Earth, the Earth exerts a gravitational force mg on the object, with the direction of the force being the same as the direction of the object’s motion. The product of the magnitude of the gravitational force mg acting on an object and the height y of the object

15. Let us now directly relate the work done on an object by the gravitational force to the gravitational potential energy of the object–Earth system. where we have used the fact that From this result we conclude that : 1- The work done on any object by the gravitational force is equal to the negative of the change in the system’s gravitational potential energy.

16. 2- This result demonstrates that it is only the difference in the gravitational potential energy at the initial and final locations that matters. This means that we are free to place the origin of coordinates in any convenient location. 3- the object falls to the Earth is the same as the work done were the object to start at the same point and slide down an incline to the Earth. Horizontal motion does not affect the value of Wg

17. CONSERVATIVE AND NONCONSERVATIVE FORCES Conservative forces have two important properties: 1. A force is conservative if the work it does on a particle moving between any two points is independent of the path taken by the particle. 2. The work done by a conservative force on a particle moving through any closed path is zero. (A closed path is one in which the beginning and end points are identical.) The gravitational force is one example of a conservative force, and the force that a spring exerts on any object attached to the spring is another.

18. Non-conservative Forces A force is non-conservative if it causes a change in mechanical energy E, which we define as the sum of kinetic and potential energies. For example, if a book is sent sliding on a horizontal surface that is not frictionless, the force of kinetic friction reduces the book’s kinetic energy. The type of energy associated with temperature is internal energy.

19. CONSERVATION OF MECHANICAL ENERGY An object held at some height h above the floor has no kinetic energy. The gravitational potential energy of the object–Earth system is equal to mgh. If the object is dropped, as it falls, its speed and thus its kinetic energy increase, while the potential energy of the system decreases. The sum of the kinetic and potential energies remains constant.

20. This is an example of the principle of conservation of mechanical energy. Note that the total mechanical energy of a system remains constant in any isolated system of objects that interact only through conservative forces.

21. Because the total mechanical energy E of a system is defined as the sum of the kinetic and potential energies, we can write We can state the principle of conservation of energy as and so we have It is important to note that this equation is valid only when no energy is added to or removed from the system. Furthermore, there must be no non-conservative forces doing work within the system.