Hook's Law

Theory

The elongation of the spring is caused by the force exerted on it by the mass due to gravity (mg). The force of gravity (or the weight) is balanced by the force exerted on it by the spring.

In terms of the stretch (or compression) of a spring, the force is defined as

F = - k y

That statement is known as the Hook's Law. Coefficient k is the spring constant that characterizes the stretchability of the spring and y is the difference in the length of a (how much the spring was stretched or compressed from its relaxed position). The unit of k is newton per meter [N/m]. Minus indicates the opposite direction of the force generated by a spring to the displacement from equilibrium.

The equation below shows the two forces, weight and the spring's reaction, in equilibrium:

-k y = m g

Figure 1.

Laboratory Procedure

1. The spring constant

Find the spring constant by adding small weights and measuring the stretch of a spring. Notice that we do not need to know the length of the entire spring; only the stretch and the force that causes the stretch.

Hook one of three springs on the force sensor as shown in the picture. Connect the sensor plug-in to the LabQuest interface. Begin stretching the spring with a small mass and continue adding more weights. If should be sufficient to plot about 5 points to find the spring constant. Read the force from LabQuest. Measure the stretch with a ruler.

One method of finding the length of a stretch is shown in the pictures below.

Figure 2.

Figure 3. Reading the prolongation of a spring

Another method is to measure the length of a spring without any weight, then measure the length with a weight added, and subtract the length of relaxed spring.

Relaxed spring

L = 7.5 cm

Spring stretched with a weight of 20g

L = 17 cm

ΔL = 9.5 cm

...weight of 30g

ΔL = 18 cm

...and so on.

Figure 4. Alternative method of measuring prolongation of the spring

Analysis

Plot the graph of Stretch vs Force (use ). Read k (the slope of the line). Remember about the units.

Determine the spring constant k for each spring using either Excel or Desmos. In order to do that, type both stretch and force data in two columns (see the screenshot below). In a cell below, paste a formula for slope (statistical formulas).

Figure 5. Determining the Slope in Excel

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