2. Conservative Forces: The forces is conservative if the work done by it on a particle that moves between two points depends only on these points and not on the path followed. This means that in moving a particle, by a conservative force, from point P to point Q (Figure 8.1) Work done along path 1= work done along path 2. as an example of a conservative force is the spring force, where the work done is As it is clear from the Equation, the work done by a spring depends only on x i and x f (the initial and final positions). The force due to gravity is another example of a conservative force.

3. Non-conservative Forces: The force is non-conservative if the work done by that force on a particle that moves between two points depends on the path taken between those points,i.e, Work done along path 1≠ work done along path 2 As an example of a non-conservative force is the frictional force. Suppose you were displaced a book between two points on a rough, horizontal surface such as a table, as shown in Figure. The work done by a friction force is –fd, where d is the distance between the two points P and Q. It is clear that the work done along path 1 is greater than the work done along path 2, since path 1 is greater than path 2.

4. In the conservative force we found that there is no loss in the work done, and the work depends only on the position of the particle (initial and final positions). That is the work done restored as potential energy U. in other words, the work done by a conservative force equals the decrease (negative) in the potential energy. If the change in potential energy is and W c, is the work done by a conservative force F along x-axis, then Or the potential energy function U(x) is 8.1

5. We can also define the relation between force and potential energy as That is the potential energy is a function of position x whose negative derivative gives the force. Finally, we conclude that there is no potential energy function associated with a non-conservative force.

6. Suppose a particle moves under the influence of a conservative force F x. Then the work-energy theorem form Chapter 8 tells us that the work done that force equals the change in kinetic energy. Since the force is conservative, we have Or The law of conservation of energy states that the change in energy is always zero. In other words, if the kinetic energy of a conservative system increases ( or decreases) by some amount, the potential energy must decreases (or increases) by the same amount. 8.2

7. Equation (8.2) can be written as Or Where E = K + U is the mechanical energy. 8.3

8. Since the force of gravity is conservative, we can define a gravitational potential energy function U g as Where y is the vertical displacement above an arbitrary horizontal level (level of zero potential energy). If the initial gravitational energy is U i = mgy i, and the final gravitational potential energy is U f = mgy f, then the change in potential energy equals the negative of the work done by the force of gravity, 8.4 8.5

9. When the displacement is upward, y f > y i, and therefore U i < U f the work done by gravity is negative, as expected. This corresponds to the case where the force of gravity is opposite the displacement. When the displacement is downward, y f U f so the work done by gravity is positive, since in this case the force due to gravity is in the same direction of the displacement. The conservation of mechanical energy expression (Equation 8.3) in the case of a freely falling body takes the form 8.6

10. The gravitational potential energy is positive if the body is above the level of zero potential energy, and negative if the body is below the level.

11. Example 8.1 A ball of mass m = 2kg is released from a height H = 10 m above the ground as shown in Figure. Using the law of conservation energy determine, a) the velocity of the ball at a distance y = 4m above the ground, b) the velocity of the ball just before it hits the ground. Solution: a) applying the conservation of energy principle between points A and B, we get Or

12. b) Conservation of energy between points A and C gives Or

13. Consider a system consists of a spring with spring constant K, and a mass slides on a frictionless surface, as in Figure. Since the force of the spring is conservative force, and since there is no external forces, the total mechanical energy must be constant. That is, the kinetic energy of the block is stored as potential energy, (elastic potential energy), in the spring, and is given by Where x is the amount of compressing, or stretching the spring. The conservation of mechanical energy of the mass-spring system can be written as Or 8.7 8.8

14. The potential energy of the spring is zero at the equilibrium point (unscratched position, = 0), and the total energy is = maximum value of kinetic energy (v is maximum). The potential energy is a maximum when x is a maximum value (when v = 0, at maximum compression). U s is always positive, since x² is always positive.

15. If there are non-conservative forces, such as friction, and these forces do work, then the total mechanical energy (kinetic + potential) is not constant, that is, the change in the total mechanical energy is not zero, or The work done by the non-conservative forces W nc equals the change in total mechanical energy. This means that W nc represents the lost in energy, so we have 8.9

16. Or Note that if W nc is zero we recover Equation 8.3 as expected. 8.10

17. (i)Select a horizontal level for the zero gravitational potential energy. (ii)Define two points: one as an initial point and the other as a final point. (iii)Find the potential energy and the kinetic energy at these two points. (iv)If there is a spring, then the total potential energy of the system is U = U g + U s. (v)If there are friction forces, then calculate W nc. If not then W nc = 0. (vi)Now use Equation 8.9 or Equation 8.10 to find the unknowns.

18. Example 8.2 A 2-kg mass slides down a rough inclined plane, as shown in Figure the mass starts from rest, and the friction force is given by f = 5N. A) use energy method to find the speed of the block at the bottom of the incline, b) If the inclined plane is frictionless find the speed at that point. Solution: The ground is chosen as the level for the zero potential energy, and the initial point is chosen at the top of the plane, while the final point is chosen at the bottom of the plane. Now

19. And Since the plane is rough, we have Now applying Equation (8.10) we get b) In this case, W c = 0, and equation 8.10 becomes

20. Example 8.3: Two blocks are connected by a light string that passes over a frictionless pulley as shown in Figure. the mass m 1 lies on a rough surface, and the system is released from rest when the spring unscratched (x=0). The mass m 2 falls a distance h before coming to rest. Calculate the coefficient of kinetic friction between m 1 and the surface. Solution In this example there are two forms of potential energy: the gravitational potential energy and the potential energy of the spring, so But ∆U g is due only to m 2, therefore we have While ∆U s is associated only to m 1, so

21. Since the initial and the final speed of the system is zero, the ∆K=0 The work done by the frictional force is given by Now applying Equation (8.9) we get or