UTC Physics 1030L: Spring Constant THE SPRING CONSTANT Objective: To determine the spring constant of a spiral spring by Hooke’s law and by its period of oscillatory motion in response to a weight. Hooke's Law Apparatus: A spiral spring, a set of weights, a weight hanger, a balance, a stop watch, and a two- Purpose: meter stick. In this lab we will measure the spring constant of two springs using two different methods. First, we will plot the force theTheory: spring exerts vs the distance the spring is stretched and second, we will measure the period for small oscillations. The restoring force, F, of a stretched spring is proportional to its elongation, x, if the Introduction: deformation is not too great. This relationship for elastic behavior is known as Hooke's law and is Part I: If onedescribed end of a springby is fixed and the other end is stretched, the spring will exert a force that tends to pull it back to its unstretched position.F = Likewise,-kx if a spring is compressed,(eq. 1), the spring will exert a force that tends to push it back to its uncompressed position. If one stays within the elastic limits of the spring, the magnitudewhere k of is thisthe constant force is directlyof proportionality proportional called to thethe spring distance constant. the spring The isspri stretched.ng’s restoring These force acts experimental resultsin the can opposite be expressed direction in to the its formelongation, of a law denoted known by as the Hooke's negative Law. sign. For a system such as shown in figure 1, the spring's elongation, x – x0, is dependent upon the spring constant, k, and the weight of a mass, mg, that hangs on the spring.F If= the−kx system of forces is in equilibrium (i.e., it has no (1)relative where F is the force,acceleration),x is the displacementthen the sum of of the forces free end down of the (the spring weight) and isk equais al constant and opposite that to depends the sum on of the the internal propertiesforces acting of the upward spring. (the The restor constanting forcek is of called the spring), the spring or constant and is always a positive number. The minus sign in the above equation keeps track of the direction of the force, the force exerted by the spring is always opposite in direction tom g the = k direction (x – x0) the spring is stretched.(eq. 2). Consider the diagram below with thefixed end of the spring on the top. Let x = 0 (xo) be the equilibrium position of the free end of the spring and let positive x values be downward. With this setupComparing it is clear equation that when 2 with the displacementthe form for thex is equation positive of the a straight spring isline stretched (y = mx and+ b), when we can the see that if displacement x weis negativeplot the force then produced the spring by is different compressed. masses Note (mg) that as a for function this to of be the true displacement the spring from is not compressed at itsequilibrium equilibrium (x-x position.0), the data Now should consider be linear Equation and the (1). slope When of thex lineis positive, will be equal the minusto the signspring makes F negative.constant, This meansk, whose that standard the spring metric exerts units a forceare N/m. in the negative direction, that is, it pulls back. When x is negative thenIf the mass minus is sign pulled makes so thatF positive. the spring This is st meansretched that beyond the springits equilibrium exerts a force(resting) in the position, the positive x direction,restoring that force is, it of pushes. the spring In thiswill labcause we an will accelera measuretion theback spring toward constant the equilibrium of two springs position by of the plotting the force each spring exerts versus the displacement the spring is stretched. We will suspend the spring in a vertical position and attach a mass holder. Here, x will be the displacement of the mass holder in the vertical direction and is measured from the equilibrium position of the holder. 66 Part II: As we will learn later in the course, the spring constant also determines the period of oscillation for a spring mass system. If a mass m is attached to an ideal spring and is released, it is found that the spring will oscillate with a period of oscillation given by rm T = 2π (2) k where k is the spring constant for the spring. 1 Hooke’s Law Laboratory Procedure: Remember to do all work in your Lab Notebook. This means copy tables into your Part I - Taking Direct Measurementsnotebook. Use correct heading and write in pen. 1. Begin by hanging one endAim: of a spring from the ring stand. Hang a mass holder from the other end of the spring. The spring mustRobert be stretched Hooke was until one the of adjacent the first to turns notice of thea relationship spring are between no longer the compressedforce applied to an against each other in orderelastic for object the elongation and its extension. of the spring This lab to varyis designed linearly to withtest and the verify applied Hooke’s force, Law in which accordance with Hooke'sstates Law. that For “the the extension brass spring, of an theelastic mass material hanger is isdirectly sufficient proportional to do this. to the For applied the force so steel spring, add 500 g masslong as to the the elastic mass hanger.limit is not exceeded”. Diagram: Apparatus: Clamp and stand Spring Metre Rule Mass holder Slotted masses Force 2. Hold the meter stick alongside the spring and measure the height of the bottom of the mass holder. This position is the equilibriumMethod: position (x = 0). Record this position in your lab notebook. Set up the apparatus as shown in the diagram above. Apply various forces to the spring and 3. Add masses in incrementsrecord of 20.0 its extension g for the (remember brass spring that and the 100.0 extension g for = the final steel length spring. í original length). 4. Record the displacementData from Coll theection: equilibrium position, x. 5. Record the uncertainty inx theRecord displacement pairs of data from for the force equilibrium and extension position, in a suitableδx, based table. on the precision of the meterstick. x Your results table and the presentation of data should include any uncertainties associated with the apparatus that you have used. 6. Calculate the fractional uncertainty (δx/x) for this measurement and record this in your data table. Data Processing and presentation: 7. Record the mass, m that was required to stretch the spring from the equilibrium position, i.e., the x Use your results to plot a suitable graph which will allow you to accurately calculate the mass added beyond that in step one. spring constant, k, for your spring. 8. Determine the uncertaintyx inSL your & HL masses, includeδm uncertaintyby adding andbars subtractingfor each data small point. amounts of mass to see how much can be addedxor HL subtracted, calculate while the uncertainty still keeping in theyour system final value in equilibrium. for the spring constant by drawing maximum and minimum slope on your graph. 9. Calculate the force (in Newtons) from your mass values. Note that there are 1000 grams , and that kg m g = 9:81 s2 . Conclusion and Evaluation: x Your evaluation should include a final value for the spring constant of your spring 10. Repeat steps 3-9 for a total of ten measurements of xi and mi, the distance and corresponding mass measured from the equilibriumtogether point, with for an one estimate spring. of the uncertainty. x Does your graph verify Hooke’s Law? 11. Repeat steps 1-10 for thex otherEvaluate spring. the procedure and result including limitations, weaknesses or errors. 12. Draw a full-page graph of the force vs. displacement. The force, F , is to be plotted along the vertical axis and displacement, x, is to be plotted along the horizontal axis. Include the data for both springs on the same graph or separateMike Dickinson graphs labeling them steel or brass. Hooke's Law 13. Fit a line to your data and determine the slopes of your lines. The slope corresponds to the spring constant, k of the spring. 2 Part II - Taking Indirect Measurements 1. Hang a small known mass, mk from the brass spring (including the mass of the holder, mh). 2. Measure the period of small oscillations of the system by measuring the time, t, it takes for the spring to make 20 oscillations (N=20). 3. Record the time and calculate the period, T , (T = t/N). As a check for blunders (miscounting, for example), repeat twice more and average. 4. Determine δT by first determining the uncertainty in your reaction time. Find your reaction time by starting and stopping the stopwatch quickly, then divide that time by your twenty revolutions. reaction time δT = N 5. Determine the fractional uncertainty (δT /T ) for this measurement and record this in your data table. 1 6. As it turns out, the mass of the spring itself does affect the motion of the system, thus we must add 3 the mass of the spring to account for this. In order to calculate the mass, m, used in equation 2 then, 1 add 3 the mass of the spring (ms), plus the mass on the hanger (mk), plus the mass of the hanger (mh). 1 m = ∗ m + m + m 3 s k h 7.