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EXP4 - Experiment 4
Physics for Development (PHY313)
BRAC University
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Principles of Physics I (PHY111)
Lab
Experiment no: 4
Name of the experiment: Determination of the spring constant and effective mass of a given
spiral spring
YOU HAVE TO BRING TWO GRAPH PAPERS (cm scale) TO DO THIS EXPERIMENT.
Theory
In this experiment a spring is suspended vertically from a clamp attached to a rigid frame work of heavy metal rods. At the bottom end (which is the free end) of the spring a load of mass, m 0 is suspended. So the force acting on the spring is the weight m 0 g of the load which acts vertically downward and the spring gets extended. Due to the elastic property of the spring, it tries to regain its initial length, hence applies a counter force on the load, which is called the restoring force of the spring. According to Hooke’s law, within the elastic limit the magnitude of this restoring force is directly proportional to the extension of the spring and the direction of this restoring force is always towards the equilibrium position. If k is the spring constant of the spring and l is the extension of the spring, then restoring force = - k l
Figure 1: a) A vertically suspended spring, b) A load of mass m 0 is attached at the bottom end of the spring and c) The spring is oscillating, the load is y distance above the equilibrium position.
Let the spring is in equilibrium with mass m 0 attached as in figure 1 (a), and so we can write
m 0 g=kl
=> m 0
k
g
l= (1)
Here k is the spring constant and g is the acceleration due to gravity.
Equation (1) is an equation of a straight line. The slope of this line is given by-
Slope =
k
g
=> k =
slope
g
(2)
We can plot l vs. m 0 graph and determine its slope and hence, the value of k.
If the load is slightly pulled down and released, the spring will oscillate simple harmonically. Suppose, at time t the velocity of the load is v and the spring is compressed by a distance y above the point C.
As you know from your earlier schools, if the mass of the spring were negligible then the period of oscillation
would be given by
k
m
T= 2 π 0
Due to the mass, m of the spring an extra term m′ will be added with the mass of the load m 0 in the above
mentioned equation. So, the period of oscillation is,
k
m m
T
+ ′
= 2 π 0 (3)
m′is called to be the effective mass of the spring. It can be shown that m′is related with the mass of the
spring by the following equation:
3
m′=m (4)
Please see appendix A (provided in the soft copy of this script in the server) to learn how to derive equations (4) and (3).
From equation (3) we get,
m
k
m
k
T = + ′
2 0
2 4 π 24 π
For different mass, m 0 of the load we find different periods of oscillation, T. If we draw a graph by plotting m 0 along X axis and corresponding T 2 along Y axis, it will be a straight line. The point where the line intersects the X axis, y-coordinate is 0, i., T 2 =0. We can find the X coordinate of the point, (i. the value of m 0 at that point) by putting T 2 =0 in the above mentioned equation.
Read carefully and follow the following instructions:
Please READ the theory carefully, TAKE printout of the ‘Questions on Theory’ and ANSWER the questions in the specified space BEFORE you go to the lab class.
To get full marks for the ‘Questions on Theory’ portion, you must answer ALL of these questions CORRECTLY and with PROPER UNDERSTANDING, BEFORE you go to the lab class. However, to ATTEND the lab class you are REQUIRED to answer AT LEAST the questions with asterisk mark.
Write down your NAME, ID, THEORY SECTION, GROUP, DATE, EXPERIMENT NO AND NAME OF THE EXPERIMENT on the top of the first paper.
If you face difficulties to understand the theory, please meet us BEFORE the lab class. However, you must read the theory first.
DO NOT PLAGIARIZE. Plagiarism will bring ZERO marks in this WHOLE EXPERIMENT. Be sure that you have understood the questions and the answers what you have written, and all of these are your own works. You WILL BE asked questions on these tasks in the class. If you plagiarize for more than once, WHOLE lab marks will be ZERO.
After entering the class, please submit this portion before you start the experiment.
Name: _____________________ ID: ______________ Sec: ___ Group: __ Date: __________
Experiment no: ___
Name of the Experiment: _______________________________________________________
_____________________________________________________________________________
Questions on Theory
*1) Draw the arrangement of this experiment. [0]
Ans:
*2) States Hooke’s law for an elastic spring. [0]
Ans:
*3) Suppose, an external force is applied on a spring to stretch it. Extension of the spring is l. If the spring constant is k then what is the restoring force of the spring due to its elasticity? [0]
Ans:
Draw the data table(s) and write down the variables to be measured shown below (in the ‘Data’ section), using pencil and ruler BEFORE you go to the lab class.
Write down your NAME and ID on the top of the page.
This part should be separated from your Answers of “Questions on Theory” part.
Keep it with yourself after coming to the lab.
DO NOT forget to bring two GRAPH PAPERS.
Data Data
A) Initial length of the spring, y 0 = AB = ____________________ cm
B) Table 1: Data of m 0 , l and T
Mass of load m 0 (gm) Extension of the spring, l (cm)
Time required to complete 20 oscillations, t (s)
Period of oscillation, T (s)
T 2 (s 2 )
C) Mass of the spring by using weight-meter, m = _____________________ gm
D) Effective mass of the spring by using the value of its mass- measured by weight-meter, m′
=__________gm
Please attach two graphs here.
- READ the PROCEDURE carefully and perform the experiment by YOURSELVES. If you need help to understand any specific point draw attention of the instructors.
- DO NOT PLAGIARIZE data from other group and/or DO NOT hand in your data to other group. It will bring ZERO mark in this experiment. Repetition of such activities will bring zero mark for the whole lab.
- Perform calculations by following the PROCEDURE. Show every step in the Calculations section.
- Write down the final result(s).
Calculations
Result:
- TAKE printout of the ‘Questions for Discussions’ BEFORE you go to the lab class. Keep this printout with you during the experiment. ANSWER the questions in the specified space AFTER you have performed the experiment.
- Attach Data, Graphs, Calculations, Results and the Answers of ‘Questions for Discussions’ parts to your previously submitted Answers of ‘Questions on Theory’ part to make the whole lab report.
- Finally, submit the lab report before you leave the lab.
Appendix A (Derivation of formula
k
m m
T
+ ′
= 2 π 0 )
Figure 1: a) A vertically suspended spring, b) A load of mass m 0 is attached at the bottom end of the spring and c) The spring is oscillating, the load is y distance above the equilibrium position.
If the load is slightly pulled down and released, the spring will oscillate simple harmonically. Suppose, at time t the velocity of the load is v and the spring is compressed by a distance y above the point C. Now,
Total energy of the spring-load system, E = kinetic energy of the load + gravitational potential energy of the load + elastic potential energy of the spring + gravitational potential energy of the spring + kinetic energy of the spring
Now,
Kinetic energy of the load = 0 2
2
1
mv
Gravitational potential energy of the load about point B = −m 0 (lg −y+a) (Here, a is the separation
between the free end of the spring and the center of mass of the load)
Elastic potential energy of the load about point B = () 2
2
1 lk −y
Now, from figure 1 c the length AD = y 0 +l−y(y 0 is the initial length of the spring)
Centre of mass of the spring lies at the midpoint of AD. The distance of centre of mass from point B =
2
0 ( 0 2/) 0
y y l
y y l y
+ −
− + − =
Gravitational potential energy of the spring about point B = ⎟
⎠
⎞
⎜
⎝
⎛ + −
2
mg y 0 y l , where m is the mass of the
spring.
Now, let’s find out an expression of the kinetic energy of the spring. At a certain time different portions of the spring have different velocities. The top most point of the spring always remains stationary. The bottom most point of the spring moves at same velocity as the load.
Say, at time t the length of the wire is AD = L. Linear
mass density of the wire, λ=m/L
It is observed that the velocity, v′ of a particular
point of the spring is directly proportion to its distance from the point of suspension (A), y. That means
v′∝y
=> v′=cy, here c is an arbitrary constant (4)
When y = L then v′=v
So, from (4) c=v/L
Therefore, from (4):
y
L
v′=v (5)
Now, let’s consider a tiny portion of the spring at distance y below the point A, having length dy.
Figure 2: Calculation of effective mass
At time t the velocity of this portion is v′. The mass of this tiny portion = dm=λdy
Therefore, the kinetic energy of this portion, () y dy
L
dy mv
L
y m
L
y dy v
L
dK v dm v 2
3
2 2 2
2 2 2
2 2 2
1
2
1 = =
⎟
⎠
⎜ ⎞
⎝
= ′ = ⎛ λ
The total kinetic energy of the spring, 2
3 3 3
2
0
3 3
2
0
2 3
2
23
1
3
0
2 2 3 2 3
v
L m
L
y mv
L
mv
y dy
L
mv
K
L L
⎟⎟=
⎠
⎞
⎜⎜⎝
⎛
⎥ = −
⎦
⎤
⎢
⎣
⎡
= ∫ =
Here,
3
m
is called to be the effective mass, m′ of the spring.
∴Kinetic energy of the spring = 2
2
1 m′v
m m
k
+ ′
=
0
ω =>
m m
k
T + ′
=
0
2 π
, Here T is the period of oscillation
=>
k
m m
T
+ ′
= 2 π 0
EXP4 - Experiment 4
Course: Physics for Development (PHY313)
University: BRAC University
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