1. Energy: What is it? Energy is an entity, concept, or whatever you want to call it that, when added to an object causes it to heat up, (heat energy Q = mcDT), speed up, kinetic energy K = .5 mv²), or change its altitude, (potential energy U_g = mgDy). An effort to quantify this concept is what is known as energy. There are many ways to transfer energy to or from an object, but one of the simplest is to apply a force to it and speed it up.

2. Work is a sub-concept of energy. Work is defined to be energy transferred to or from an object by means of a force. More specifically, Work = force (in the direction of motion) x distance (over which the force is applied). Since this applied force causes the object to speed up, the Law of conservation of energy, (which comes from the law of conservation of momentum), requires that the net work done on the object equals the amount of change in its Kinetic energy DK, hence, the

Work-Energy theorem W_net = DK

3. But what happens when the force is not in the direction of motion? For instance, suppose the force is applied at an angle Uwith the horizontal. What force contributes to the acceleration of the object? Conclude W = Fcos U. x

Define the Dot Product of two vectors F . x = FxcosU Where Uis the angle between F and the direction of motion, thus, W = F*x, where x and F are vector quantities.

4. Work due to friction force. Since friction is always a reaction to the applied force, it will always be in the opposite direction to the direction of motion. i.e. the angle between Fk and displacement x is always 180o, thus,

W_friction = Fkx cos 180o = -Fkx. i.o.w. Work due to friction is always negative !

5. Work done by varying force. How to find F*X if F is changing?

W = limit…Wi: = limit …FiDx = ÝF dx ! Moreover,
Dx Æ0 Dx Æ0

W_net = ÝF_netdx = Ýma dx = Ým(dv/dt)dx = Ým dv(dx/dt) = Ýmvdv = .5mv² - .5mv_o² = DK

The RBD Work is area under curve!

6. Work done by a Hooke's Law spring.

F_s (x) = - Kx = -F_app (see figures at left), thus,

W_app= ÝF_appdx = Ýkxdx = .5kx², and
W_s = ÝF_sdx = -ÝF_appdx = -W_app= - .5kx²

7. Now suppose F = (3i + 2 j) and r = 2i - 5j. How's an easy way to find W = F . r ?

Recall A . B = AB cosU. How could one find A.B without knowing A,B and U ? i.e. How to find (A_x i + A_y j) . (B_x i +B_y j)= ? Just multiply it out !

A_x A_y + B_x B_y = AB cos U So, AB cos U = A_xB_x + A_yB_y, or

cos U = (A_xB_x + A_yB_y)/AB
r2
8. If F = F_x i + F_yj and displacement is dr = dr_x i + dr_y j, then W =Ýr1 F ^r2 dr = ?

9. Power_avg = work/time = W/D t = P
P = limit dW/Dt = dW / dt = d (F . r) / dt =(d/dt) ?
Dt Æ0

DISCUSS limitations of the P = F v formula , Pg 187 :
units of power = J / s = Watts (W)
1 hp = 746 W. = 550 ft lb.

Chapter 8: Potential Energy

Def. Pot - ENERGY - Energy an object has by virtue of its location relative to some equilibrium position. Note: Since it is the energy obtained from some force applied to the system, for both gravity and springs, potential energy = Wapp = DU_g or DU_s . Specifically,
U_g = mgh or mgDy, and Us = .5kx² or .5k(x² - x_o²)
so the work done by gravity or a spring is W_g or W_s = -DU_{s or g}

Def. Conservative Force (Fc) - All energy expended goes into savings and can be retrieved. NO ENERGY IS LOST, so total energy E_o = E_f . Work done by g when the pendulum in the figure moves from A to B gets stored in Kinetic Energy at B and is used to raise ball to C.
E_o = E_f ,
K_o + U_o = K_f + U_f
Work done by a conservative Force is independent of path.

F_g= -mg = F_c , so Wg = -mgDy = -DU = Ý-mgdy = Ý F_g dy = -DUg.

Notice the two objects at right travel different displacements, but both travel the same Dy, the distance parallel to the direction of the conservative force Fg.

So, W_g = -mgh = -DU or if U (x_o) = 0, DU = U = -ÝF_gdy so dU/dy = -F_c (y) or F_c = -dU/dy

This is a characteristic of a conservative force in general: U_c = - Ý F_c dx = -W_c.

F_nc - nonconservative force, all energy lost forever. --(friction, in our case).
W_nc = Work done by F_nc, so W_net = W_nc + W_c.

Work-- Energy Theorem revisited W_net = W_nc + W_c = DK.
But since W_c = -DU, W_net = W_nc - DU = DK, so
W_nc = DK + DU =DE, (Remember E represents total energy.)
Spring Pot. Energy
F_s = -Kx, so W_s = ÝF_sdx = Ý-kxdx = -.5k(x² - x_o²) = -DU_s , so DU_s = .5k(x² - x_o²) , or,
if x_o = 0, U_s = .5kx² , so the work done by gravity or a spring is W_g or W_s = -DU_s or g