Phy212: General Physics II Laboratorypage1 of 6

Instructor: Tony Zable

Laboratory: Good Vibrations

Preliminary Questions:

1. Consider a mass attached to a string that is connected to the ceiling (as shown in the figure).

a. Predict how the period (T) of the oscillations should vary with the size of the mass. Explain.

b. How might the period vary with the length of the string? Why?

c. If you have 2 identical pendulums, one on the surface of the Earth and the other on the surface of the moon, which one would have the greater period? Why?

2. A mass is attached to an “ideal” spring that is connected to the ceiling (as shown in the figure).

a. Predict how the period (T) of the oscillations might vary with the size of the mass. Explain.

b. How might the period depend on the stiffness of the spring? Why?

c. If you have 2 identical spring systems, one on the surface of the Earth and the other on the surface of the moon, which one would have the greater period? Why?

Introduction:

In this laboratory you will study the motion of a mass suspended from a string, and a mass suspended from a spring. The mass-string combination is called a simple pendulum, and the mass-spring combination is called a Simple Harmonic Oscillator (SHO for short), although what is simple about it may not be immediately apparent.

These two systems exhibit remarkably similar properties. First …

Part 1: The Simple Pendulum

A drawing of a pendulum, which consists of a mass at the end of a string, is shown below.

The two forces that act on the mass are the weight,mg, and the tension force, FT (both indicated by bold arrows). The weight of the pendulum has a component along the direction of the circular arc,the mgsin term, and this is the force component that generates the torque which causes the pendulum to swing back and forth.

The magnitude of this torque is

and since Newton's 2nd law says this must equal I we have

For small angles, sin  ≈  (in radians), therefore

(1)

In the above equation, is the angular acceleration of the pendulum. Since the angular displacement,, varies with time, equation (1) suggests that the angular acceleration also changes with time. Clearly, this is a very complex kind of motion.

Equation 1 is a second order differential equation, where the angular frequency for the swinging pendulum (not to be confused with the angular velocity of the pendulum!!) is:

(2)

Question 1: What is the equation for the corresponding swing frequency for this simple pendulum?

Question 2: What is the equation for the corresponding period, T, for the pendulum swing?

Your first objective will be to use measurements of pendulum length and swing period to calculate the acceleration of gravity, g. You will need to measure the period as accurately as possible. For small values of m (see above figure) the period should be independent of the maximum value of m. You can use this fact to get an accurate measurement of the period by observing the time taken to swing through a number "N" of round trips, and then dividing this observed time by "N".

Definitions:

Procedure: (Pendulum)

  1. Set-up a simple pendulum.
  2. Measure the length of the pendulum (L), from fulcrum to the center of the hanging mass. Record in Table 1.
  3. Using a stopwatch, measure the time it takes for 20 complete swing cycles (t). Record time in Table 1.
  4. Calculate the period, T, and record the value in table.
  5. Shorten the pendulum length then repeat steps 2 through 4.
  6. Repeat steps 2-5 for a total of 5 trials.

Table 1
Trial / Pendulum Length, L / # of cycles, N / Time, t / T / T2 / g
gavg=
g =

Analysis: (Pendulum)

  1. Using the equation you obtained in Question 2 above; calculate g for each of your trials. Record values in Table 1.
  2. Calculate the average value of g, gavg. Record value in Table 1.
  3. Estimate the experimental uncertainty of gavg using the min-max method. Record value in Table 1.

Part 2: The Simple Harmonic Oscillator (SHO)

The simplest model of a vibrating system is called the simple harmonic oscillator (or SHO). You can make a SHO by hanging a mass on a spring and letting the mass come to rest as shown in figure below.

The distance the spring is stretched (from the zero mass length) is often labeled x. Since the hanging object is at rest the force that the spring exerts on the mass just equals the weight of the mass. The more weight that you hang on the spring the longer it stretches, and the force that the spring exerts on the mass increases linearly with the amount of stretch (remember Hooke’s Law??). In general, the following relationship applies:

Fspring = -kx

where the constant k in this equation is the spring constant.

Now if you displace the mass slightly from its equilibrium position, and then release it, the mass oscillates up and down as it tries to return to equilibrium. The oscillation is called simple harmonic motion, and it too has a period, T, that is the time for the mass to make one cycle of oscillation.

Applying Newton’s 2nd Law to the oscillating mass:

where x is the distance of the oscillating mass from its equilibrium position.Note that the acceleration in the above equation reflects both the x and x. The acceleration of the masscan be written as therefore

.(3)

Note:

Equation 3 is a second order differential equation, where the angular frequency for the oscillation is:

(5)

Question 3: What is the equation for the corresponding oscillation frequency, f, for the simple harmonic oscillator?

Question 4: What is the equation for the corresponding period, T, for the simple harmonic oscillator?

Procedure: (SHO)

  1. Set-up a simple harmonic oscillator using a gram mass.
  2. Record the mass in Table 2.
  3. Using a stopwatch, measure the time it takes for 20 complete swing cycles (t). Record time in Table 2.
  4. Calculate the period, T, and record the value in table.
  5. Replace the hanging mass with a different weight then repeat steps 2 through 4.
  6. Repeat steps 2-5 for a total of 5 trials.

Table 2
Trial / Mass, m / # of cycles, N / Time, t / T / T2 / k / k (adjusted)
kavg=
k =

Analysis: (SHO)

  1. Using the equation you obtained in Question 2 above, calculate k for each of your trials. Record values in Table 2.
  2. Calculate the average value of k, kavg, for your data. Record value in Table 2.
  3. Estimate the experimental uncertainty of kavg using the min-max method. Record this value in Table 2.
  4. An important concern in this set of experiments is the fact that the spring is not really mass-less. Thus, depending on the mass of the spring to the mass of the hanging weight, the spring’s mass may be significant. If the mass of the spring is not neglected, how would the mass of the spring alter your calculation of k? Explain.
  5. Observe your k values above. Do you notice a tend in the k values as the hanging mass is increased? If so, then you will need to correct your k values to account for the spring mass.
  6. In your answer to question 5 is yes, measure the mass of the spring then re-calculate the k values to account for the mass of the spring. Record values in Table 2.
  7. Determine the average adjusted k value and the uncertainty of this value. Does the average k value change significantly? Does the uncertainty in k decrease? Explain.

One Last Thing: The Unification

The motions of the pendulum and the SHO are very similar, one oscillates back and forth and one oscillates up and down. To confirm the similarity of these motions construct a pendulum and SHO combination such that both oscillators have the same period.

  1. Select a combination of g, L, k and m values that produce a common oscillation period for your pendulum and SHO.
  2. Record values for g, L, k and m in Table 3.
  3. Using the values in step 2, calculate the predicted period for the pendulum and the SHO. Record values in Table 3.
  4. Construct a pendulum and SHO that correspond to the values in step 2.
  5. Measure the period of oscillation for the pendulum and the SHO. Record the values in Table 3.
  6. Calculate the % Error between predicted and measured period values for the pendulum and SHO, respectively. Record in Table 3.

Table 3
Experimental Parameters
g / (predicted) Tpendulum / % Error
L / (measured) Tpendulum
k / (predicted) TSHO / % Error
m / (measured) TSHO
  1. Calculate the % Error between the measured Tpendulum and TSHO values.