Professional Development Series

June 18, 2005

The Physics of Tops

presented by

Luis A. Martínez-Pérez, Ph.D.Associate Professor

Science EducationFlorida International University

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.

AverageVelocity

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

InstantaneousVelocity

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.

AverageAcceleration

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

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.

InstantaneousAcceleration

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

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.

Instantaneous angular velocity

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 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.

Moment of Inertia

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).

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

Rotational kinetic energy = ½ moment of inertia x

(angular speed)2

is the resistive force that occurs when two surfaces travel along each other when forced together. It causes physical deformation and heat buildup.

Friction

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

Just Click on the Star of Your Choice

Universal Law of Gravitation

CONTACT INFORMATION

Luis A. Martínez-Pérez, Ph.D.Associate ProfessorMiddle and Secondary Science EducationCollege of Education EB 348BFlorida International UniversityUniversity ParkMiami, Florida 33199Office Telephone: (305) 348-3215Office Fax: (305) 348-2086Home Fax: (305) 385-0005

E-mail: martinel@fiu.edu

Web Page: http://www.fiu.edu/~sste/sste_index.htm