1. 1 5. Work and Energy 5.1. Work The work done by a constant force is given by the dot product of that force and a displacement vector (5.1) F t – projection of force on the displacement direction If the force varies along the displacement, then for an infinitesimal displacement it can be written (5.2) For the displacement between A and B the work is given by the integral (5.3) Taking into account that and, integral (5.3) can be expressed in the cartesian coordinate system as follows: (5.4)

2. 2 Example Work, cont. O(b) xixi x O (c) xfxf x O (a) x

3. 3 5.2. Power The rate at which work is done is called power (5.5) The average power in a time interval Δt is defined as (5.6) where P(t) – instantaneous power The SI unit of power is the watt Commonly used non-SI power unit is the horsepower (hp), equal 746 W.

4. 4 5.3. Kinetic energy The work of a net force can be calculated as (5.7) Then for a constant mass one obtains (5.8) where is the kinetic energy From (5.8) it follows that (5.9) Eq. (5.9) is a base of the work – kinetic energy theorem: Change in the kinetic energy of a particle equals to net work done on the particle.

5. 5 Example Kinetic energy, cont. A particle with mass m attached to the spring of spring constant k is stretched by x from the initial, relaxed position and then released, moving under the influence of the spring force. What is the speed of a particle in the relaxed position.

6. 6 5.4. Conservative forces, Potential energy The work done by a conservative force on a particle moving between two points A and B does not depend on the path taken by the particle but only on the positions of the initial and final points. On the closed path, when A ≡ B, the work done by a conservative force is zero Examples of conservative forces: gravitational, spring (elastic), electrostatic. Other examples of conservative forces: central force, constant force. Typical nonconservative force: friction force. (6.0) (6.1)

7. 7 Example Conservative forces, cont. Calculate a work done by the gravitational force close to the Earth surface ( ) during sliding a particle along a frictionless track from point A to point B (figure below). From Eq. (5.4) one obtains: The same result would be obtained for calculation of the work along the path ACB. Along AC we have mg (h 1 -h 2 ) cos0 = mg (h 1 -h 2 ). Along CB we have mgd cos(π/2)=0. And finally we have W ACB = mg(h 1 -h 2 ) what is in accordance with (6.2). The force is then conservative. C (6.2)

8. 8 Potential energy Generally the conservative force is a function of the particle position and in this field of a conservative force one introduces the new quantity, potential energy U. The stored potential energy can be recovered and changed for the kinetic energy. The work of a conservative force is equal to the negative change in the potential energy Integrating Eq. (6.3) one obtains: The work of a conservative force is equal to the difference in potential energies in the initial point A and the final B. (6.3) (6.4) Taking point A as a reference, one can determine pot. energy for any other point (6.5)

9. 9 Potential energy, cont. In one dimension Eq. (6.5) can be written as Differentiating both sides of Eq.(6.6) one obtains The negative derivative of a potential energy is equal to the field force. (6.7) Eq. (6.7) in three dimensions in the vector form is written as (6.6) (6.8) or in a more abbreviated form where the operator of gradient is defined as (6.9) The sign is called „nabla”, - partial derivative

10. 10 Conservation of mechanical energy From the properties of a conservative force it follows that or equivalently The sum of kinetic and potential energies, called the mechanical energy, is constant when only conservative forces cause energy changes. This is the principle of conservation of mechanical energy. (6.10a) Eq. (6.10a) can be integrated what gives (6.10) Eq. (6.11) can be rewritten as dE k = -dU d(E k + U) = 0 E k + U = const = E m (6.11) Δ E k + Δ U = 0 ΔE k = - Δ U or

11. 11 Motion of a particle under the influence of a spring force The spring force F = - kx is a conservative force. Potential energy associated with the state of compression or extension of an elastic object is equal The mechanical (kinetic + potential) energy of the system mass + spring is constant, because only a conservative force does work For a given energy E the particle motion proceeds between the turning points – x 0 and x 0, for which E k (-x 0 ) = E k (x 0 ) = 0 (between these points E k >0). The force acting on a particle is the negative of the slope of the potential energy curve (F = - dU(x)/dx). At the position x = 0 the force is also zero and additionally this position is always restored. This is a stable equilibrium. Example U(x)+ E k (x )= E = const