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CONSERVATIVE FORCE SYSTEMS
Purpose
a. To investigate Hooke’s law and determine the spring constant. b. To study the nature of conservative force systems using a spring-mass system as an example.
Theory
I. Hooke’s law and Spring constant
When an object of mass m is attached at the lower end of a vertical spring, it elongates and comes to equilibrium. From Hooke’s law, the spring force F = - k Δ X, where Δ X is the displacement ( elongation ) of the spring from its unstretched position as shown in the Figure 1, and k is the spring constant. The minus sign shows that the force F acts in such a direction as to reduce the magnitude of the displacement. At the equilibrium position the spring force is balanced by the weight of the object attached. We are assuming an ideal (non-dissipative) spring with negligible mass. Thus
k Δ X = mg. (1)
In the first part of this lab, we will investigate this relation and determine the spring constant for a spring.
II. Conservation of Energy
In a conservative force system the work done by the force can be expressed as the negative of the change in the potential energy ( Wc = - Δ U ). Potential energy decreases (increases) when a conservative force does positive (negative) work. Thus the total mechanical energy (kinetic plus potential energy) is always conserved.
We can calculate the kinetic and potential energies by measuring the velocities and positions of a mass attached to the spring using a motion detector.
a. What other information do you need to calculate the kinetic energy?
b. What other information do you need to calculate the spring potential energy?
c. What other information do you need to calculate the gravitational potential energy?
When the hanging mass is stretched down and released, it will oscillate about the equilibrium position. The kinetic energy is given by
KE = 1 2 mv
Unstretched
Equilibrium
Arbitrary position
Detector (origin)
+ x
x
X
Figure 1. Spring-mass system in (a) equilibrium and (b) oscillating. xo, xem and x(t) are positions of the bottom of the hanger.
(a) (b)
Since the hanging mass is oscillating vertically, it involves both the gravitational and the spring potential energies. The spring potential energy, us , is given by
us =
1 2 𝑘(Δ x )^2 , (3)
where Δ x is the displacement ( elongation ) of the lower end of the spring relative to its unstretched position x 0 at any time t and is measured positively upward (see Figure 1). Note that the spring potential energy ( us ) is zero when it is unstretched. Now, assuming the gravitational potential energy is zero when x = xem (We can make this assumption since it is only differences in potential energy that will be physically significant.), at arbitrary x is given by ug = mg ( x - xem ). (4) See Figure 1a. In our experiment x at any time t will be measured positively upward from a motion detector to the bottom of a hanger. Thus, the total mechanical energy is
Total Energy = 1 2 mv
(^2) + mg ( x - x em ) +^
1 2 𝑘( Δx )
The spring force and the gravitational force are conservative forces. If there are no other forces acting on our system then, from the principle of conservation of energy, the total energy is conserved; i.e., the total energy does not change. In part II of this lab we will investigate this experimentally. One can show that the total energy is equal to 1 2 kA
2 k ( ΔX )
(^2) + mgc = constant (5.1)
where A is the amplitude of the oscillation, Δ X is the elongation of the spring at equilibrium, and c is the distance from the bottom of the hanger to the c.m. of the weights and hanger.
To take into account the mass of the spring, heretofore ignored, replace m in 1 2 mv
(^2) in Eqs. (2)
and (5) by an effective mass meff = m + 1 3 msp ,^ (6) where msp is the mass of the spring.
Apparatus Jolly balance, scale, spring, set of slotted weights (50 and 100 grams), 50 gram slotted mass hanger, rulers, graph paper, motion detector, Vernier data acquisition system, Logger Pro software.
Description of Apparatus The Jolly balance was invented by the German physicist
Philipp von Jolly in 1864. This and the other apparatus that will be
used in this laboratory are shown in Figure 2. It consists of a
movable arm at the top of a stand pipe. The movable arm has an
engraved scale and can be moved up or down by rotating the knurled wheel at the bottom of the balance. A Vernier scale is also attached
to increase the resolution of the measurement. A spring is fastened at
the top of the movable arm and a weight hanger is hung at the lower
end of the spring. It has a movable pointer attached on the stand to
mark the position of the spring. It also has a pan on which to put the
motion detector.
Motion detector
LabQuest
Knurled wheel
Pointer
Hanger
Spring^ Scale
Figure 2. Jolly balance and accessories.
From the velocity versus time graph, can you tell how the acceleration changes with time? Locate the positions of the bottom of the hanger where the velocities are zeroes and maxima (in magnitude). Record them on the data sheet. From the position graph find the amplitude, A, of the oscillation. Record it. Determine the time period T of the oscillation from position versus time graph by fitting the data with a sine function thereby obtaining ω = 2π/T. Record your values of ω and T.
The spring constant, mass, and period of oscillation, T , are related by means of the equation
𝑇 = 2𝜋√
𝑚𝑒𝑓𝑓 𝑘 (7)
Find the spring constant k from Eq. (7) above. Record it.
Calculate the elongation ∆𝑋 = 𝑚𝑔 𝑘. Record it.
- From the diagram (Figure 1) you should be able to see ( with some thought ) that Δ x , the elongation of the spring at any time t, is given by Δ x = x – ( xem + Δ X ) (8) where x is the distance from the motion detector (i.e., from x = 0). ( Hint : Note that in this diagram Δ x is negative, and we take Δ X to be a positive number.)
Part III. Calculating and plotting kinetic energy and potential energies versus time
You can calculate kinetic energy, spring potential and gravitational potential energies from the data you collected from the motion detector using the equations (2, 3, 4, and 8). You have to plot the graphs of kinetic, spring potential and gravitational potential energies versus time for further analysis. Include the graphs in your report. You can copy your data and use Excel for calculating energies and plotting the graphs for further analysis. OR better yet follow the instructions given below. Print the graphs in 3, 5, 6, 7, and 8 below after saving them. a. Plotting Kinetic Energy versus time graph
- On the computer in Logger Pro window, click ' New Calculated Column ' under ' Data ' menu. Name it 'Kinetic energy' (KE for short name) and 'Joules' in the units box.
- Under the ‘expression’, type the right hand side of the formula for kinetic energy (Eq. 2) as in any computer language. ( Hint : Use * for the multiplication sign and ^ for power .) Use meff for the mass. You may directly use the value for meff or use it as a parameter and then substitute the value. For v , click 'Variables (Columns)'and choose (velocity). Do not forget to square the velocity in the expression! Once it is done, this will add a column for you with calculated kinetic energy values corresponding to the time column.
- Now 'Insert' a new graph to display the Kinetic Energy (KE) versus time (t) graph and save the graph.
b. Plotting Potential Energy versus time graph You will be repeating the previous steps to plot graphs for potential energies.
- Let’s do it for the gravitational potential energy first. Add a 'New Calculated Column' under 'Data' menu. Name it 'Gravitational Potential Energy' (GPE for short name) and 'Joules' in the units box. In the expression, this time you are going to use the equation for gravitational potential energy (Eq. 4). For x , click 'Variables (Columns)' and choose (position). This will add a column for you with calculated gravitational potential energy values corresponding to the time column.
- 'Insert' a new graph to display the Gravitational Potential Energy (GPE) versus time (t) graph and save the graph.
- Similarly, plot a graph for Spring Potential Energy (SPE). Use Eqs. 3 and 8 to write expression.
c. More plots
- Plot and save a graph of Total Potential Energy (GPE+SPE) versus time.
- Plot and save a graph Total Energy (KE+GPE+SPE) versus time.
Computations
From your data in Table 1, plot a graph of weight ( mg ) vs. spring elongation Δ X. Does the graph suggest a straight line for mg = k Δ X? Find the slope. Determine the spring constant. Record.
Go to Part II.
Calculate the time period from the value of k obtained from Part I. Explain sources of error. Analyze the graphs. You should find the answers for all the questions given below.
Questions
- How well does the spring obey Hooke’s law?
- Look carefully at the plots of KE vs t and PE vs t. Locate the positions where the kinetic energy is maximum. Mark them on your graph. What can you say about them?
- Locate the positions where the kinetic energy is zero. Mark them on your graph. What can you say about them?
- How does the gravitational potential energy vary in one cycle?
- How does the spring potential energy vary in one cycle?
- Compare the graphs of kinetic energy and of total potential energy (GPE+SPE). Explain if there are any interesting features.
- How does the total energy (kinetic plus potential energies) change with time? (In theory it should be constant. One could show that it is equal to the sum of the KE and the spring and gravitational potential energies at the equilibrium position.)
Data Sheet
Date experiment performed:
Names of the group members:
Part I.
Table 1. Hooke’s law and finding Spring constant m (kg)
mg (N)
Δ X
(m)
mg /Δ X (N/m)
Spring constant, k from procedure of part I = __________________ N/m
Part II. Measuring position and velocity
m =____________ kg Mass of spring msp = ___________ kg
Effective mass meff = m + 1 3 msp^ = _________________ kg Position of equilibrium from motion detector, xem = _____________ m Zeroes of velocity at x = _______, _______, _______, _______ m Maxima (magnitude) of velocity at x = _______, _______, _______, _______ m A (amplitude) = _________ m ω = ___________ s-1^ T = _____________s k (from Eq. (7) = ____________ N/m Elongation Δ X = __________m Time period from value of k from part I = Measured total energy = _________________ J c = ___________________ m Theoretical total energy (Eq. 5.1) = ___________________ J % difference = ____________________
