Physics 202A-Lab 10
Table of Contents
Experimental Verification of Hooke’s Law, Determining Spring Constant and Probing into Nature of Spring Force
Apparatus: Spring, set of masses, mass holder, meter ruler, ruler, straight edge, graph papers, stop and calculator.
Introduction: In this experiment we study spring force, spring force constant (k) and extension (change in its length) of the spring (x) produced when an external force (Fapplied) is applied to the spring. We apply a force of Fapplied =mg N by suspeneded a known mass, m, at the lower end of the spring (see Figure 1). We measured the extension (x) of the spring by a ruler. The quantitaive relationship between Fapplied and extension of the spring it produces (x) was experimentally discovered by Oxford University professor, Robert Hooke, in 1660. In fact, the relationship is known as Hooke’s Law or Hooke;’s law of eleasticty. As Hooke put it in Latin: Ut tensio, sic vis or “As the extension, so the force.” We can think of the Hooke’s Law like this: the change in the length the spring is directly proportional to the applied force that pulls the spring and produces the extension: x a F or F a x and Fapplied= kx —-(1) where k is the constant of proportionalty called the spring constant and F = Fapplied . A stiffer spring has higher k than an easily stretchablke one. Extension, x, of a spring can be positive (increase in its original length) or negative (decrease in its original lenth, which occurs when a copression force/push force is applied to it). Note that the equation Fapplied= kx works for a spring that is stretched vertivcally and for a spring that is stretched horizontally as shown below.
A vertically suspended/arranged spring
with its original (un-stretched) length
A horizontally suspended/arranged spring with its original (un-stretched) length
When a mass, m, is suspended, it is pulled downward by gravity force (=mg) and it exerts a force of Fapplied = mg, which pulls the spring and extends its length by x m. As the mass is equilibrium, there must be an upward force whose magnitude is =mg, but it is in the upward direction, which is opposite to the direction of extension of the spring often called displacement of the spring (refer to the sketch below). The greater the applied force is, greater this upward force. As applied force is directly proportional to x, the upward force is proportional to –x. What is the source of this upward force acting on the suspended mass that is in equilibrium? Of course, the source is our spring. This upward force is called the spring force and it points in upward direction when the mass, m, is in equilibrium-it has zero acceleration).
During lab demo, make sure you notice and understand the direction of the spring force when the mass is set into up and down oscillations at the end of the spring. Now the spring force changes its direction and its magnitude twice in each cycle/oscillation. Have you ever known any such force that continuously changes its direction and magnitude?
We can thus state Hooke’s Law like this: The force of the spring is directly proportional to the extension of the spring, but in opposite direction, i.e., Fspring a – x or Fspring = -kx
In this activity, we conduct a experiemnet similar to that of Hooke’s. We apply a pull force of varying magnitudes to a pring and measure the change in length of the spring that each force produces. We analyze our date by graphical analysis and determine the spring force constant (k) for the spring used.
Theory/Initial Calculations: Answer the following questions before taking any data. Show your work in the Theory section of your report.
- Write down Hooke’s law in words and as an equation. What are the SI units of k?
- Write the expression/equation relating spring force, Fs, k, and extension (x) of a spring.
- When a mass, m, hangs at rest from a spring of spring constant, k, the spring is stretched a distance, x. Start with a FBD of the above description, apply Newton’s second law to this static set-up, and find the relationship between m and x.
- Explain why a graph of m versus x will be a straight line with a slope of k/g.
Instructions
Verification of Hooke’s Law: Nature of Spring Force, the applied force versus the extension (stretch) of the spring and determining spring constant, k, of the spring
- Suspend a spring from a laboratory stand and align its lower end with zero-cm line of a meter ruler, which is arranged vertically behind the spring. Make sure that higher-diameter end of the spring is down. This arrangement is displayed in Figure 1 to which you are expected to add a meter ruler and a figure caption.
- Hang/suspend various masses indicated in the Table 1 (shown on the following page) from the bottom of the spring. Do not forget that the mass holder has a mass of 50 gm. Record the equilibrium position of the bottom of the mass hook or bottom of the mass holder/hanger. We are interested only in change in the length of the spring. Each mass is pulled by gravity force (mg) and mass is thus pulling the spring as it is attached to it. This is our applied force, Fapplied=mg
- Make a graph of applied force Fapplied versus Extension of the spring, x on a graph paper. Draw the best-fit line and find the slope of this line. Make sure you do this work completely showing which two points you use for determining the slope, slope calculations shown on the page of the graph, units of the slope. Recall that the SI unit of the spring constant (k) is N/m.
- Calculate uncertainty for k, i.e., dk. The only uncertainty we have in this activity is the uncertainty in measured extension (dx) of the spring, x. dx is ±0.3mm=0.03 cm=________ m (fill in this blank)
Figure 1. Write the caption for this figure here. (select the figures you like best and use it for your lab report)
Table 1. Data showing mass (m) suspended at the end of a spring and extension (x) in the length of the spring produced by the pull force (F=mg) and g=9.8m/s2
| Suspended Mass, m
(± 1 gram) |
Suspended Mass, m, in kg
(± 1x 10-3 kg) |
Pull Force Applied to the Spring,
F= mg (N) |
Extension/Elongation, x, in the length of the spring (±0.1 cm) | Extension/Elongation, x, in the length of the spring in meters
(± 1 x 10-4 m cm) |
| 0 | 0 | 0 | 0 | |
| 50 | 0.05 | 0.49 | 19.94 | 0.1994 |
| 100 | 0.1 | 0.98 | 39.72 | 0.3972 |
| 150 | 0.15 | 60.61 | 0.6061 | |
| 200 | 0.20 | 1.96 | 79.85 | Complete this |
| 250 | 0.25 | Complete all the blanks | 100.17 | Complete this |
| 300 | 0.30 | 120.88 | Complete this | |
| 350 | 0.35 | 3.43 | 140.75 | Complete this |
| 400 | 0.40 | 160.14 | Complete this | |
| 450 | 0.45 | 181.10 | Complete this |
_________________________________________________________________________________
Answers to Questions: At the end of your lab report: write the following questions in the section of your report titled “Answers to Questions” and answer each (write question in and then answer it before writing the next question. Do not copy answer(s) from any Internet resources. This is known as plagiarism! I am interested in your own answers.
- If this experiment is performed on the Earth’s surface and then on the Moon using the same spring, would the results of the spring constant be the same or different. Explain.
- What is the physical significance of the spring constant, i.e., what does it tell you?
- A car accelerates gradually to the right with most of the power on the two rare wheels. What is the direction of the friction force on the two rear tires? Show this direction by an arrow and also show the velocity of the car and its direction in a carefully sketched, simple, and well-labeled diagram/figure. Hints: In what direction that portion of each of the rear tire in contact with the road exerts the friction force on the road). The friction force of the road on each tire must then be opposite to that of the friction force that tires exet on the road car is traveling on.
- The force constant of a Spring is 125 N/m. (a) Find the magnitude of the applied external force required To compress the spring by 2.47 cm from its original/un-stretched length.
- A force of 19 N is required to stretch an ideal spring a distance of 40-cm from its rest position (original length/un-stretched position. What force (in newtons) is required to stretch the same spring ____________
(i) twice the distance? (ii) three times the distance? (iii) one-half the distance?
- When a box of unknown mass is placed into the trunk of a car, both rear shocks (called shock absorbers) are compressed a distance of 5 cm. If we assume the two rear shocks are made from springs, each with a spring constant value of = 30,000N/m, what is the mass of the box? (Assume g = 9.80m/s2).
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