Announcements

1
Announcements

• Homework 5 due Wednesday at end of class

2
Work Done by Spring Force

• Spring is an example of a variable force
• Cant use
• Need to integrate work done as spring stretches
• Example align coordinate system so spring
  stretches along the x axis, x 0 corresponds to
  position of object attached to spring in the
  relaxed state when no force is applied
• Calculate work done on object in moving from its
  relaxed state to a position x along the spring
  axis

3
Example Bungee Jumping

• You have to be a little crazy to jump off a
  perfectly fine bridge with only a little cord
  tied to you
• Will you survive?
• 80 m above rocky chasm dearly want to avoid
  going splat on rocks
• Relaxed length of bungee cord (y0) is 50 m
• Your mass (m) is 50 kg
• Spring constant (k) is 100 N/m, bungee cord
  stretches a distance x
• Calculate total work (0 when bungee cord stops
  your fall)
• y0y 77.6m you survive (thanks to Ph5 but
  dont try this in lab!!)

y0
0
y
4
Work and Power

• Power is the rate at which work is being done
• SI unit of power Watt
• 1 Joule/s 1 W
• 1 horsepower 746 W
• Watt most often associated with electrical power,
  but can be used whenever work is being done
• Example Electricity measured in kilowatt hours
• 1 kW-hour 1000 W/kW 3600s/hour 3.6x106 Ws
  (or Joules)
• Example pedaling for power
• Force pushing/turning bicycle pedals does work
• Work done turning generator converted to
  electricity

5
Work and Power

• Power is an instantaneous quantity
• Go back to work done in a small displacement ds
• Example What velocity can a 1 HP motor pull a
  100 kg object on a flat surface with a
  coefficient of friction m 0.2

6
Kinetic Energy and Work Summary

• For a constant force, define work as
• If force varies, need to compute line integral
• Conservative forces have work that doesnt depend
  on path
• Doing work on an object changes its kinetic
  energy
• Spring force is a common example of a
  non-constant force
• Power measures rate work is being done

7
Potential Energy

• Example A large boulder teeters on the edge of a
  high cliff
• The slightest push would send it crashing down
• As it fell, gravity would do work on the boulder,
  increasing its kinetic energy
• Once at the bottom of the cliff, the boulder has
  lost the potential energy it had at the top of
  the cliff
• Example Stretched spring connected to air track
  glider
• If the spring is released, it will do work on the
  glider, giving kinetic energy to the glider and
  reducing the potential energy of the spring
• The kinetic energy reaches its maximum when the
  spring is in its relaxed state
• Further movement of the glider compresses the
  spring, doing negative work on the glider,
  decreasing the kinetic energy of the glider and
  increasing the potential energy of the spring
• As glider moves back and forth, we have potential
  energy being converted to kinetic energy (and
  vice versa)

8
Definition of Potential Energy

• We define the change in potential energy based on
  the work done
• We previously saw that there is a change in
  kinetic energy when work is done
• If we do negative work, then the kinetic energy
  decreases and the potential energy increases
• Example upward motion of a ball tossed in the
  air
• If we do positive work, then the kinetic energy
  increases and the potential energy decreases
• Example downward motion of the ball

9
Conservative and Non-Conservative Forces

• Examples we have used utilize conservative forces
• These forces have the property that the work done
  in going from A ? B can be recovered by returning
  to the starting point (B ? A)
• Total work done in going from A ? B ? A is W1
  W2
• No work is done if I just stay at point A
• For conservative forces, work is path independent
• Thus, W1 W2 0 or W1 -W2
• Concept of potential energy isnt really
  applicable to non-conservative forces
• We could certainly calculate the quantity defined
  as potential energy, but unless we can get the
  energy back it isnt really potential energy
• Example Work done by friction or drag forces
  heats up object - Thermodynamics tells us that
  its impossible to undo such processes

10
Taking Advantage of Path-Independence

• Use the path-independence to your advantage!
• Consider the rather tame roller coaster drawn
  below
• Assume a car starts at rest at the top of the
  roller coaster
• The work done by gravity is the same for paths A
  and B
• Do the easier calculation of path B
• Horizontal segment does no work
• Vertical segment does work mgh where h is the
  vertical drop
• The velocity of the car at the end point is
• Example balls rolling on tracks

11
Gravitational Potential Energy

• Calculate change in gravitational potential
  energy for a mass m from the work done

12
Absolute Potential Energy

• Formulation of potential energy only describes
  changes in potential energy DU
• You might ask When is the potential energy 0? If
  I know that, I can measure the potential energy
  by the change from this value
• The answer is Anywhere you want it
• We only know how to measure changes in potential
  energy, so the point where the potential energy
  is 0 is arbitrary
• Use this to your advantage pick a point that
  makes the calculation easier
• For gravitational potential energy, one common
  choice is to pick U 0 for the floor
• The potential energy a distance y above the floor
  is then U(y) mgy
• Dont hesitate to pick other locations just be
  clear where U 0 is located

13
Potential Energy of a Spring

• Calculate the potential energy of a spring
  aligned along the x axis
• If we choose U 0 for x 0,