Lecture 12 Goals Analyze situations with Work and a Constant Force Analyze situations with Work and a Varying Force Introduce Hooke’s Law springs Relate external or “net” work to speed Introduce concepts of Kinetic and Potential energy 1

Definition of Work, The basics Ingredients: Force ( F ), displacement (  r )   r displacement F Work, W, of a constant force F acts through a displacement  r : W ≡ F · r (Work is a scalar) Work tells you something about what happened on the path! Did something do work on you? Did you do work on something? If only one force acting: Did your speed change?

Net Work: 1-D Example (constant force) A force F = 10 N pushes a box across a frictionless floor for a distance x = 5 m. F q = 0° Start Finish x Net Work is F x = 10 x 5 N m = 50 J 1 Nm ≡ 1 Joule and this is a unit of energy

Net Work: 1-D 2nd Example (constant force) A force F = 10 N is opposite the motion of a box across a frictionless floor for a distance x = 5 m. Start Finish q = 180° F x Net Work is F x = -10 x 5 N m = -50 J

Work: “2-D” Example (constant force) An angled force, F = 10 N, pushes a box across a frictionless floor for a distance x = 5 m and y = 0 m x F q = -45° Start Finish Fx (Net) Work is Fx x = F cos(-45°) x = 50 x 0.71 Nm = 35 J

Infinitesimal Work vs speed along a linear path x F Start Finish Fx

Work vs speed along a linear path If F is constant DK is defined to be the change in the kinetic energy

Work in 3D…. x, y and z with constant F:

Examples of External Work DK = Wext Pushing a box on a smooth floor with a constant force; there is an increase in the kinetic energy Examples of No External or Net Work Pushing a box on a rough floor at constant speed Driving at constant speed in a horizontal circle Holding a book at constant height This last statement reflects what we call the “system” ( Dropping a book is more complicated because it “potentially” involves changes in the “potential” energy. The answer depends on what we call the system. )

Exercise Work in the presence of friction and non-contact forces A box is pulled up a rough (m > 0) incline by a rope-pulley-weight arrangement as shown below. How many forces (including non-contact ones) are doing work on the box ? Of these which are positive and which are negative? State the system (here, just the box) Use a Free Body Diagram Compare force and path v 2 3 4 5

Work and Varying Forces (1D) Consider a varying force F(x) Area = Fx Dx F is increasing Here W = F · r becomes dW = F dx Fx x Dx Start Finish F F q = 0° Dx Work has units of energy and is a scalar!

A Hooke’s Law spring Most springs have a force which increases linearly with displacement from the equilibrium Displacement (m) 1.0 2.0 3.0 4.0 20 30 10 40 50 Force (N) spring at an equilibrium position m x F spring compressed x F Slope: spring constant k

Example: Work Kinetic-Energy Theorem with variable force How much will the spring compress (i.e. x = xf - 0) to bring the box to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (vo) on frictionless surface as shown below ? spring compressed x vo m to F spring at an equilibrium position V=0 t

Example: Work Kinetic-Energy Theorem with variable force How much will the spring compress (i.e. x) to bring the box to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (vo) on frictionless surface as shown below ? x vo m to F spring compressed spring at an equilibrium position V=0 t Notice that the spring force is opposite the displacement For the mass m, work is negative, Ws For the spring, work is positive, Wapp

What happens when I release the spring? How fast will the block be moving when it loses contact with the spring? V=0 F m spring compressed vo m spring now at an equilibrium position

Work done by friction SFy = 0 = -mg + N SFx = m ar = m v2 /R A small block slides inside a hoop of radius r. The walls of the hoop are frictionless but the horizontal floor has a coefficient of sliding friction of m. The block’s initial velocity is v and is entirely tangential. How far, in angle around the circle does the block travel before coming to rest ? SFy = 0 = -mg + N SFx = m ar = m v2 /R Ff = m N = m mg W = Ff Ds = DK = ½ mvf2 – ½ mvi2 = – ½ mv2 - m mg Ds = – ½ mv2 s = v2 / 2mg = q r q = v2 / (2mgr) v Start Finish R

Looking down on an air-hockey table with no air flowing (m > 0). Friction….again Looking down on an air-hockey table with no air flowing (m > 0). WPath 1 friction = Ff Dxpath 1 WPath 2 friction = Ff Dxpath 2 Dxpath 2 > Dxpath 1 so W2 > W1 Path 2 Path 1

The spring and friction have an important difference For the spring I can reverse the process and recover the kinetic energy The compressed spring has the ability to do work For the spring the work done is independent of path The spring is said to be a conservative force In the case of friction there is no immediate way to back transfer the energy of motion In this case the work done is dependent on path Friction is said to be a non-conservative force

Potential Energy (U) For the compressed spring the energy is “hidden” but still has the ability to do work (i.e., allow for energy transfer) This kind of “energy” is called “Potential Energy” The gravitation force, if constant, has the same properties.

½ m vyi2 + mgyi = ½ m vyf2 + mgyf = constant Mechanical Energy If only “conservative” forces, then total mechanical energy (potential U plus kinetic K energy) of a system is conserved For an object in a gravitational “field” ½ m vyi2 + mgyi = ½ m vyf2 + mgyf = constant U ≡ mgy K ≡ ½ mv2 Emech = K + U Emech = K + U = constant K and U may change, but Emech = K + U remains a fixed value. Emech is called “mechanical energy”

For Tuesday Read Chapter 8 Start HW6