Objective: To prove energy is conserved in a vertically-
oscillating mass-spring system.
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| The apparatus |
In this lab we are trying to find the total energy of this
vertically oscillating spring- mass system. We firstly derive the formula for
each type of energy, then we use force sensor and motion detector and with the
help of loggerpro we come up with the numerical value of this spring mass
system.
First of all, there are five types of energy present in this
system
1.
Elastic potential energy (EPE)
2.
Gravitational Potential Energy (GPE)
3.
Kinetic energy (KE)
4.
Gravitational Potential Energy of the spring (GPEspring)
5.
Kinetic energy of the spring (KEspring)
The first three types of energy are common. EPE is there
because we have a spring and the energy from spring is elastic potential
energy. GPE is present because the spring is hanging at a certain height; there
is GPE since we set the ground to be the origin. KE is there because at the end
of the spring connects the mass, velocity is created from the oscillation of
the spring.
The last two types of energy are not so easily to identify.
In order to identify them, we need to see the spring itself as the system.
GPEspring is one of the energy in this system
because different part of the spring has different height, as the spring
oscillates, the height of the different part of the spring changes. When the
height is changing, we have different potential energy from different part of
the spring. In order to calculate the total potential energy of the spring, we
need to use integration to come up with a function with a variable, y0
. After we have the function, we can then use loggerpro to help complete the
process.
The above picture is the integration to find the function of
energy. We begin by finding dm.
a)
dm is a piece of mass we chosen to represent one
part of the spring. Since the spring is constantly changing, we need to choose
one part to integrate. This piece of mass, dm, has a height, we called it dy. The
piece of mass, dm, is divided by the height of the piece of mass, dy. This is
equal to the total mass of the spring, M, divide by its changing length, h-y0.
b)
We write out the function of potential energy
using dm. Note that the height is y0 not h.
c)
We integrate from y0 to h because
that is the length that the spring is changing its position. y0 is
referring to the end position of the spring at each position. It is a
constantly changing variable since the end of the spring is constantly changing
its position as it oscillates.
d)
Eventually, our result involves the sum of two
fractions. One fraction is constant and the other one includes a change in
position, y0. We are interested only on the one with y0 because
we are looking for the function that would describe the energy at different
position during oscillation. The constant fraction just gives us the energy at
a constant height.
KEspring is part of the total energy of this
system because as the spring oscillates, it creates different downward velocity
at different part of the spring. The top of the spring is not going to move a
lot, it would have less velocity than the part of the spring that is at the end
, which would have more velocity. This change of velocity at different part of
the spring also requires the use of integration. Our goal would be to come up
with a function describing the energy of the spring with a variable, vend.
From then on, we can use loggerpro to help find the kinetic energy of the
spring.
Note that the origin is at the top so that the downward
velocity would be positive.
a)
In this calculation we begin by finding dm. To
find the mass of a piece of spring dm, we divide the mass of the spring, M, by
the length of the spring, L. Just as in the last calculation, dm is divided by
dy, its height.
b)
Next, we find the function of kinetic energy.
The formula for kinetic energy is ½ mv2. We start by substituting dm
into m and vpiece into v. The
reason it is vpiece is because we are integrating the change of
velocity of a piece of spring. Then, we replace vpiece by vend.
This vend is multiplied
by y/L. Since velocity is not constant throughout the spring, we cannot just
take the velocity at the end of the spring as our velocity. Therefore, we need the ratio of the spring
to find the velocity at certain position. y/L is a good ratio because y is the
change of length, and it is divided by the total length of the spring.
c)
We separate the constant from variable. Then we
integrate y2 from 0 to L. This covers the whole spring. We want to
find the velocity of each pieces of the spring.
d)
Finally, we rearrange our function so that it
looks like the formula of kinetic energy.
Now, we have our five functions of energy written out in a
format that we may plug into loggerpro.
1.
Elastic
potential energy (EPE )= ½ *k* “∆y”2
ü
We may
obtain the spring constant, k, through loggerpro or measures by meterstick.
ü
∆y is the
unstretched position of the spring
2.
Gravitational
Potential Energy (GPE) = m*g* “position”
3.
Kinetic
energy (KE) = ½ *m* “velocity”2
4.
Gravitational
Potential Energy of the spring (GPEspring) = (Mspring /2) * g * "position"
5.
Kinetic
energy of the spring (KEspring ) =( ½) * (Mspring/3) * “velocity”2
Note that m is the
mass connected to the end of the spring; it is not the mass of the spring!
The mass is 0.3 kg and the mass of the spring, Msping, is 0.087 kg.
The mass is 0.3 kg and the mass of the spring, Msping, is 0.087 kg.
The Set-Up:
We secure a long rod vertically by the rim of the table
using a table clamp. On the long vertical rod we secure a shorter rod
horizontally. Then we put a force sensor at the end of the horizontal rod. The
force sensor is zeroed and calibrated so downward motion is positive. A spring
is hanged on the force sensor. On the ground, below the spring is a motion
detector. Both the force sensor and motion detector is connected to the
loggerpro.
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| Force vs position graph |
We begin by finding the spring constant, k, of the spring in
the apparatus. We keep the spring still then measure it using the force sensor.
The force sensor gives us a force vs time graph, we linearized the graph. Here the slope is the spring constant. The
spring constant is 8.487 N/m.
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| Position vs time graph (Top) & Velocity vs time graph (Bottom) |
We find the position of the mass using motion detector. We
hang a mass at the end of the spring and let it oscillates, through loggerpro,
the motion detector records a sinusoidal graph. With the position data, we
manage the loggerpro to give us the velocity graph as well.
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| Table of all five energy and total energy |
Notice that the far
right corner is the total energy. The total energy is not constant, it is
changing but the difference is just a little, it is this way because there is
some uncertainty to this spring- mass system, such as air resistance, that
could alter our result by a little.
Also, all five energy, doesn’t matter what time increment we
pick, add up to be around the same value , thus, we can say that energy is
conserved in this vertically oscillating spring-mass system.
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| Total energy vs time graph |
Finally, we put the total energy data into a graph, we get a
sinusoidal graph, almost straight line. The mean is 1.699 J, which the total
energy of this system.
Summary:
In this lab, we explore the topic of energy conservation. We
try to find the sum of all energy in a mass-spring system, and we prove that
the total energy is the same at every time increment.
We start the lab by finding the function of all the energy
that exists in the mass-spring system. We come up with five energy, EPE, GPE,
KE, GPEspring, and KEspring. We use integration to find
the function for GPEspring and KEspring. After having all
five functions for energy, we begin finding the velocity and position at every
time increment. We also find spring constant. Then, we input the functions into
loggerpro. We obtain a table that has the amount energy for all five energy
types at each time increment. We look at the total energy, it all revolves
around 1.699 J. We have successfully prove that energy is conserved in this
vertically- oscillating spring- mass system!
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