LABORATORY: OSCILLATIONS AND WAVES Objectives: 0 to investigate the relationship between the mass attached at the end of a spring, the spring constant, and the resulting stretching of the spring. 0 to observe relationships between frequency, wavelength, speed of waves in a rope, and observe how these change with changing tension in the rope, Materials Required: Computer with Excel and access to simulations: 0 Masses and Springs simulation: https://phet.colorad0.edu/en/simulation/massspring-lab 0 Wave on a String: httpsz/lphet.colorado.edu/en/simulation/wave-on-astring Software Requirements: Windows Macintosh Chromebook Linux iPad Mobile Phone Chrome, Edge Chrome, Safari Chrome Not recon'nnended Safari Not recommended Introduction: There is no perfect rigid body. The stiffest of metal bars can be twisted, bent, stretched, and compressed. Under certain circumstances (typically, when forces are not too large), a body deformed by forces acting upon it will return to its original size and shape when the forces are removed, a property known as elasticity. Permanent distortion due to large forces is referred to as plastic deformation. Within the elastic limit, according to Hooke's Law, if a mass m placed at the end of a spring is displaced from its equilibrium position by an amount )6, the force 13' exerted by the spring on the mass, to bring it back to its equilibrium posmon IS 13" = k;? (1) where k is the spring constant. The negative sign reflects the fact that the force E and the displacement 3? always point in opposite directions. A wave is an oscillation accompanied by a transfer of energy which displace particles of the transmission mediumithat is, with little or no associated mass transport. Mechanical waves propagate through a medium, and the substance of this medium is deformed. Electromagnetic waves do not require a medium and can therefore travel through a vacuum. A wave can be transverse, where a disturbance creates oscillations that are perpendicular to the propagation of energy transfer, or longitudinal: the oscillations are parallel to the direction of energy propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse in free space. Mathematically, the most basic wave is the (spatially) one-dimensional sine wave with an amplitude ymax described by the equation: y( x , t ) = ymax Sin (k x a) t + Q5 ), where: ymax is the maximum displacement from equilibrium (maximum distance from the highest point of the disturbance in the medium (the crest) to the equilibrium point during one wave cycle); x is the space coordinate, t is the time coordinate, k is the wavenumber, a) is the angular frequency, and qb is the phase constant. The wavelength 2 is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber k, the spatial frequency of the wave in radians per unit distance (typically per meter), can be associated with the wavelength by the relation k = 2 n 1. The period T is the time for one complete cycle of an oscillation of a wave. The frequency f is the number of periods per unit time (per second). The frequency and period of a wave are reciprocals. The angular frequency @ is related to the frequency or period by w = 2 nf = 2n /T . The wavelength a of a sinusoidal waveform traveling at constant speed v is given by: 1 = v f , where v is called the phase speed (magnitude of the phase velocity) of the wave. Activity 1: Experimental Determination of the Spring Constant of a Spring The spring constant of a spring can be experimentally determined by applying different forces to stretch the spring different distances. If the force is plotted versus distance, the slope of the resulting straight line is equal to k. 1. Open the Lab tab in the Masses and Spring simulation. Get familiar with the simulation. Energy Graph a Mass Ism Spring Constant Lingo Natural Length CONTROL SHOW or HIDE Mass Equilibrium O Movable Line gravity; the energy in the Period Trace DETERMINE the Gravity 9.8 m's system gravity on a mystery planet Damping MEASURE the 00:01.54 What is the value of gravity? period Velocity Planet X O Acceleration DISCOVER the period with EXPERIMENT Period Trace O Norma with mystery Height = 0 m Slow masses Masses and Springs PHET 2. Hide the Energy Graph window and Select the Natural Length and Movable Line tools. Attach a 50.0 g mass at the end of the spring. Let the mass hang freely and wait for it to come to rest or press to stop the mass from oscillating. 3. Drag the movable line and place it at the end of the spring, and use the ruler to measure the spring-end's displacement (distance between the natural line and the movable line, i.e. the magnitude of the green displacement vector). Record the resting equilibrium position (x) (i.e. the stretch of the string from its equilibrium position) in Table 1. 4. Repeat the procedure for the other masses listed in Table 1. 10 20 Table 1: Hanging Mass Stretch Force 2m (kg) x (m) my (N) 0.050 0.100 0.200 0.250 0.300 5. Using the data from the Table l, in Excel, create a Force vs. Stretch scatter plot graph (nclude titles, axes, and the appropriate units). 6. What dependence of the springs stretch on the applied force does the graph show? 7. What are the units of the proportionality constant? 8. Using a linear t (y = mx + b ) write down the corresponding equation. The slope of the F orce v3. Stretch graph is known as the spring constant. Include appropriate units. 9. Include a copy of the graph here: 10. Knowing the spring constant of the spring, we can determine the unknown mass of a hanging object by measuring the stretch experienced by the spring. Attach the red block at the end of the spring. Record the resting equilibrium position (x) (i.e. the stretch of the string from its equilibrium position) in Table 2. ll. Calculate the block's mass m = g. and list the value in Table 2 below. 12. Repeat the procedure for the blue block 0. unknown mass. Table 2: Stretch kx = k Block x(m) m g ( at) Red Blue Activity 2: Speed of \\Vave and Tension 13. Open the Waves on a String simulation. Get familiar with the simulation. CREATE a wave Manual Restart Fixed End Loose End with an oscillator O Pulse NO En or pulse PLACE reference generator, or by line anywhere moving a wrench Slow Motion Normal MEASURE Frequency CONTROL wave Amplitude 1.00 cm 1.50 H2 Damping Tension distance or time Reference Line properties O Wave on a String PhET = 14. Set to "Pulse"; "Fixed End"; Damping = None; Normal; Tension = Low. Turn the Ruler and Timer on. Use the ruler to measure the length of the string. Record the data in Table 3 below. 15. Click the button on the pulse generator to send a pulse through the rope. 16. Use the timer to measure how long it takes for the pulse to travel back and forth 5 times. Record the data in Table 3 below. 17. Calculate the speed of the wave: length of the string V = - time it takes the pulse to travel the length of the string 18. Repeat the procedure for moderate and high tensions. Table 3: Tension Length of the Time for pulse to Time to travel Speed of wave = string travel 5 lengths one length distance/ time (m) (s) (m/s) Low Medium High 19. Repeat for moderate and high tensions. 20. How does tension affect the speed of a wave in a rope? Activity 3: Relationship between the Wavelength and the Frequency of a Wave 21. Set simulation to "Oscillate" and "No End." Keep the ruler and timer on.22. Set the frequency to 1.0 HZ and tension to low. Measure the time it takes for a wave to travel the length L of the string. Record it in Table 4 below. 23. Calculate the speed of the wave and record it in Table 4 length of the string L I] = = time it takes the pulse to travel the length of the string t 24. Measure the distance between two consecutive crests (or troughs). This is the Wavelength it of the wave. Record it in Table 4. 25. Repeat the measurements for 2.0 H Z and 3.0 H Z. and -or moderate and high tension. Table 4: Tension Time to travel the Speed Frequency length of the string Wavelength (L/t) f - it KHZ) t5(5) Mm) 1L'(m/S) (Tn/S) Low 1 Low 2 Low 3 Moderate 1 Moderate 2 Moderate 3 High 1 High 2 High 3 26. If the tension remains constant and the frequency increases, what happens to the wavelength? 27. Based on the pattern in the last two columns of the table (1) and f - A), What is the relationship between frequency. wavelength and speed of a wave