assignment

shm_labExplorationA.pdf

PHYS 242 Lab Week 11 Lab San Francisco State University

Simple Harmonic Motion In this lab we will investigate the properties of a mass on a spring, using a simulation from the PhET team. The simulation is available at the following link: https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html

The simulation can be run in a browser. If you have issues with the simulation, try using another browser. If you are unable to run the simulation, your TA will provide you with remote assistance. When you run the simulation, choose the “Lab” option.

NOTE: For this activity, you do not need to write a full lab report. Instead, answer the questions that are included in this lab. Please type up your answers, if possible. For the sketches, you can take pictures and include the images in your write up. Submit your answers to these questions on iLearn as a PDF. If you need help with this process, please ask your TA for assistance.

Part I: Masses and Springs

This lab will investigate the motion of a mass hanging vertically on a spring, as shown in Figure 1.

Question: Draw a free body diagram for the hanging mass.

Figure 1: A 100 g mass hanging vertically on a spring, as shown in the PhET simulation. The spring constant is set to the default value.

PHYS 242 Lab Week 11 Lab San Francisco State University

Part II: Familiarize yourself with the simulation

Again, make sure that you choose the “Lab” option for the simulation. At the very bottom of the screen you will see the other options for the simulation, including a home button, “Intro,” “Vectors,” “Energy,” and “Lab.” If you accidentally navigate to another area, you can return to the Energy option by clicking the “Lab” button.

The simulation provides a spring with a certain default spring constant, as shown in Figure 1. By moving the 100 g mass up to the spring, we can attach it to the spring. As you showed in Part I, if the system is not in equilibrium the mass will begin to oscillate up and down. We can pause or slow down the simulation with the buttons in the lower right corner of the screen. We can also drag the ruler and the timer (initially in the box on the right-hand side of the screen) over to the system to measure the displacement of the mass at different times.

The quantities in the right-hand box allow us to change the gravity and the damping for the system. Make sure that the gravity is set to Earth’s gravity, and the damping is set to “None.”

On the left side, we can expand the “Energy Graph” box to see how the kinetic, potential, and total energy of the system vary with time.

Part III: Simple Harmonic Motion

You will investigate the motion of the mass. Use the following initial parameters for your simulation:

Damping = “None”

Gravity = 9.8 m/s2 (Earth)

Mass = 100 g

Spring Constant = Default value

Mass Equilibrium box = Checked

The dashed black line will now show the equilibrium position of the system. Set up your ruler so that 50 cm falls on the dashed black line. Pause the simulation, and set your timer to play. Move the mass so that the center is at about 90 cm. Your setup should now look similar to that in Figure 2.

When you hit play, the timer will begin to count down. You can reset the timer by hitting the bottom left arrow button on the timer.

Change the animation speed to “Slow.”

PHYS 242 Lab Week 11 Lab San Francisco State University

Question: Record the displacement of the mass at the times that are given in Table 1. (This is simplest to do by running the animation at “slow” speeds, pausing it when the clock hits one of the values in Table 1, and recording the displacement at that time.) Find y’ by subtracting 50 cm from y. Make a plot of the adjusted y-displacement (y’ in Table 1) as a function of time, t. What shape does this plot have?

Figure 2: The initial setup for your experiment in Part III.

PHYS 242 Lab Week 11 Lab San Francisco State University

Table 1: Displacement versus time. Note that y’ = 0 cm when y = 50 cm.

t (s) y (cm) y’ (cm) = y – 50 cm

0.1

0.2

0.3

0.4

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

PHYS 242 Lab Week 11 Lab San Francisco State University

The movement of the mass can be described as “simple harmonic motion,” where the mass obeys the equation:

!′ = %&'((*+ + -) where A is the amplitude, ω is the angular frequency (in rads/s), and φ is a phase constant (don’t worry about this too much for now). The angular frequency is also related to the frequency and period of the system.

Question: What is the amplitude of the system?

Question: The period of the oscillation is the time it takes to complete one full cycle. What is the period of this system? What are the frequency and the angular frequency?

The angular frequency of a mass on a spring is directly related to the mass and the spring constant, according to the equation:

* = /01

Question: What is the spring constant for this system?

Part IV: Velocity, Acceleration, and Energy

Recall that the maximum velocity and acceleration of a single harmonic oscillator can be found by taking the first and second derivatives, respectively, of the displacement equation and setting them equal to zero. If we do that, we find:

2!"# = *% 3!"# = *$%

Question: Where in the mass’s oscillation is the velocity at a maximum? Where is the acceleration at a maximum? Show these points on your plot of y’ versus t and explain.

PHYS 242 Lab Week 11 Lab San Francisco State University

Looking at the graph on the left, you can see how the kinetic and potential energy vary as the mass oscillates. Note that PEelas is the potential energy of the spring, whereas PEgrav is the potential energy due to gravity.

Question: Where is the kinetic energy at a maximum? Where is the potential energy at a maximum? Show these points on your plot of y’ versus t and explain.

Part V: Mystery Masses

You will now determine the masses for the blue and red weights at the bottom using the same spring constant.

Question: Given what we have learned from the 100 g mass, how would you determine the masses of the other two blocks?

Question: What are the masses of the other two blocks?