N ame Partner(s): Experiment 9: Simple Harmonic M otion Objectives Equipment Pre-Lab Study the simple harmonic motion (SHM) of a mass on a spring. spring, ruler, weight hanger, hook, masses, timer, motion detector A list of Activities is to be completed before this Lab. Completing these Pre-Lab assignments w ill expedite your progress during the lab period. A s part of your preparation you are also expected to read through the lab manual before the lab. You will be pressed for time during the lab. Since successful completion of all lab activities counts towards your final lab grade it will be important to be well prepared by doing Pre-Lab assignments and reading the entire lab before attending the lab. Points earned today Lab Challenge Total Instructor Initials IX-1
Pre-Lab for LAB#9 Complete the following before you attend class: Introduction: Simple harmonic motion (SHM) is characterized by a displacement x from an equilibrium point. The object's motion is a sinusoidal function of time. Such motion occurs in systems in w hich there is a restoring force w hich increases linearly w ith distance from equilibrium: the farther the object is from its center, the harder the restoring force pulls back on it. In class, you will study simple harmonic motion of a mass on a spring. Recall that the force F needed to stretch a spring a distance x (labeled as Dx in the figure below) from its equilibrium length is described by the equation F = k x, w here k is the spring constant, or "stiffness" of the spring. The constant k may be found by applying a force to the spring, and measuring x. The SI units for k are N/ m. If the end of the spring is not accelerating, the spring exerts a restoring force that is equal in magnitude and in the opposite direction. 1. In the figure below, draw a free body diagram for the forces acting on a mass m hanging from the bottom of a spring. Figure 1: Stretching of a spring. 2. A spring stretches by x = 10 cm w hen a mass of 75 g is hung from it. What is the spring constant k of the spring? IX-2
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3. To more accurately measure k, one can construct a graph of force as a function of displacement. From the spring equation, such a graph should be a straight line. What variable or constant does the slope represent? Below is a graph of displacement vs. time for an object that experienced SH M. 4. What is the amplitude of the oscillation of the motion? amplitude = cm IX-4
List of Today s Activities Introduction Problem Solving Lab Activity Problem Solving Lab Challenge Brief review of key concepts and Pre-Lab questions Observe sinusoidal oscillations using the motion detector. Discuss concept questions. Simple harmonic motion of a mass on a spring M easure spring constant k M easure Period T Group Work Problem SH M of molecules Find the unknown mass. IX-5
Laboratory Activity 1 Lab Activity and Group Work Observe sinusoidal oscillations using the motion detector. Concept Questions Use the motion detector to record the simple harmonic motion of a mass on a spring. Record position, velocity, and acceleration and then answer the concept questions below. 1. A block on frictionless surface is connected to a wall by a spring. The block oscillates betw een extremal points A and E. Identify the point(s) w here each of the following is true: a. maximum speed: b. zero speed: c. maximum acceleration: d. zero acceleration: A B C D E 2. Circle the direction of the force exerted by the spring on the block for the following situations: v v A B C D E F sp on blk : 0 A B C D E F sp on blk : 0 IX-6
Activity 2 Lab Activity M easuring the spring constant, k: Record the equilibrium length of the spring with no additional mass attached. Then measure the distance the spring is stretched when hanging a series of masses up to 170 g. Choose a reference point somew here near the bottom of the spring or masses from w hich to record the differences. The displacement is the scale reading minus the reading for zero added mass. IMPORTANT: The springs you are working w ith are quite delicate. Do not hang masses greater than 200 grams from the springs, or they may be permanently stretched out of shape. Hanging Mass m (kg) 0.000 0.050 0.080 0.110 0.140 0.170 Scale Reading x (m) Table 1 Displacement x (m) Stretching Force F = mg (N) 0.000 0.000 N ow open the LoggerPro template Spring plot.cmbl on the desktop. Enter the mass and displacement data into the worksheet. Do not include the point with zero hanging mass. LoggerPro will automatically generate a plot of stretching force F on the vertical axis versus displacement, x on the horizontal axis. LoggerPro can calculate the best linear fit to your data (click on the appropriate button near the top of the window). The slope of this line is the spring constant k. k = ( ) include units! IX-7
Activity 3 Lab Activity Period versus amplitude for a spring: In general, the period (T) of any object engaged in simple harmonic motion is related to the stiffness parameter (k) and the inertia of the system (m). When a mass hanging on a spring is displaced from its equilibrium, the mass oscillates with a period: T = ( 2π ) This equation can be derived from the spring equation using Newton s 2 nd law. The mass m in this equation is the total mass, which includes the hanging mass, as well as the hook (30 g) and the spring itself (~1 g). Note that the equation does not contain the amplitude (A) of the oscillation. Test this relation by measuring the period of oscillation for a hanging mass of 120 g for several different amplitudes. To start the mass oscillating, pull the spring dow n by a few cm, then release. Use a stopwatch to record the time it takes the spring to complete 25 cycles of oscillation. m k Amplitude A (cm) Total time for 25 cycles 25T (s) Period T (s) Does the period depend on the amplitude? Compare the period measured above w ith the period calculated using the value of the spring constant you determined above. T measured = T calc = % difference = f = 1/T = ( ) include units! Calculate the angular frequency of this motion: ω = 2π f = 2π / T = ( ) include units! IX-8
Activity 4 Group Work and Lab Activity 1. Predict the ratio of the periods of tw o masses M 1 and M 2 = 4 M 1 that oscillate in SHM on springs that have the same spring constant k. Explain the reasoning behind your prediction. Prediction: T 2 T 1 = 2. Devise an experiment to test you answ er then carry out the experiment. Give details of w hat you did, and show how the results either support or do not support your prediction. IX-9
Activity 5 Group Work Problem SH M of molecules M olecules vibrate at frequencies that can be predicted using simple harmonic motion. For example, the chemical bond between hydrogen atoms in the hydrogen molecule (H 2 ) acts like a spring: So f ar y ou've only considered the motion of one mass attached to a fixed support by a spring. A hydrogen molecule (w hich has two masses connected by a spring), is equivalent to a one-mass system if you use the r educed mass m r educed : m r educed = m 1 *m 2 / (m 1 +m 2 ) Given that the mass of a hydrogen atom is 1.67 x 10-27 kg, and the hydrogen molecule vibrates w ith a frequency of 1.32 x 10 14 Hz, calculate the following: m r educed = k H2 = Deuterium is an isotope of hydrogen; the nucleus has 1 proton and 1 neutron, instead of only 1 proton as for hydrogen. Thus, a deuterium atom has twice the mass of a hydrogen atom. Calculate the reduced mass and force constant for the deuterium molecule D 2, given an observed vibrational frequency of 9.34 x 10 13 Hz: m r educed = k D2 = How do the k values compare for H 2 and D 2? What might this tell you about the relative chemical reactivity? IX-10
Activity 6 Lab Challenge: Find the unknown mass. Use the equation for the period of oscillation to find the mass of an object. unknown mass spring = Use the scales in the lab room to measure the unknown mass. unknown mass scale = Analysis H ow w ell did you do? Calculate the percent error to compare the tw o values of the unknown mass: Error = (unknown mass spring - unknown mass scale )/ unknown mass scale ] x 100 = % End of Lab 9 When you are finished, close Logger Pro. D o not save any changes. IX-11