COMPLETE AND HAND IN BEFORE LAB
Name:
PRE-LAB EXERCISE
You will derive the expression relating the period 𝑇 of the oscillating mass-spring system to the
physical parameters of the system: the mass 𝑚 and the spring constant 𝑘.
This derivation will be performed using conservation of energy in the oscillating system.
Note that, while the derivation will be done for a horizontal oscillation, the result applies
equally to vertical or (as in this lab) inclined oscillations.
Energy in the oscillation mass-spring system
Figure 1 – Ideal oscillation of mass-spring system (Knight, 2022).
Consider the oscillatory motion represented above (Figure 1).
The total energy 𝐸𝑡𝑜𝑡 in the oscillating system is the sum, at any time, of the kinetic energy 𝐾 of
the mass and the elastic (spring) potential energy 𝑈𝑠 stored in the stretched or compressed
spring.
𝐸𝑡𝑜𝑡 = 𝐾 + 𝑈𝑠
Eq. 2
For SHM, the total energy is conserved in time.
We will calculate and compare this total energy at two moments in time:
• When the spring is fully stretched, and the mass is instantaneously at rest.
•
When the spring is instantaneously unstretched.
Mass at rest (spring fully stretched)
Since the equilibrium point of the oscillation is also the unstretched position of the spring,
the amplitude 𝐴 of SHM is equal to the maximum extension of the spring.
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SN3 Lab #1
Marianopolis College
COMPLETE AND HAND IN BEFORE LAB
Question 1. a) From Mechanics, state the elastic potential energy 𝑼𝒔 stored in a spring that is
stretched by an amount 𝒙 with spring constant 𝒌?
b) Determine the elastic potential energy 𝑼𝒔 stored in the spring (in terms of, possibly, 𝒎, 𝒌
and 𝑨) when the spring is fully stretched.
c) At the instant when the spring is fully stretched, what is the kinetic energy 𝑲? Briefly
explain.
Spring unstretched
The mass has its maximum speed (and therefore maximum kinetic energy) when the spring is
unstretched.
Question 2. Using the general equation for SHM (Eq 1), derive […hint] the general velocity
function, 𝒗𝒙 (t), for SHM.
Question 3. From this function, determine the maximum speed of the mass (in terms of 𝑨 and
𝝎).
Question 4. a) From Mechanics, determine the kinetic energy 𝑲 of the mass (in terms of,
possibly, 𝒎, 𝑨 and 𝝎) when the spring is unstretched.
b) Determine the elastic potential energy 𝑼𝒔 at the instant when the spring is unstretched.
Briefly explain.
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SN3 Lab #1
Marianopolis College
COMPLETE AND HAND IN BEFORE LAB
Conservation of energy
Question 5. Considering the two moments mentioned above, use conservation of energy to
obtain an expression for (i) angular frequency 𝝎, (ii) period 𝑻, in terms of 𝒎 and 𝒌.
Experimental determination of 𝑘
In this lab, we wish to obtain an experimental value for the spring constant from measurements
of the oscillation period and the parameters of the system. This will involve measuring the
period 𝑇 for different values of 𝑚, and obtaining a linearized graph of this variation.
Question 6. On the graph to the right, label what axes can be used to
obtain a linear graph and show the expected trendline. Use the
expression found in Q5.
Question 7. Show how the spring constant 𝒌 can be obtained from the
slope of the graph. State what you expect the vertical intercept to be.
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SN3 Lab #1
Marianopolis College
Python graphing code
The code in Appendix A will be made available to you and can be modified accordingly for
graphical analysis. Read over the code and familiarize yourself with it before coming to the lab.
Question 8. Suppose that lab partners Alice and Bob have used the code to plot their data,
but they’re not satisfied with the min/max lines created by default.
By visual inspection, they judge that the min line is a good fit, but they would like to increase
the slope of the max line. To do this, they would like to move the red point up by 0.6 units
and move the cyan point down by 0.5 units. Which line(s) of code in Appendix A should they
modify and how? Rewrite the line(s) of code with the desired modifications.
Write down the useful equations from this prelab in the Notes section on p.10.
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SN3 Lab #1
Marianopolis College