PHYSICS 1: Classical Mechanics

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PHYSICS 1Classical Mechanics

• Ronald L. Westra PhD
• Dept. Mathematics
• Maastricht University

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Classical Mechanics

• Statics
• Dynamics
• Energy
• Simple Machines

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Classical Mechanics

• Classical Mechanics is oldest part of Modern
  Physics
• Galileo Galilei, Christiaan Huygens, Isaac
  Newton, Leibniz,
• Both fundamental laws of Nature as well as valid
  for accurately describing macroscopic objects

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Classical Mechanics

• Statics
• Kinematics or translational Dynamics
• Rotational dynamics
• Many-body dynamics
• Conservation Laws (momentum, energy,)

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1. STATICS

• Forces and friction
• Normal force
• Inclined Planes
• Decomposition of forces
• Torque ( momentum or couple)

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1. STATICS

• Forces
• A force is a physical concept that works on
  material objects
• A force has a magnitude and a direction hence
  it is a vector
• Either forces cancel exactly and the object
  remains in rest (STATICS), or there is a net
  resulting force that causes a uniform
  acceleration of the object (Newtons laws)

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1. STATICS
FN (Normal force)
Fg (weight) m.g
FN Fg (magnitude) FN Fg 0
(vector) No net force the object remains in
rest
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1. STATICS
FN (Normal force)
Fg (weight) m.g
Fg ? 0 (vector) Net resultant force the
object accelerates in direction of the force
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1. STATICS
Q1 At what force will the object start to move?
FN (Normal force)
F1 (pushing force)
Ff (friction force)
Fg (weight) m.g
FN Fg F1 Ff 0 (vector) No net force
the object remains in rest Ff f. FN
with f the friction coefficient of the
surface
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(No Transcript)
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Inclined Plane
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1. STATICS
Q2 At what angle a will the object start to
slide down?
INCLINED PLANE
angle a
Answer When tan a f
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1. STATICS

• LEVERS and TORQUES
• The concept of torque, also called moment or
  couple, originated with the studies of Archimedes
  on levers.
• The rotational analogues of force, mass, and
  acceleration are torque, moment of inertia, and
  angular acceleration, respectively.

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The torque is a vector
1. STATICS

• The force applied to a lever multiplied by its
  distance from the lever's fulcrum ( the length
  of the lever arm) is its torque.
• A force of three newtons applied two meters from
  the fulcrum, for example, exerts the same torque
  as one Newton applied six meters from the
  fulcrum. This assumes the force is in a direction
  at right angles to the straight lever.
• The direction of the torque can be determined by
  using the right hand grip rule curl the fingers
  of your right hand to indicate the direction of
  rotation, and stick your thumb out so it is
  aligned with the axis of rotation. Your thumb
  points in the direction of the torque vector.

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1. STATICS
TORQUE (Momentum, Couple)
F (force)
r (arm)
T r x F The Torque is a vector with
size r.F and perpendicular to r and F.
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1. STATICS
TORQUE (Momentum, Couple)
F (force)
a
r (arm)
r- (perpendicular projection of arm)
T r - F r. F
.sin(a) T r x F (vector) The Torque is
a vector with size r.F and perpendicular
to r and F.
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Levers
Equilibrium condition No net resultant
torque Ttotal 0 (vector) D1.F1 (-
D2.F2) 0
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Levers
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First Class Lever
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Second Class Lever
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Third Class Lever
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1. STATICS
F4
F2
F1
F3
CONDITION for equilibrium Sk Fk 0 (No net
force vector) Sk Tk 0 (No net torque
vector) the object remains in rest
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1. STATICS
TORQUE statics
F4
F2
r2
C
r4
r3
r1
F1
F3
1. Select a center C (any center!) 2. Determine
the torque relative to C 3. Sk Tk 0 r1 x
F1 r2 x F2 r3 x F3 r4 x F4 ?
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1. STATICS
Q3 How large must F be such that the system is
in equilibrium?
F ?
B
armlength OB 1.5 m
A
armlength OA 1 m
FR (reaction force)
Fg (weight) m.g 100 N
angle a 30o
O
Here Fg 100 N, armlength OA 1 m ,
armlength OB 1.5 m, a 30 o
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1. STATICS
Q3 How large must F be such that the system is
in equilibrium?
F ?
F2 F.cos a
B
armlength OB 1.5 m
A
armlength OA 1 m
F1 F sin a
FR (reaction force)
FR2
angle a 30o
Fg (weight) m.g 100 N
O
FR1
STEP 1 Select reference frame O as origin,
horizontalvertical coordinates STEP 2
decompose the forces STEP 3 static forces
X-axis FR1 (- F sin a) 0
1 static forces Y-axis FR2 F cos a -
Fg 0 2 STEP 4 static
torque relative to O F.OB FgOA.cos a 0
3 Three linear equations with three variables
(F, FR1, FR2) solvable!
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Q3 How large must F be such that the system is
in equilibrium?
FR1 (- F sin a) 0
1 FR2 F cos a - Fg 0
2 F.OB FgOA.cos a 0 3 These
solve to FR1 Fg sin(2a)/3 ( 34.2 N) FR2
Fg (1 2cos²(a)/3) ( 35.3 N) F
2(Fgcos a)/3 ( 57.7 N) Notice that FR2/FR1
(2 - cos 2a)/sin 2a gt tan a, so the reaction
force FR does not point in the direction of arm
OB but above it!
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Simple Machines
1. STATICS
Five classical simple static mechanical machines
are

• Wedge
• Screw
• Lever
• Wheel and Axle
• Pulley

These are all mechanical devices that change the
direction and/or magnitude of a force. Examples
are shown later in Section 5
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2. KINEMATICS

• Newtons Laws
• Uniform velocity
• Forces and acceleration
• Ballistic trajectories

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The basic concepts of classical mechanics

• For simplicity real-world objects are modeled as
  point particles.
• Thus, the motion of an object is defined by a
  small number of parameters its position, mass,
  and the forces applied to it.

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The basic concepts of classical mechanics

• Objects with non-zero size have more complicated
  behavior than hypothetical point particles,
  because of the additional degrees of freedomfor
  example, a baseball can spin while it is moving.
• However, the results for point particles can be
  used to study such objects by treating them as
  composite objects, made up of a large number of
  interacting point particles.
• The center of mass of a composite object behaves
  like a point particle.

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Displacement and its derivatives

• The displacement, or position, of a point
  particle is defined with respect to an arbitrary
  fixed reference point O in space, located at the
  origin of the coordinate system, is defined as
  the vector r from O to the particle

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Velocity and speed

• The velocity, or the rate of change of position
  with time, is defined as the derivative of the
  position with respect to time

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Canonical momentum

• The (canonical) momentum p of a point particle
  with mass m and speed v is

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Acceleration

• The acceleration, or rate of change of velocity,
  is the derivative of the velocity with respect to
  time (the second derivative of the position with
  respect to time)

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Frames of reference

• Classical mechanics assumes the a special family
  of reference frames in which the laws of Nature
  take a simple form.
• These special reference frames are called
  inertial frames. They are characterized by the
  absence of acceleration of the observer.
• Consequences about the perspective of an event in
  two inertial frames, S and S', where S' is
  traveling at a relative velocity of to S.
• 1. the velocity of a particle from the
  perspective of S' is slower by than its velocity
  from the perspective of S
• 2. the acceleration of a particle remains the
  same regardless of reference frame
• 3. the force on a particle remains the same
  regardless of reference frame
• 4. the speed of light is not a constant in
  classical mechanics.

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Forces Newton's Second Law

• In his second law of motion Newton defined the
  relationship between force and momentum.
• It can be seen as a definition of force and mass,
  or as a fundamental postulate, a Law of Nature.

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Forces Newton's Second Law

• In words If a force is applied to a material
  object, the object will change its momentum such
  that its rate of change exactly equals the
  magnitude and direction of the force.

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Forces Newton's Second Law

• Notice that
• an object that loses or gains mass also changes
  its velocity.
• If there is no force, the total momentum is
  conserved

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Classical Mechanics
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Classical Mechanics
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2. KINEMATICS
Q4 Describe the motion of this object.
rain 0.01 kg/sec
FN (Normal force) 10 N
v 10 m/s
Ff (friction force) 0 N
Fg (weight) 10 N
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Q4 Describe the motion of this object.
There is only change in the horizontal
coordinate as the mass slowly increases as
m(t) 1 0.01.t, (NB g 10 m/s²) and there
is no net force. Now apply Newtons second law
d(m(t)v(t))/dt 0 ? m(t)v(t) constant
m(0)v(0) 10 v(t) 10/(1 0.01.t)
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2. KINEMATICS
Q5 Describe the motion of this object.
Ff (friction force) -0.1v
Fg (weight) 0.1 N
A rain drop falls freely under the force of
gravity. The force of air friction Ff is
directed opposite to the velocity of the droplet,
and has a size that is a constant (0.1 kg/sec)
times the magnitude of this velocity Ff
-0.1 v
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Q5 Describe the motion of this object.
There is only movement in the vertical direction
choose there your axis. The force that act
are gravity (0.1N) and air friction (-0.1.v), so
the total force points downward (hence is
negative) and equals Ftot - 0.1 0.1.v.
Let g 10 m/s² and the initial velocity at t0
be 0 m/s. Now apply Newtons second
law d(m(t)v(t))/dt Ftot - 0.1 0.1.v ?
use some calculus v(t) -100
100exp(-0.1.t) So, after some time the rain
drop falls freely under the force of gravity
with a constant velocity of 100 m/s
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Q5 Describe the motion of this object.
v(t) -100 100exp(-0.1.t)
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2. KINEMATICS
Q6 Describe the motion of this object.
target
100 m
A ballista launches a rock with 50 m/s in a
uniform gravitational field with g -10 m/s2.
What is the safest angle a with the horizon
in order to hit a target at a distance of 100
m? And what information is (not) required?
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Q6 Describe the motion of this object.
Fg (weight) 10 N g 10 m/s2
Here we have two independent movements Horizont
al Ftot 0, so d(mv1(t))/dt 0 ? v1(t)
constant v1(0) v cos a Vertical Ftot -10,
so d(mv2(t))/dt -10 ? v2(t) v2(0) 10t (v
sin a) 10t Therefore the vertical position z
is z(t) (v sin a)t 5t² . The bullet hits
the ground if z0, so when (v sin a)t 5t² 0,
i.e. at t0 (trivial, the start) and tmax (v
sin a)/5. The vertical position is then x(tmax)
(v cos a)tmax v² sin(2a)/10 250 sin(2a).
This must be equal to the position of the goal
100 m, so x(tmax) 250 sin(2a) 100 ? sin(2a)
0.4. This means that 2a 23.5782 or 2a 180 -
23.5782 156.4218. So a 11.7891 or a
78.2109. The smaller angle is better, for it
remains less long in the air and therefore is
more accurate.
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3. ROTATIONAL DYNAMICS

• Equivalence to Newtons Laws
• Angular momentum
• Torques and angular acceleration
• Precession and nutation
• Milankovitch cycles

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The equivalence to translational dynamics
3. ROTATIONAL DYNAMICS

• The rotational analogues of force, mass, and
  acceleration are torque, moment of inertia, and
  angular acceleration, respectively.

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Angular position

• The angular position of a point particle is
  defined as the angle of a line through the point
  particle and the origin O of the coordinate
  system with a fixed line originating from O

f
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Angular velocity

• The angular velocity is defined as the derivative
  of the angular position with respect to time

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Angular acceleration

• The angular acceleration, or rate of change of
  angular velocity, is the derivative of the
  angular velocity with respect to time (the second
  derivative of the angular position with respect
  to time)

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Angular momentum

• The angular momentum L of a point particle with
  momentum p at radius r from the axis of rotation
  is

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Angular momentum

• The angular momentum on a rigid body can be
  written in terms of its moment of inertia I and
  its angular velocity ?

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Torque and angular acceleration

• The torque on a body determines the rate of
  change of its angular momentum

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Torque and angular acceleration

• This relates to the inertia and angular
  acceleration as

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Comparison translational and angular movement
f
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Complex angular motion

• The torque caused by the two opposing forces Fg
  and -Fg causes a change in the angular momentum L
  in the direction of that torque. This causes the
  top to precess.

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Complex angular motion

• Precession and Nutation are also observed in the
  earths motion around the sun.
• They partially explain the cycles in the ice
  ages.
• (Milankovitch cycles)

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Complex angular motion

• Precession and Nutation of the gyroscope

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Milankovitch cycles

• Milankovitch cycles are the collective effect of
  changes in the Earth's movements upon its
  climate. The eccentricity, axial tilt, and
  precession of the Earth's orbit vary in several
  patterns, resulting in 100,000-year ice age
  cycles of the Quaternary glaciation over the last
  few million years.

The Earth's axis completes one full cycle of
precession approximately every 26,000 years. At
the same time, the elliptical orbit rotates, more
slowly, leading to a 21,000-year cycle between
the seasons and the orbit. In addition, the angle
between Earth's rotational axis and the normal to
the plane of its orbit moves from 22.1 degrees to
24.5 degrees and back again on a 41,000-year
cycle. Currently, this angle is 23.44 degrees
and is decreasing.
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4. ENERGY

• Forces and Work
• Potential Energy
• Kinetic Energy
• Rotational Energy
• Conservation of Energy

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Energy
Energy is the ability to do work. The Law of
Conservation of Energy states the total energy
of an isolated system remains constant despite
any internal change. Forms of energy
include kinetic, potential, thermal, elastic,
gravitational, sound, electromagnetic, chemical,
and nuclear.
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Work
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Kinetic energy
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Rotational energy
A uniform rotating point particle with constant
angular velocity ? and inertia I, has a
rotational energy Erot equal to Erot ½ I.?2
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Potential energy
For a point particle, the potential energy Ep
relates to a (so-called conservative) force field
that can perform an amount of work on the
particle. For a uniform gravitational field with
accelaration g the potential energy of a point
particle with mass m increases in the vertical
direction and at an altitude of z meters above
the origin (where the energy is defined as zero)
is Ep m.g.z
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4. ENERGY
Q7 Describe the motion of this object.
A elastic ball of mass 0.05 kg falls from a
height of 2 m to a floor, and loses 20 of its
kinetic energy when bouncing to the floor. We
are interested in the maximum heights in between
successive bounces, and the lengths of the
successive time intervals.
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Examples of Energy

• Kinetic Energy
• Harnessing
• Storing
• Potential Energy
• Gravitational
• Elastic

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Kinetic Energy
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Harnessing Kinetic Energy
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Harnessing Kinetic Energy
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Storing Kinetic Energy
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Storing Kinetic Energy
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Potential Energy
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Gravitational Potential Energy
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Elastic Potential Energy
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Elastic Potential Energy
Compression Spring Extesion Spring Constant-Fo
rce Spring Torsion Spring
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5. SIMPLE STATIC MECHANIC MACHINES

• Wedge
• Screw
• Lever
• Wheel and Axle
• Pulley

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Simple Machines
The five classical simple machines are

• Wedge
• Screw
• Lever
• Wheel and Axle
• Pulley

These are all mechanical devices that change the
direction and/or magnitude of a force.
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Wedge
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Screw
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Screw
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Wheel and Axle
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Pulley
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Fixed Pulley
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Movable Pulley
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Movable Pulley
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Compound Pulley
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Compound Pulley
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Compound Pulley
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Pulley and Belt
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Many Uses of the Pulley
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Other Mechanical Devices and Motions
There are practically infinite variations of
these simple machines. They can be combined with
each other and with other mechanical devices to
create a wide variety of movements.

• Gear
• Cam
• Crank
• Hydraulic Pump