PHYSICS | CONSERVATION OF ENERGY
THE CLIP
This video clip, shows Newton’s cradle which (usually) consists of an odd number of identical
steel balls each suspended by a bifilar suspension from a sturdy frame. It is commonly used
to demonstrate “the Conservation of Momentum” which states that, within some problem
domain, the amount of momentum remains constant; momentum is neither created nor
destroyed, but only changed through the action of forces as described by Newton's laws of
motion.
THIS DEMONSTRATES
… the Conservation of Momentum as described by Newton’s laws of motion.
What is Momentum? (Henderson 2004)
Definition: The momentum of an object is defined as its mass
multiplied
by its velocity (FHSST 2005).
Momentum refers to the quantity of motion that an object has. If an object is in motion then
it has momentum. Momentum can be defined as "mass in motion." All objects have mass; so
if an object is moving, then it has momentum - it has its mass in motion. The amount of
momentum which an object has is dependent upon two variables: how much stuff is moving
and how fast the stuff is moving. Momentum depends upon the variables mass and velocity.
In terms of an equation, the momentum of an object is equal to the mass of the object times
the velocity of the object.
Momentum = mass * velocity
In physics, the symbol for the quantity momentum is the small case "p"; thus, the above
equation can be rewritten as
p=m*v
where m = mass and v=velocity. The equation illustrates that momentum is directly
proportional to an object's mass and directly proportional to the object's velocity.
The units for momentum would be mass units times velocity units. The standard metric unit
of momentum is kg*m/s. Momentum is a vector quantity, i.e. a quantity which is fully
described by both magnitude and direction.
The Momentum of a System
Definition: The total momentum of a system is the sum of the momenta
of
each of the objects in the system (FHSST 2005).
The Law of Momentum Conservation
The Law of Action-Reaction (Henderson 2004)
A collision is an interaction between two objects which have made contact (usually) with each
other. As in any interaction, a collision results in a force being applied to the two colliding
objects. Such collisions are governed by Newton's laws of motion. Newton's third law of
motion says that...
... in every interaction, there is a pair of forces acting
on the two interacting objects. The size of the force on
the first object equals the size of the force on the second
object. The direction of the force on the first object is
opposite to the direction of the force on the second
object. Forces always come in pairs - equal and opposite
action-reaction force pairs.
Newton's third law of motion is naturally applied to collisions between two objects. In a
collision between two objects, both objects experience forces which are equal in magnitude
and opposite in direction. Such forces cause one object to speed up (gain momentum) and
the other object to slow down (lose momentum). According to Newton's third law, the forces
on the two objects are equal in magnitude. While the forces are equal in magnitude and
opposite in direction, the acceleration of the objects are not necessarily equal in magnitude.
In accord with Newton's second law of motion, the acceleration of an object is dependent
upon both force and mass. Thus, if the colliding objects have unequal mass, they will have
unequal accelerations as a result of the contact force which results during the collision.
Collisions are governed by Newton's laws. The law of action-reaction (Newton's third law)
explains the nature of the forces between the two interacting objects. According to the law,
the force exerted by object 1 upon object 2 is equal in magnitude and opposite in direction to
the force exerted by object 2 upon object 1.
Momentum Conservation Principle (Henderson 2004)
The law of momentum conservation can be stated as follows:
For a collision occurring between object 1 and object 2 in an isolated system, the total
momentum of the two objects before the collision is equal to the total momentum of the two
objects after the collision. That is, the momentum lost by object 1 is equal to the momentum
gained by object 2.
The above statement tells us that the total momentum of a collection of objects (a system) is
"conserved" – i.e. the total amount of momentum is a constant or unchanging value.
This Principle of Conservation of Linear Momentum is one of the most fundamental
principles of physics and it alone justifies the definition of momentum. Since momentum is
related to the motion of objects, we can use its conservation to make predictions about what
happens in collisions and explosions. If we bang two objects together, by conservation of
momentum, the total momentum of the objects before the collision is equal to their total
momentum after the collision (FHSST 2005).
Principle of Conservation of Linear Momentum
The total linear momentum of an isolated system is constant.
or
In an isolated system the total momentum before a collision
(or explosion) is equal to the total momentum after the
collision (or explosion).
Newton’s Cradle (Wikipedia 2005, Kurtus 2005)
Newton's cradle is a device that demonstrates conservation of momentum and energy. It is
constructed from a series of pendulums (usually 5) abutting one another. Each pendulum is
attached to a frame by two strings of equal length angled away from each other. This string
arrangement restricts the pendulums' movements to the same plane.
Newton's Cradle was invented in the late 1600s by scientist and mathematician Isaac
Newton. Since the 1970s, Newton's cradle has been a popular desk toy.
End balls move
When an end ball is pulled up and let go, it swings down as a pendulum and hits the next
ball. The energy and momentum from that ball is transmitted through the remaining balls at
rest to the ball on the other end. That ball is propelled forward at the same speed as the first
ball had due to the force of the first collision.
If two or more balls are pulled up and let go, the same number of balls will be propelled
forward due to the collision.
Slowly slows down
Since it is a form of pendulum, the action will go back and forth until it slowly slows down
due to losses from friction and the elasticity of the balls.
Explanation and equations
When one or more balls are pulled up and let go, they swing down as a pendulum. The
velocity (v) of the balls, when they hit the resting balls is determined by the height (h) from
where they were let go and the acceleration of gravity (g). The velocity is approximately
v = SQRT(2*h/g).
The kinetic energy of the balls is KE = m*v2/2 and the momentum is p = m*v, where m is
the mass of the balls in motion.
Because the momentum and energy must be maintained in this system, the balls that move
will have the same velocity and total mass as the balls that were in the initial motion. Thus, if
two balls are let go, two balls will be sent in motion after the collision.
Of course, friction and losses due to elasticity will slowly reduce the speed of the balls.
References:
FHSST Authors 2005, The Free High School Science Texts: A Textbook for High School
Students Studying Physics [Online], http://www.nongnu.org/fhsst/fhsstphy.pdf, 29, Dec
2005.
Henderson, T. 2004, Momentum [Online],
http://www.glenbrook.k12.il.us/gbssci/phys/Class/momentum/u4l1a.html, 29, Dec 2005.
Kurtus, R. 2005, Newton's Cradle [Online], http://www.school-forchampions.com/science/newtons_cradle.htm, 29 Dec, 2005.
Wikipedia 2005, Wikipedia, the free encyclopedia [Online],
http://en.wikipedia.org/wiki/Main_Page, 29 Dec, 2005.