In this lab you will explore the motion of a mass attached to a spring hanging vertically from a fixed support and the motion of a simple pendulum.

Open a Microsoft Word document to keep a log of your procedures and your results. This log will form the basis of lab report. Address the points highlighted in blue. Answer all questions.

A spring has an equilibrium length L. When a spring is compressed, then a
force with magnitude proportional to the **decrease** in its length from its equilibrium length is pushing each end away from the
other. When a spring is stretched, then a force with magnitude
proportional to the **increase** in its length from its equilibrium length is pulling each end towards the other.

If one end of a spring is fixed to a support and an object is attached to the
other end, then the force exerted by the spring on the object has a magnitude
proportional to the displacement ΔL of the free end
from its equilibrium position. The direction of the force is always
opposite to the direction of the displacement. If the x-axis of a
coordinate system is chosen parallel to the spring and the equilibrium position
of the free end of the spring is at x = 0, then the displacement ΔL
is equal to x, and the force is F = -kx. The proportional constant k is
called the **spring constant**. It is a
measure of the spring's stiffness.

If an object attached to the free end of the spring is displaced along the x-axis (in the positive x-direction) and then released, the force exerted by the spring on the object will accelerate it towards the equilibrium position (in the negative x-direction). When it reaches the equilibrium position it will have its maximum speed. As it passes through the equilibrium position, the force the spring exerts on the object changes direction. The object now decelerates, until its speed is zero and it displacement is maximum in the negative x-direction. The acceleration is in the positive x-direction and the object again accelerates towards the equilibrium position. The object oscillates about the equilibrium position.

- The position of the object as a function of time t is
x = x
_{max}cos(2πft + φ).

x_{max}is the amplitude of the oscillations and f is the frequency.

φ is the phase of the oscillations.

The period T of the oscillations is T = 1/f. - The velocity of the object as a function of time t is v = -v
_{max}sin(2πft + φ).

v_{max}is the maximum speed of the object as it passes the equilibrium position. - The acceleration of the object as a function of time t is a = -a
_{max}cos(2πft + φ).

a_{max}is the magnitude of the acceleration of the object at maximum displacement.

The force acting on the object is F = ma.

You will measure the period of the oscillations of different masses attached to the same spring. For a mass on a spring we expect

ω = √(k/m), T = 2π√(m/k).

To find the spring constant k you will plot T^{2} versus m. The
slope of the best fitting straight line equals 4π^{2}/k.

Procedure:

You will analyze 4 video clips, spring_x.mp4, x = 1 - 4. To play a video clip or to step through it frame-by-frame click the "Begin" button. The "Video Analysis" web page will open.

Analyze each of the clips to find the period of oscillation. Track
the point where the mass attaches to the spring. (See Lab
9, hints.)

The weight of the object attached to the spring is given. Find its
mass. Construct a spreadsheet as shown below.

Video Clip | mass m (kg) | period T(s) | T^{2} (s^{2}) |
---|---|---|---|

1 | |||

2 | |||

4 | |||

4 |

- Construct a plot of T
^{2}versus m. - Use the spreadsheet's regression function to find the slope of the best fitting straight line to this plot and the uncertainty in the slope.
- Find the spring constant k and the uncertainty in this value, Δk. Use Δk/k = Δ(slope)/(slope).
- Optional: If you have completed extra credit 6, compare your value for k found in this lab with the value found in exercise 6. Both experiment use the same type of spring.

Paste your table and plot of T^{2} versus mass
and your value for the slope of the best fitting straight line into your log.

What value did you find for k and what is the uncertainty in k?

If you have completed exercise 6, did you find the same value for k in exercise 6?

You will analyze one video clips. The clip shows a swinging simple pendulum. You will measure the period T of the pendulum and compare your measured value to the theoretically predicted value T = 2πSQRT(L/g). For a review, click here.

Procedure:

To play the video clip or to step through it frame-by-frame click the "Begin" button. Choose the pendulum_2.mp4 video clip. Choose to track the x- and y-coordinate. Calibrate x and y, and choose the suspension point as the origin of your coordinate system (the first point you click when you calibrate x or y). You will have to do some measuring or estimating to get the correct distances.

Copy the data table into Excel and add a column displaying θ(rad) = tan^{-1}(x/y).
(In cell D2 type =ATAN(B2/C2).) The first wo rows of your
spreadsheet should lokk similar to the table below. Plot θ as a
function of time. Use your data to determine the period and the length of
the pendulum.

A | B | C | D | |
---|---|---|---|---|

1 | time | x (m) | y (m) | theta (rad) |

2 | 0 | 0.017 | 0.520 | 0.0324 |

3 | 0.0334 | -0.033 | 0.513 | -0.0658 |

4 | 0.0667 | -0.072 | 0.505 | -0.1412 |

Assume θ(t) = θ_{max}cos(ωt +
φ). Find the most
negative and the most positive angular displacements. Average the absolute
value of these numbers to find the amplitude θ_{max}.
The time interval between the most negative and the most
positive angular displacements is one half period, T/2.

Paste your graph into your log.

What is the period of the pendulum as
determined from your data?

Estimate the length of the
pendulum using the scale provided in the video clip. Record it.

Using this length, calculate the theoretically predicted period of the pendulum for small
oscillation. Record it.

The equation describing the motion of the pendulum is θ(t) = θ_{max}cos(ωt + φ).
What value did you obtain for the amplitude θ_{max}?

Convert your log into a lab report.

**Name:E-mail address:**

**Laboratory 9 Report**

- In one or two sentences state the goal of this lab.
- Insert your log with the requested table, graphs, and the answers to the questions in blue font.

Save your Word document (your name_lab9.docx), go to Canvas, Assignments, Lab 9, and submit your document.