Student Exploration: Determining a Spring Constant (ANSWER KEY)

Student Exploration: Determining a Spring Constant
Vocabulary: displacement, equilibrium, Hooke’s law, restoring force, slope, spring, spring constant, weight
Prior Knowledge Questions (Do these BEFORE using the Gizmo.)
At the grocery store, you put a watermelon on a produce scale. This causes the spring to stretch as shown. How far will the spring stretch if you add another watermelon of equal mass?
What property allows springs to be used in scales?
Gizmo Warm-up
When you put a grapefruit on a grocer’s scale, the scale may bounce up and down a bit, but eventually it settles into an equilibrium state. At this point, the force pulling the spring down is equal to the restoring force pulling the spring up. You can explore these forces in the Determining a Spring Constant Gizmo™.
To begin, check that Spring 1 is chosen and nothing is hanging from the spring.
What is the level of the bottom of the spring?
Place the scale on the bottom of the spring. The scale has a mass of 20 grams. Wait for the spring to stop moving. At this point it has reached equilibrium.
What is the level of the spring now?
How much did the spring stretch? This is the displacement of the spring.
Place mass C (20 grams) on the scale. What is the level of the spring?
What is the displacement of the spring? 
Activity:

The spring constant
Get the Gizmo ready:
  • Remove all weights from Spring 1.
  • Select the TABLE tab.
 

Question: How is the displacement of a spring related to the weight it bears?
Predict: In this activity, you will create a graph of the displacement vs. the weight on the spring. What do you think this graph will look like?
Calculate: Place the 20-g scale on the spring.
Convert the mass of the scale in grams to kilograms by dividing by 1,000.
What is the mass of the scale in kg?
To find the weight of the scale, multiply the mass (in kg) by gravitational acceleration, 9.81 m/s2. (Note: The units for weight are kg·m/s2, or newtons (N)).
What is the weight of the scale in newtons?
Gather data: Record the position and displacement for each mass on the spring. Click Record data each time the spring reaches equilibrium. (Note: You will have to figure out which combination of objects adds up to each of the masses listed in the table.)
Analyze: What patterns do you notice in your data? (Hint: What happens to the displacement when the weight is doubled?)
(Activity continued on next page)
Activity (continued from previous page)
Interpret: Select the GRAPH tab. What do you notice? The graph is linear and increases 1 cm per 0.30 N.
Measure: Turn on Show line. The slope of the line (rise divided by run) is given by the value of m. Adjust the m slider until the line is aligned with all four points on your graph.
What is the slope of the line?
Calculate: On your data table on the previous page, multiply each displacement value by the slope of the line recorded above. What do you notice?
Infer: The slope of the line is a measure of the stiffness of the spring. The greater the slope is, the stiffer the spring because it indicates that more force is required to stretch the spring a given amount. The slope of the line is called the spring constantand given the symbol k.
Based on your data, create an equation that relates the force on the spring (F), the displacement (x), and the spring constant (k).
This relationship is known as Hooke’s law. Usually, Hooke’s law is written for the restoring force (FR) rather than the force on the spring. Because the spring is in equilibrium, the restoring force is equal to the negative of the force that is pulling the spring.
Apply: How far will Spring 1 stretch with a mass of 70 grams?
Use the Gizmo to check your answer. Show your work below.
Practice: Find the spring constant for each of the other springs in the Gizmo. Show your work on a separate sheet of paper.
Spring 2: k =
Spring 3: k =
Spring 4: k =

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