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) Ft – 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 definite integral (5.3) Taking into account that and , integral (5.3) can be expressed in the cartesian coordinate system as follows: (5.4)

Work, cont.

Work, cont. Special cases 1. The work done by a constant force perpendicular to the displacement is zero 2. Only the component of a force parallel or antiparallel to the velocity changes the speed. These components do positive or negative work. 3. For a constant force the work depends only on the displacement, not on details of the path 4. When many forces are doing the work, the work can be calculated for each individual force, but the total work done on a particle depends on the resultant force. The work is a scalar quantity.

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.

Power – sample problem A force of 5 N acts on 15 kg body initially at rest. Compute the work done by the force in (a) the first second, (b) the third second and (c) the instantaneous power due to the force at the end of the third second. a) b) c)

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.

Kinetic energy, cont. Example 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.

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. (6.0) On the closed path, when A ≡ B, the work done by a conservative force is zero (6.1) Examples of conservative forces: gravitational, spring (elastic), electrostatic. Other examples of conservative forces: central force , constant force . Typical nonconservative force: friction force.

Conservative forces, cont. Example 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: C (6.2) The same result would be obtained for calculation of the work along the path ACB. Along AC we have mg (h1-h2) cos0 = mg (h1-h2). Along CB we have mgd cos(π/2)=0. And finally we have WACB= mg(h1-h2) what is in accordance with (6.2). The force is then conservative.

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 (6.3) Integrating Eq. (6.3) one obtains: (6.4) The work of a conservative force is equal to the difference in potential energies in the initial point A and the final B. Taking point A as a reference, one can determine pot. energy for any other point (6.5)

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

Gravitational potential energy . yi y yf mg m A particle of mass m moves along y-axis from yi to yf. The gravitational force –mg does work, which changes the potential energy of the particle-Earth system. Assuming that yi = 0 and Ui = 0 one obtains

5.5. Conservation of mechanical energy From the properties of a conservative force it follows that dEk = - dU (6.10) or equivalently (6.10a) d(Ek + U) = 0 Eq. (6.10a) indicates that differential of energy is zero. This equation can be integrated what gives Ek + U = const = Em (6.11) 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. Eq. (6.11) can be rewritten as or ΔEk + Δ U = 0 ΔEk = - Δ U

Motion of a particle under the influence of a spring force Example 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 U(x)+ Ek(x )= E = const For a given energy E the particle motion proceeds between the turning points – x0 and x0, for which Ek(-x0) = Ek(x0) = 0 (between these points Ek>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.

5.6. Work done by an external force External forces can act on a system of objects. In this case mechanical energy of the system increases by this work W W = ΔEm (6.12) When friction is involved additional energy is lost for heating up of the sliding bodies W = ΔEm + Δ Eth (6.13) If a system is isolated, i.e. no energy is transferred to or from the system, one obtains from (6.13) 0 = ΔEm + Δ Eth or ΔEm = - Δ Eth (6.14)

Sample problem A body slides down the ramp of a height y = H and slope angle a and then moves a distance d along a flat surface. What is that distance d if the coefficient of kinetic friction on the total path is f = 0.50 ? T1 = f R1 T2 = f R2 According to (6.14) ΔEm = - Δ Eth Em2 – Em1 = - Δ Eth 0 – mgH = - Δ Eth where Ek1 = 0, U1 = mgh, Ek2 = 0, U2 = 0 Δ Eth = T1 H/ sina + T2 d = f N1H /sina + f mg d = f mg cosa H /sina + f mg d Hence mgH = f mg d + f mg H/ tana d = H(1/ f – cot a )