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The Physics of Tops
presented by
Luis A. Martínez-Pérez, Ph.D.
Associate Professor
Science Education
Florida International University
Professional Development Series
June 18, 2005
Conceptual Ideas
TOPS
Newton's 1st Law
Newton's first law applies to an object at rest or
else moving at constant velocity. It states
simply that there can be no acceleration without
forces. More technically:
An object at rest or moving at a
constant velocity will continue to do
so unless acted upon by an external
force.
Newton's 2nd Law
The 2nd law of Newton describes
what happens quantitatively when
a force does act upon an object:
Force of an object = mass x acceleration
We thus see that the acceleration of an object
is always proportional to the force acting on it.
Double the force, and the acceleration
doubles; halve the force and the acceleration
is cut in half.
Newton's 3rd Law
Newton's 2nd law deals with a single
object on which a force is exerted. The
3rd and last law of motion discovered
by Newton explains what happens to
the object that is exerting the force. The
3rd law can be summarized by stating
that:
For every action there is an equal and
opposite reaction.
Motion of an Object
Position
Velocity
Acceleration
Position
The first idea needed in describing the
motion of an object is its position relative
to some fixed reference point. This entails
two ideas: the distance the object is away
from the reference point, and also the
direction relative to that reference point.
Average
Velocity
For an object in motion the concept of velocity
is important. There are two types of velocity that
can be defined. Suppose an object moves from
one point to another in a certain time interval.
The average velocity during that time interval is
defined as:
Average Velocity = Change in position divided by time
Instantaneous
Velocity
The average velocity only depends on the initial
and final points of the travel - it knows nothing of
the details of the motion between these points.
For this reason one introduces the concept of
instantaneous velocity, which applies to a
single point, or time of the motion.
Average
Acceleration
The acceleration of an object describes the rate of
change of velocity. Also just as with velocity, there
are two types of acceleration. The first - the average
acceleration - involves the change in instantaneous
velocity over a given time interval:
Average acceleration =change of velocity over time
Instantaneous
Acceleration
We can also define the instantaneous
acceleration at a point A as the limit of the
average acceleration between point A and a
nearby point B as the interval between A and B
becomes zero:
The instantaneous acceleration at A is the
average acceleration between A and B as A
approaches B.
Rotational Motion
Circular motion
Motion of a body about a fixed axis
When we discussed linear motion, we
introduced three measures of such motion:
position, velocity, and acceleration. For
circular motion we can define analogous
quantities:
■ Angular Position
■ Angular Velocity
■ Angular Acceleration
The first is the angular position,
conventionally denoted by θ, as in the figure
below. This is the angle at a particular
instant in time that the object makes with
respect to some fixed reference axis.
Angular Displacement
One of the reasons that the description
of circular motion in terms of angular
displacement is so useful is that it can
be applied to the motion of extended,
rigid objects rotating about a fixed axis
Angular velocity
In analogy with the concept of velocity for linear motion, the
angular velocity for rotational motion can be defined. One first
introduces the average angular velocity over a time by which
the object moves from a point A on a circle to point B:
Average angular velocity = change in angular position over time
Instantaneous angular velocity
The instantaneous angular velocity at a point B is
then defined to be the average angular velocity
between point A and a point B as the point A
approaches point B:
The instantaneous angular velocity at A is the average
angular velocity between A and B as A approaches B.
Angular acceleration
The last analogy with linear motion is the concept of angular
acceleration. As with angular velocity, one first introduces the
average angular acceleration between two points A and B as A
approaches B.
Average angular acceleration = change in angular velocity over time
Instantaneous angular acceleration
Instantaneous angular acceleration at A is defined to be the average
angular acceleration between point A and a point B as the point A
approaches B:
The instantaneous angular acceleration at A is the average
angular acceleration between A and B as A approaches B.
GRAVITY
Newton's Law of
Gravity
Every particle attracts every other particle with
a force that is proportional to the product of
their masses and inversely proportional to the
distance between them.
Sir Isaac Newton:
The
Universal Law of
Gravitation
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Center of Gravity
The center of gravity is a geometric
property of any object. The center of gravity
is the average location of the weight of an
object.
Center of Gravity
Activity
BALANCE
A force is a push or a pull. A force can give
energy to an object causing the object to
start moving, stop moving, or change its
motion. Forces occur in pairs and can be
either balanced or unbalanced. Balanced
forces do not cause a change in motion.
They are equal in size and opposite in
direction.
Moment of Inertia
Moment of inertia is the name given
to rotational inertia, the rotational
analog of mass for linear motion. It
appears in the relationships for the
dynamics of rotational motion. The
moment of inertia must be specified
with respect to a chosen axis of
rotation.
For a point mass the moment of
inertia is just the mass times the
square of perpendicular distance to
the rotation axis, I = mr2. That point
mass relationship becomes the basis
for all other moments of inertia since
any object can be built up from a
collection of point masses.
Rotational Kinetic
Energy
Kinetic energy is the energy of
motion. An object which has motion
- whether it be vertical or horizontal
motion - has kinetic energy. There
are many forms of kinetic energy vibrational (the energy due to
vibrational motion), rotational (the
energy due to rotational motion),
and translational (the energy due to
motion from one location to
another).
Rotational kinetic energy
Recall that an object of a certain mass
moving with particular speed will have an
associated kinetic energy mass x speed2. An
object spinning about an axis will also have
associated with it a kinetic energy, composed
of the kinetic energies of each individual part
of the object. These individual contributions
may be summed up to give an expression for
the total kinetic energy of the spinning object:
Rotational kinetic energy
= ½ moment of inertia x
(angular speed)2
Friction
is the resistive force that occurs
when two surfaces travel along
each other when forced together.
It causes physical deformation and
heat buildup.
Catapult at the Miami Museum of
Science
Projectile motion refers to
the motion of an object
projected into the air at an
angle.
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Physics Content Outline
The Physics Classroom
Newton’s Laws of Motion
The Scribbling Top
Projectile Motion
Universal Law of Gravitation
CONTACT INFORMATION
Luis A. Martínez-Pérez, Ph.D.
Associate Professor
Middle and Secondary Science
Education
College of Education EB 348B
Florida International University
University Park
Miami, Florida 33199
Office Telephone: (305) 348-3215
Office Fax: (305) 348-2086
Home Fax: (305) 385-0005
E-mail: [email protected]
Web Page:
http://www.fiu.edu/~sste/sste_index.htm