1.8: Hooke’s Law and Simple Harmonic Motion

1. Measure the spring constant and potential energy of a spring and confirm the relationship between the mass and oscillatory period T.

1. Obtain a spring with a known spring constant, a stand to attach the spring to, at least 5 weights of varying masses that can be attached to the spring, a meter stick, and a stopwatch.
2. Secure the stand to a solid foundation and attach the spring to the stand. Make sure that there is enough room below the spring for it to stretch without hitting the table or ground.
3. For each of the masses, calculate the force exerted on the spring by the Earth's gravitational force (F = mg). Start with the least-massive weight. Record these values in Table 1.
4. Measure how high above the surface of the table the spring is while in its un-stretched position.
5. Attach the least-massive weight to the spring and measure the displacement Δy1 (see Figure 1). Record this displacement in Table 1.
6. With the weight attached, slightly raise the weight before releasing it. Observe the oscillatory motion. Measure the period T with a stopwatch. For a more accurate measurement, record the time for multiple periods and divide that time by the number of periods observed. Do this multiple times and record the average time measured for the period T in Table 1.
7. Repeat steps 1.5-1.6 for all of the masses, in order of increasing mass.
8. Calculate the potential energy of the spring for each of the different masses and record them in Table 1.
9. Plot the force F as a function of displacement Δy. According to Equation 1, this should be linear. Fit a slope to the line. This slope will correspond to the spring constant k. Compare the measured value to the known value of the spring.
10. Using the known spring constant and Equation 5, calculate what the period T of oscillation should be for each of the masses; report them in Table 1. Compare them to the T that was measured with a stopwatch in step 1.6.

Figure 1: Srping oscillation,

Hooke's Law and the phenomenon of simple harmonic motion help in understanding the physics associated with elastic objects.

Hooke's Law implies that in order to deform an elastic object, like a slingshot, a force must be applied to overcome the restoring force exerted by that object. This restoring force is a product of the elasticity constant k of the object and the displacement ?y but in the opposing direction to the displacement or applied force.

Clearly the elastic object stores energy that has the potential to do work. After the work is done the elastic object undergoes oscillation. If we plot this oscillatory behavior as the object's position versus time, then the graph represents simple harmonic motion.

In this video, we will demonstrate an experiment that uses springs and weights to validate the concepts behind Hooke's law and simple harmonic motion.

Before demonstrating how a spring behaves, let's revisit the concepts behind its oscillation. Imagine, applying a force to the spring, like a weight, that causes it to stretch from its initial non-deformed position until an opposing restoring force eventually balances it and equilibrium is established.

As per Hookes' law, this restoring force is equal to the spring constant k, which depends on the elasticity or stiffness of the material being deformed, times the displacement of the spring from its initial position, or ?y.

Therefore, knowing ?y and recalling that the restoring force is equal and opposite to the applied force, which is the weight in Newtons, the spring constant can be determined. Also, plotting F-applied versus ?y gives a line passing through the origin with a slope that represents k.

Now, with the spring at its equilibrium position, if you introduce an external force and lift the attached weight to a certain height, you allow the spring to gain some elastic potential energy PE. This potential energy is given by this formula, where k is the spring constant and ?y is the distance from the equilibrium position.

Now when you release the spring, it undergoes a periodic motion, known as simple harmonic motion. If plotted on a graph of position versus time, the motion yields the sinusoidal waveform of simple harmonic motion.

The period of oscillation T is given by this formula, which shows that T is inversely proportional to k -- the elasticity constant, and directly proportional to m -- the mass of the weight attached. Therefore, the larger the mass, the longer the spring would take to complete one cycle of oscillation.

If this system was isolated - unaffected by external forces, the oscillations would go on indefinitely as the kinetic and potential energies, KE and PE, would be continuously converted to one another. But in the real world there are always some frictional forces that cause damping and therefore the spring will ultimately come to a halt.

Now that you have an idea about the laws that govern spring oscillation, let's see how to test them in a physics lab. This experiment consists of a spring with a known spring constant, a stand, a set of weights with different but known masses, a meter stick, and a stopwatch.

Secure the stand to a solid foundation, such as a table. Attach the spring to the stand making sure there is enough room to stretch the spring without contacting the top of the table.

Using the meter stick, note the non-deformed position of the spring, or the distance between the bottom of the spring and the tabletop. Make a note of this starting position on the meter stick.

Now, starting with the smallest mass, calculate and record its gravitational weight. Attach the weight to the spring and measure the distance between the bottom of the spring denoting the equilibrium position and the starting position noted earlier. Record this displacement value.

Next, raise the weight slightly from its loaded position and release it to observe simple harmonic motion. Using the stopwatch, measure the oscillation period by dividing the time required for multiple periods by the number of periods. Repeat this procedure three times to obtain an averaged period. Since the period does not depend on the amplitude of oscillation, the values should be consistent.

Repeat the measurements of the spring displacement and the oscillation period for each additional weight, in order of increasing mass, and record all the readings.

Using the values from the displacement measurements, plot the gravitational weight as a function of the displacement distance. As expected from Hooke's Law, the dependence is linear and the slope of the line gives the spring constant. Comparing this measured value to the known spring constant value of k = 10 N/m reveals good agreement to within the error expected for this type of measurement.

Now calculate the potential energies for each weight using the known spring constant and the measured displacements. Given the equation, a plot of potential energy versus displacement square demonstrates linear proportionality.

Using the known spring constant, calculate the oscillation period for each weight. A comparison with the measured periods reveals strong agreement and confirms the expected relationship; that is, the period is proportional to the square root of the mass.

The restoring force that an elastic object exerts when it is deformed can be observed in several everyday events.

The suspension of modern vehicles consists of shock absorbers, which help minimize impact when driving over rough roads. The shock absorbers act as damped springs, absorbing the kinetic energy at impact and then dissipating it. Reducing the spring constant makes the ride smoother or mushier while increasing it is preferred in high performance vehicles for better handling.

Another application of these concepts would be harmonic oscillators - systems that undergo simple harmonic motion and experience continuous energy exchange. For instance, mechanical clocks convert potential energy stored in a torsion spring into mechanical energy to drive the gears and move the hands of the clock. Another example would be an LC circuit, which exhibits oscillation between electric potential energy, stored in the capacitor C, and magnetic potential energy, stored in the inductor L. This oscillation happens over a very specific period given by this formula, making LC circuit an integral part of many electronic devices.

You've just watched JoVE's introduction to Hooke's Law and Simple Harmonic Motion. You should now understand the concepts of the elastic potential energy, the restoring force, and how this force results in simple harmonic motion. Thanks for watching!