Dynamics

1
Dynamics

• Chapter 4

2
Expectations

• After Chapter 4, students will
• understand the concepts of force and inertia.
• use Newtons laws of motion to analyze situations
  involving force, inertia, and acceleration.
• understand the meaning of dynamic equilibrium and
  identify objects or systems that are, or are not,
  in equilibrium.

3
Expectations

• After Chapter 4, students will
• recognize the properties of, and perform
  calculations involving, particular kinds of
  forces
• gravitational
• normal
• frictional
• tension

4
Force and Inertia

• Newtons second law of motion If an object
  experiences a net (unbalanced) force, it
  accelerates in the direction of the force. The
  magnitude of its acceleration is directly
  proportional to the magnitude of the force, and
  inversely proportional to its inertia (mass).

5
Force and Inertia

• Newtons second law of motion If an object
  experiences a net (unbalanced) force, it
  accelerates in the direction of the force. The
  magnitude of its acceleration is directly
  proportional to the magnitude of the force, and
  inversely proportional to its inertia (mass).
• In mathematical terms

6
Force and Inertia

• Sir Isaac Newton 1642 1727
• Natural philosopher
• (scientist / mathematician)
• the original mathematical
• physicist, and inventor of
• the calculus. He was also
• greatly interested in theology.

7
Force and Inertia

• We need more definition of the terms force and
  mass.
• Force a push or a pull, which tends to produce
  acceleration.
• Mass the property of an object which resists
  acceleration by a force (inertia), and which is
  determined by the amount of matter that
  constitutes the object.

8
Force and Inertia

• Mass a scalar quantity
• SI unit the kilogram (kg) a basic unit
• Force a vector quantity. We get the dimensions
  and units of force from Newtons second law
• dimensions
• SI units

9
Forces and Free-Body Diagrams

• We said that an object accelerates as a result of
  the net or unbalanced force acting on it.
• More than one force can act on a body at the same
  time.
• Forces are vectors. We can add them together as
  vectors to determine the net or total force
  acting on an object. The net force is simply the
  vector sum of all the forces acting on an object.

10
Forces and Free-Body Diagrams

• A convenient method for keeping track of the
  forces acting on an object is to draw a
  free-body diagram for it.
• A free-body diagram
• is drawn of a single object, and shows that
  object.
• shows every force that acts on an object.
• does not show any force that acts on anything
  besides the object.

11
Forces and Free-Body Diagrams

• Example a wooden crate is being dragged across a
  level floor by a rope that makes an angle of 30
  with the horizontal.

12
Forces and Free-Body Diagrams

• Resolve forces into x and y components (as
  needed)

13
Forces and Free-Body Diagrams

• Add the x-direction forces, and also the
  y-direction ones. The resultant or net force is
  entirely horizontal, toward the left. The crate
  accelerates in that direction. If its mass is
  17.9 kg

14
Equilibrium Newtons First Law

• Why did we discuss Newtons second law before
  discussing his first law?
• Because the first law is really a special case of
  the second.
• In words an object remains at rest, or moves in
  a straight line with constant velocity, unless it
  is acted on by a net force.

15
Equilibrium Newtons First Law

• Expressed mathematically, the second law
• and the first law
• An object is in equilibrium if the sum of the
  forces (net force) acting on it is zero. Its
  velocity is constant. It does not accelerate.

16
Action - Reaction Newtons Third Law

• If object A exerts a force on object B, Newtons
  third law says that object B exerts a force,
  having the same magnitude and the opposite
  direction, on object A
• FAB and FBA are an action-reaction pair.

17
Kinds of Forces

• Three fundamental categories of forces
• Gravitational
• Weak (electromagnetic)
• Strong nuclear
• They are listed here in ascending order of
  strength.

18
Gravitational Force

• As with many other things, we have Isaac Newton
  to thank for the first mathematical description
  of the operation of the gravitational force.
• Newtons law of universal gravitation any two
  objects exert equal attractive forces on each
  other. The magnitude of the forces is
  proportional to the product of the objects
  masses, and inversely proportional to the square
  of the distance separating them.

19
Gravitational Force

• Newtons law of universal gravitation
  (mathematically)
• We can make this proportionality into an equation
  by introducing a constant of proportionality, G
• G is called the universal gravitational constant.

20
Gravitational Force

• Do not confuse G, the gravitational constant,
  with g, the acceleration due to gravity!
• If we use SI units, the value of G is 6.6710-11
  Nm2/kg2.
• If one of the masses is the Earths mass, and r
  is the Earths radius, the gravitational force is
  called the weight of an object.

21
Gravitational Force

• We can use this expression for the gravitational
  force to calculate the value of g, the
  acceleration due to gravity at the Earths
  surface.
• Ordinarily, we express an objects weight as

22
Normal Force

• The atoms of which objects are made are
  surrounded by electron clouds. Due to the
  electromagnetic (weak) force, these electron
  clouds resist being brought close together.
  Their resistance to being actually merged is
  extremely powerful.
• When objects are in macroscopic contact, each
  exerts a force on the other to prevent their
  occupying (macroscopically) the same space at the
  same time. These forces are called normal forces.

23
Normal Force

• If objects A and B are in contact
• They exert equal and opposite normal forces on
  each other. , in accordance with
  Newtons third law.
• The directions of the forces are perpendicular to
  the contact surface. (Normal is another way of
  saying perpendicular.)
• The magnitude of the forces is as large as
  necessary to prevent the objects from merging.

24
Normal Force

• Example a box rests (equilibrium) on a table.
• Since the box is in equilibrium
• (does not accelerate), the sum of
• the forces acting on it must be zero.
• The magnitude of the normal force
• on the box is equal to its weight.

25
Apparent Weight (Nonequilibrium)

• Frame of reference another name for a
  coordinate system
• A coordinate system that does not accelerate is
  also called an inertial reference frame.
• What happens in a frame of reference that does
  accelerate (a non-inertial frame)?

26
Apparent Weight (Nonequilibrium)

• Consider an elevator car that can accelerate up
  or down or can move with constant velocity or
  be at rest.
• When it accelerates, it is a non-inertial frame
  of reference. When it does not accelerate, it is
  an inertial frame.
• We place a scale inside the elevator, and an
  object on the scale. A scale reports the normal
  force that it exerts on an object (and that the
  object exerts on it).

27
Apparent Weight (Nonequilibrium)

• When the elevator moves with
• constant velocity and is an inertial
• frame
• the normal force exerted
• (and reported) by the scale
• is equal to the weight of the
• object

28
Apparent Weight (Nonequilibrium)

• When the elevator accelerates upward
• the normal force exerted
• and reported by the scale
• (the objects apparent weight)
• increases

29
Apparent Weight (Nonequilibrium)

• When the elevator accelerates downward
• the normal force exerted
• and reported by the scale
• (the objects apparent weight)
• decreases

30
Frictional Forces

• Another force that objects in contact exert on
  each other is the frictional force.
• Two kinds of frictional force
• Static when the objects are stationary with
  respect to each other.
• Kinetic when the objects move relative to one
  another.

31
Frictional Forces

• Where do frictional forces come from?
• Even objects whose surfaces are macroscopically
  smooth have microscopic textures that tend to
  interlock.

32
Frictional Forces

• Frictional forces are exerted parallel to the
  contact area between objects.
• Their direction is such as to oppose motion
  between the objects.
• The static frictional force is as large in
  magnitude as necessary to prevent motion but
  not larger than a calculable maximum magnitude.

33
Frictional Forces

• Once motion between the contacting objects takes
  place, the frictional force become a kinetic one.
  The kinetic frictional force between two objects
  is smaller than the maximum static frictional
  force.
• The magnitude of the frictional force depends on
  the normal force, and on a coefficient of
  friction which is determined by the materials in
  the objects, as well as the condition / texture
  of their contacting surfaces

34
Frictional Forces

• Rearranging as a defining equation
• We see that the coefficient of friction is a
  dimensionless ratio of the magnitudes of two
  forces.
• There are two coefficients of friction for any
  combination of surfaces, yielding two frictional
  forces

35
Tension Forces

• Tension means both the forces applied to the
  ends of a rope, string, belt, ribbon, wire,
  chain, cable, thread, etc. and the force that
  the rope-like object (R-LO) exerts to resist
  being pulled apart.
• The reluctance of R-LOs to be pulled apart means
  that when you pull on one end, the other end
  pulls on whatever it may be attached to. We can
  think of R-LOs as being transmitters of pulling
  forces.

36
Tension Forces

• If the R-LO has mass in a non-equilibrium
  situation, the tension force is diminished in
  transmission

37
Important to Remember

• If an object or system does not accelerate (a
  0), it is in equilibrium, and
• If the object or system does accelerate (a ? 0),
  it is not in equilibrium, and
• In either case, Newtons second law applies. In
  equilibrium, the right-hand side becomes zero,
  and the second law becomes the first law.

38
Important to Remember

• Velocity and acceleration are vector quantities.
  So, Newtons laws apply separately in the X and Y
  directions
• Often (as in the case of a projectile), an object
  is in equilibrium in one direction and not in
  equilibrium in the other. Both the dynamic and
  kinematic situations must be considered
  separately in each axis.