Ideas of Modern Physics

1
Ideas of Modern Physics
When you can measure what you are speaking about,
and express it in numbers, you know something
about it but when you cannot measure it, when
you cannot express it in numbers, your knowledge
is of a meagre and unsatisfactory kind....
William Thomson, Lord Kelvin (1891)
2
Review

• Free objects move uniformly (constant speed and
  direction) in inertial frames of reference
• Simplest accelerated motions uniform
  acceleration in a straight line and uniform
  circular motion
• All objects freely falling on the surface of the
  earth have constant acceleration a g 9.8 m/s/s

3
Today

• Inertial mass
• Quantities of motion Momentum and kinetic energy
• A fundamental mechanical law conservation of
  momentum

4
Inertial Mass

• Example Consider a heavy sliding object in
  uniform motion. A force is required to change its
  velocity in magnitude (increase or decrease) or
  in direction (sideways)
• Inertial mass M (kg) determines resistance to
  change in velocity and is proportional to weight.

5
Force, mass, and weight

• A force F produces an acceleration in inverse
  proportion to the mass aF/m
• Weight is the force of earths surface gravity
  and is proportional to inertial mass
• WF(gravity) m g (kg-m/s/s newton)
• 1 kg (mass) weighs 9.8 newton on earth.
• A large mass is subject to a large gravitational
  force but has correspondingly large inertia
• Gravitational acceleration aF(gravity)/m
  (mg)/m g is independent of mass!

6
Inertia of extended objects

• A free object maintains its straight line motion
• A free spinning object maintains its spin
• An extended object has inertia in translation and
  rotation.

7
Linear momentum

• Define linear momentum p as the product of mass
  and velocity p m v
• Linear momentum has magnitude and direction and
  is a measure of the inertia in translation
• A free object maintains its linear momentum
• A force is required to change linear momentum

8
Angular momentum

• If mass (element) m moves in a circle of radius
  r, angular momentum L is the product of distance
  and linear momentum L r p r m v
• Angular momentum is a measure of the inertia
  under rotation
• A spinning object has angular momentum about the
  axis of rotation
• A force with a lever arm (torque) is required to
  change L

9
Kinetic energy

• Define K (1/2) m v2
• K has magnitude but not direction - the sign of v
  doesnt matter because v is squared.
• A translating or spinning object has kinetic
  energy.

10
Systems of objects

• Consider two masses m and M with speeds v and V
• Total momentum P p1p2mv MV
• Total angular momentum L L1L2
• Total kinetic energy K K1K2

11
Momentum conservation

• A closed system is a collection of objects which
  interact only with themselves.
• Objects exchange momentum but total linear and
  angular momentum remain unchanged!

12
Example collision

• M with velocity V strikes m of velocity v.
• After the collision M has velocity V and m has
  velocity v.
• MVmv MVmv
• Special case M hits a stationary mass m and
  sticks to it. If v0 and Vv then MV(Mm)Vgt
  V MV/(Mm)

13
Energy conservation

• Kinetic energy may be converted to internal
  atomic motion so not appear to be conserved. If
  it is conserved, a collision is called elastic
  (think billiard balls). If it is not conserved, a
  collision is called inelastic.

14
Rocket science

• Propeller driven planes swim through air.
• To accelerate in empty space, throw mass
  backwards. Momentum conservation implies forward
  recoil.
• Thrust/propulsion force is equal to the rate at
  which momentum is ejected.

15
More on collisions

• Consider a billiard ball striking headon another
  which is at rest and suppose the collision is
  elastic.
• Momentum and energy are conserved.
• The struck ball must carry away all of the
  momentum and energy and the striking ball be
  brought to rest.

16
Angular momentum conservation

• Consider a spinning skater whose outstretched
  arms have angular momentum L mvr. If arms are
  pulled in (r decreased) the speed v and spin
  frequency increase to keep L constant.

17
Example

• Equal masses on a dumbbell of length 2r rotating
  about in center
• P mv m(-v) 0
• L mvr mvr 2mvr
• K (1/2)mv2 (1/2)mv2 mv2

18
Center of mass

• Suppose mass m position x, velocity v
• Suppose mass M position X, velocity V
• MV (mv) rate of change of MX(mx)
• MVmv constantgt MXmx changes uniformly.
  MVmv0gt
• X(cm) (mxMX)/(mM) is constant.
• SO, a closed system can not (self) accelerate its
  center of mass!

19
Summary

• We can quantify inertial motion in terms of
  linear and angular momentum and energy.
• Momentum and energy(with caveats) are conserved
  in all interactions!
• There is something universal in ALL mechanical
  processes. Hmm