Minilab 1 Standing Waves Department of Physics & Astronomy Page 1

Today, we will observe standing waves on a string in order to learn and verify how the formation of standing waves depend on: Excitation Frequency Tension of the string Linear mass density of the string PURPOSE Department of Physics & Astronomy Page 2

When a wave moves along a string, we say the wave is propagating along the string, as shown. The linear mass density is related using: 𝝁= 𝒎 𝒍 The density and the tension of the string (T) affect the velocity that it propagates at: 𝒗= 𝑻 𝝁 = 𝑻 𝒎 𝒍 v THEORY Department of Physics & Astronomy Page 3

Note that waves reflections depend on how the string is attached at one end. End of string is fixed, the wave gets inverted End of string is loose, the wave is not inverted Theory Department of Physics & Astronomy Page 4

We use the term superimposed to mean two waves that are overlapping We use the term superimposed to mean two waves that are overlapping. Below, these two waves are travelling in opposite directions. Moving to right Moving to left The sum of the two waves (“superposition”) theory Anti-Nodes Nodes Department of Physics & Astronomy Page 5

If the length remains unchanged, standing waves only occur at specific frequencies. In our case, we have strings with nodes at both ends, which produces the following: l/2 l 3l/2 theory Department of Physics & Astronomy Page 6

equipment Node on top of the pulley wheel Node somewhere beyond the wave driver Mechanical Wave Driver creates waves (Frequency and Amplitude controlled by Capstone Software) Mass creates tension in string: T = mg equipment Department of Physics & Astronomy Page 7

procedure Experimental Determination of Speed of Wave The velocity of the wave can be calculated as follows V = f l (or ) (read off frequency in Capstone Software, measure l when you see a standing wave pattern). Start from low frequency and observe several different standing waves (different f and l). Plot f versus 1/ l  The slope of this graph equals v. Repeat the procedure using a different tension in the string (use a different mass at the end of the string). V should be different because it depends on the tension T. procedure Department of Physics & Astronomy Page 8

Once you have collected your data, you will need to plot f versus 1/λ in Excel. The slope of your graph is equal to v. If you are still having struggles with plotting in Excel, please refer to the Excel Tutorial online, or make sure your lab partner can explain it to you. procedure Department of Physics & Astronomy Page 9

procedure Theoretical Determination of the Speed of the Wave Important: The string is elastic and gets stretched under tension.  However: It is easier to determine Put the loose string on the scale and measure the mass Measure the full length of the un-stretched string with a ruler Calculate procedure Department of Physics & Astronomy Page 10

procedure Next, you need to determine how and are related. Imagine two situations: 1) Stretched string (with tension) in our setup (no mass hanging on string): Mass of the part of string between the pulley and the rod: M Length of string between the pulley and the rod: procedure Department of Physics & Astronomy Page 11

Take off the mass to release the tension  the string shrinks a bit. 2) Un-stretched string (no tension) in our setup (no mass hanging on string): Mass of the same string portion: M Length of the same string portion: procedure Department of Physics & Astronomy Page 11

procedure So, we have two equations relating to these two situations: We can combine these and eliminate M (the mass of that string portion): Then we can get an equation for the speed of the wave of the stretched string procedure Department of Physics & Astronomy Page 11

procedure To compare the theoretical and measured velocities, use the % difference calculation: procedure Department of Physics & Astronomy Page 12

Final hints Homework Policies Lab Report Policies You must do your homework BEFORE CLASS, and everyone must turn in their own work. Lab Report Policies Submit one lab report per group. Groups consist of two or three people. Make sure all members of the group write their name on the lab report! Final hints Department of Physics & Astronomy Page 14