Lab 10 Simple Harmonic Motion - Mass on a Spring (Force Sensor, Motion Sensor) GENERAL EDUCATION COMMON ASSESSMENT TEMPLATE 1. TITLE & NUMBER OF THE COURSE: 2. TITLE OF THE ASSIGNMENT: Simple Harmonic Motion Lab 3. GENERAL EDUCATION CORE OBJECTIVES TO BE ASSESSED WITH THIS ASSIGNMENT (List specific general education outcomes and any student learning outcomes): A. Critical Thinking B. Quantitative and Empirical Skills C. Communication D. Teamwork 4. DESCRIPTION OF ASSIGNMENT (as it would appear in your syllabus or class handouts) Use the Force Sensor to measure the spring constant k for the spring. Use the Motion Sensor to measure the period of motion for the spring.

Simple Harmonic Motion - Mass on a Spring (Force Sensor, Motion Sensor) Equipment: Force Sensor 1 Hanger and Masses: 6-50 g; 2-100 g Motion Sensor 1 Support Rod Spring 1 Clamps: right angle; spring clamp; C-clamp

A spring that is hanging vertically from a support with no mass at the end of the spring has a length L (called its rest length). When a mass is added to the spring, its length increases by L. The equilibrium position of the mass is now a distance L + L from the spring's support. What happens when the mass is pulled down a small distance from the equilibrium position? The spring exerts a restoring force, F = -kx, where x is the distance the spring is displaced from equilibrium and k is the force constant of the spring (also called the 'spring constant'). The negative sign indicates that the force points opposite to the direction of the displacement of the mass. The restoring force causes the mass to oscillate up and down. The period of oscillation depends on the mass and the spring constant.

As the mass oscillates, the energy continually interchanges between kinetic energy and some form of potential energy. If friction is ignored, the total energy of the system remains constant. Equipment Setup-1: Finding the Spring Constant (K)

1. Mount the C-clamp to the edge of the table, put the rod in the clamp, mount the Force Sensor vertically so its hook end is down. Note, for this online lab, you are calculating the force by using F=mg. So, you are not using the values from the force sensor. 2. Suspend the spring with hanger from the Force Sensor's hook so that it hangs vertically. Procedure-1: Data Recording

1. Add 50 grams of mass to the end of the hanger. 2. Measure the new position of the end of the spring at equilibrium. Enter the difference between the new position and the equilibrium position as the x, in the table under displacement column. The value of the displacement for each run is provided in the table. 3. Add 50 grams to the hanger (for a total of 100 g additional mass). Measure the new position of the end of the spring, enter the x value in the table under displacement column. 4. Continue to add mass in 50 grams increments until you have added 350 grams. Each time you add mass, measure and enter the new displacement value from equilibrium. Complete the following table and calculate the average spring constant. Table:1 Mass (g) Mass (kg) Weight(N) F=m*g Displacement x (cm) Displacement x (m) Spring constant, k=F/x Average Spring constant 50 1.5 100 3.1 150 4.7 250 7.4 300 8.9 350 10.8 5. Choose an appropriate scale and plot Force vs Displacement graph (that's Force on the Y-axis and Displacement on the x-axis) using your data from the table 1. Find the slope of the graph from your plot. Explain the physical meaning of the slope. Activity-2: Use the Motion Sensor and 850 Interface to record the motion of a mass on the end of the spring and determine the period of oscillation than compare the value to the theoretical period of oscillation. Computer Setup-2: 1. Unplug the Force Sensor's plug from the interface. 2. Connect the Motion Sensor's stereo phone plugs into Digital Channels 1 and 2 of the interface. Plug the yellow-banded (pulse) plug into Digital Channel 1 and the second plug (echo) into Digital Channel 2. Equipment Setup-2: 1. Using a support rod and C-clamp, suspend the spring from the spring clamp so that it can move freely up-and-down. Attach it firmly to the outermost position. Put a mass hanger on the end of the spring. 2. Find the mass of the spring. Add 153 g to the hanger. 3. Record the mass of the hanger, the 153 g, and 1/3 the mass of the spring (in kg) in the Data section. Return the hanger and masses to the end of the spring. 4. Place the Motion Sensor on the floor directly beneath the mass hanger. 5. Adjust the position of the spring so that the minimum distance from the mass hanger to the Motion Sensor is greater than the Motion Sensor's minimum distance (15 cm) at the maximum stretch of the spring. Procedure-2: 1. Pull the mass down to stretch the spring about 5 cm. Release the mass. Let it oscillate a few times so the mass hanger will move up-and-down without much side-to-side motion. 2. The plots of the position vs time of the oscillating mass will be displayed. Continue recording for about 10 seconds. This plot is provided below for your analysis and calculations. 3.

The position vs time curve resembles the plot of a sine function. Analysis 2: 1. Find the average period of oscillation of the mass by taking the difference in time between any two peaks.

Position vs time plot

1.1.

Observe the first peak in the plot of position versus time and read the time value for the first peak. Record the value of time in the Data Table 3. Note: the time values for peak 1 and 2 are recorded for you. 1.2. Now study each consecutive peak in the plot and record the value of time shown for each peak (up to peak 7). 1.3. Calculate the time period of oscillation of the mass by taking the difference time between any two consecutive peaks; for consecutive peaks 1 and 2 the time period is calculated for you. continue this for up to peak 7. Find the average of the time period. Data Table-2 Item Value Equilibrium Position 30 Spring Constant (slope)

Data Table-3 Mass = _________0.153 kg Peak 1 2 3 4 5 6 7 Time (s) 0.400 0.84 Period (s) N/A 0.45 Average period of oscillation = ________ sec Critical Thinking and Quantitative and Empirical Skills Assessment (Complete Individually) 1. Calculate the theoretical value for the period of oscillation based on the measured value of the spring constant of the spring and the mass on the end of the spring.

2. How does your calculated value for oscillation compare to the measured value of the period of oscillation? What is the percent difference? 3. When the position of the mass is farthest from the equilibrium position, what is the velocity of the mass? 4. When the absolute value of the velocity of is greatest, where is the mass relative to the equilibrium position? 5. A mass of 225 g is suspended from a vertical spring. It is then pulled down 15 cm and released. The mass completes 10 oscillations in a time of 32 seconds. What is the force constant for the spring? 6. A block of unknown mass is attached to a spring with a force constant of 6.50 N/m and undergoes simple harmonic motion with an amplitude of 10.0 cm. When the block is halfway between its equilibrium position and the end point, its speed is measured to be 30 cm/s. Calculate a) the mass of the block, b) the period of the motion, and c) the maximum acceleration of the block. Critical Thinking/Written Communication (Complete Individually) 7. Conclusion: Describe the physics principles studied in this lab. Discuss the uncertainties involved in the measurements and possible errors which made the experimental results different from the theoretical results. Suggest possible improvements in the experiment which could reduce these uncertainties.