Download Centre of Mass

QUESTIONS:
Centre of Mass
Reference
point
•
The point at which all the mass within a system (two
objects) can be considered “concentrated” is called
the centre of mass
•
Since all objects that have mass exert a gravitational
force on each other, there will be a point somewhere
between them where consider the mass to be most
‘concentrated.
•
Centre of mass is always closer to the larger object
unless they are the same mass. If they have same
mass the centre of mass will be equal distance from
both objects.
•
We always take the distance between any two
objects from their centre of mass. So when finding
the distance from an object to the centre of mass of
a system we always take the distance from the
centre of the object to the centre of mass
•
We always take every distance from some reference
point that we give the
•
Most questions asked are two-dimensional and
involve momentum. Problems involving the position
of centre of mass rarely come up.
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Practice Question
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QUESTIONS:
Momentum
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Practice Question
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QUESTIONS:
2011; 3
2008; 4
2006; 2
Gravitational Forces
Orange
Force on an Orange
Force on Earth
Earth
•
When an object is near the Earth’s surface, a
downward force is extorted on the object that will
attract the object to Earths surface. This force is
called weight force.
•
All things that have mass attract each other. Most of
the time we don’t notice it as every day objects have
little mass.
•
This means every object exerts an attractive
gravitational force on other object no matter of size
difference.
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Practice Question
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QUESTIONS:
2011; 3
2008; 4
2006; 2
Satellites
Satellite in orbit
•
In order to launch a satellite into a stable orbit
around Earth. A rocket must propel the satellite into
space with such a force that is great enough to
overcome to force of gravity due to the Earth.
•
We call the velocity of an object going fast enough to
‘break free’ from the gravitational attraction of a body
to be called the Escape Velocity.
•
We call the velocity of the object that revolves
around the Earth in circular path, the Orbital Velocity.
•
Since a satellite takes a circular orbit the centripetal
force required to act towards the centre of the Earth
is produced by the Gravitational Force of Attraction.
•
A satellite in orbit can be thought of as constantly
falling towards the Earth but missing, due its
tangential velocity.
•
A satellite with an orbital period of 24 hours is called
‘Geostationary’, and always appears in a fixed point
in space as it rotates about Earths axis at same rate
as Earth
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Practice Question
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QUESTIONS:
Apparent Weight
•
Apparent weight is experienced as a result of the
reaction force on an object from a supporting
surface. Such as a man in an elevator.
•
In an elevator, when the passenger is accelerated
vertically (either by starting or stopping the elevator)
a net force is experienced by the passenger. This is
called the Apparent Weight.
•
During an acceleration, Weight stays the same as
gravity doesn’t change. Although the reaction force
will change.
•
The passenger may experience feeling heavier if the
lift starts moving vertically up suddenly or stops
suddenly.
•
The passenger may experience feeling lighter if the
lift starts moving vertically downwards suddenly
•
A satellite in orbit can be thought of as constantly
falling towards the Earth but missing, due its
tangential velocity.
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Practice Question
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QUESTIONS:
2014; 2
2013; 3
Circular Motion
Max G.P.E
Min K.E.
Min Speed
Vertical
circles
only
Min G.P.E
Max K.E.
Max Speed
•
An object travelling in a circular path has its velocity
changing all the time since its direction will always be
changing.
•
The object has a constant speed, however since
velocity is always changing there must be an
acceleration
•
Since there is an acceleration there must be net
force causing circular motion. So every object
travelling a circular path will have an unbalanced
force acting on it.
•
We the force acting on the object the “Centripetal”
Force always points to the centre of the circle from
the object, and is always resultant from other forces
e.g. gravitational attraction force in satellite motion.
•
Travelling once around the circular path is called one
revolution.
•
Objects following a vertical circular path have
maximum gravitational potential energy and
minimum kinetic energy and speed at the top of the
path. Vice versa for the bottom of the path
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Practice Question
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QUESTIONS:
2009; 1
2004; 3
Banked Tracks
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Practice Question
o
h
a
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QUESTIONS:
2013; 1
2012; 1
2009; 2
Rotational Motion
•
When solving rotational motion problems, linear
motion equations apply except we use different
variables to represent the rotational quantities.
•
To convert linear values to rotational values we
simply divide by the radius at which the motion
occurs from a centre of mass or pivot.
•
Angular displacement is the angle at which an object
as moved through at a distance r from the axis of
rotation or pivot.
•
We always take angular displacement anti-clockwise
from the x-axis.
•
We consider angular velocity to be a vector quantity
like velocity. We say the direction of angular velocity
is along the axis of rotation. We can find the axis of
rotation by applying the right hand rule with our
thumb as the axis of rotation.
•
Every point on a rotating object has the same
angular velocity, but will have a different tangential
velocity depending on its distance from axis of
rotation.
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Practice Question
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QUESTIONS:
2012; 1
2008; 1
2006; 3
Rotational Inertia
•
The rotational inertia of an object depends on its
mass, and its distribution of its mass at a distance r
from the axis of rotation.
•
Objects that have high rotational inertia generally
either have a lot of mass, or mass that is located at a
large distance from the axis of rotation.
•
If we want to make an object rotate, we need to
apply a torque to the object. How much the object
angular acceleration increases by depends on the
size of the torque and he objects rotational Inertia.
High rotational inertia means harder to accelerate.
•
•
Rotational inertia varies from shape to shape
depending on the distribution of mass from the axis
of rotation.
Angular momentum of an object is defined as the
rotational inertia multiplied by the angular velocity of
the object. Which is basically its linear momentum at
a point, multiplied by its distance from the axis of
rotation.
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Practice Question
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QUESTIONS:
2014; 2a
2013; 3a,b
S.H.M - Pendulum
•
Simple Harmonic Motion is a special kind of periodic
motion which requires two conditions:
1) The restoring force acts in the opposite
direction to any displacement.
2) The restoring force is always proportional the
magnitude of the displacement.
•
Some examples of SHM are a simple pendulum,
vibrating spring, or a test tube bobbing in water.
•
In a simple pendulum, it is accelerating centripetally
towards where it suspended from, and towards is
equilibrium position.
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Practice Question
1.
The period of oscillation for a
pendulum depends on the
length of the string and is
independent of mass.
2.
We can use the length of the
string to find a value for T by
substituting the given values and
solving.
3.
We need to discuss the
conditions required for SHM.
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QUESTIONS:
2009; 3
S.H.M – Spring-mass
M
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Practice Question
1.
The restoring force on the
springs is the weight force of the
plane. We can calculate this
from Newtons second law.
2.
Since the springs are displaced
we need to find the change in
length before and after
compression.
3.
The force on spring and its
change in length can be used to
express the spring constant k.
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QUESTIONS:
2013; 3d
Resonance
Light
Damping
Heavy
Damping
•
Every system has its own particular resonant/natural
frequency.
•
If you drive a system at its particular natural
frequency, the amplitude of the oscillations will
increase in size due to the occurrence of resonance.
•
For resonance to occur the driving force must have
the same frequency and phase as the natural
frequency.
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Practice Question
1.
The period of oscilation only
depends on the length of the
string, so we need not worry
about using mass
2.
We can find the period of
oscillation from the following
equation.
3.
The resonant frequency is the
inverse of the period.
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QUESTIONS:
2013; 3c
2012; 2c,
SHM - Energy
Total Energy
Kinetic Energy
Potential Energy
•
When a simple harmonic motion is present in a
system. At any one time the total energy of the
system is made up of the sum of the kinetic and
potential energies.
•
A system will continue to oscillate at the resonant
frequency forever since energy is conserved. This is
always true unless energy is lost due to some kind of
damping.
•
At maximum displacement from the equilibrium point,
the Total Energy of the system is made up of only
potential energy.
•
•
When the system is at minimum displacement from
the equilibrium position (assuming there is motion),
the total Energy of the system is made up of only
Kinetic energy
There are two kinds of damping that can remove
energy from a system:
1) Natural Damping – such as internal forces
within a spring, fluids exerting viscous drag.
2) Artifical Damping – such as shock absorbers
•
Damping results in decreasing amplitude of
oscillations.
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Practice Question
1.
We need to find the potential
energy due to the spring being
expanded/compressed.
2.
Since Ep is expressed in terms
of displacement and spring
constant. We can solve for Ep
3.
Total energy of the system
comprises of both Kinetic and
Potential energy. So we just
need to add the energies
together.
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