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

**Part 1:**

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. |

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

, .

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

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. (Lab
9, hints) | |||||||||||||||||||||

The weight of the object attached to the spring is given. Find its mass. | |||||||||||||||||||||

Construct a spreadsheet as shown below.
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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). | |||||||||||||||||||||

If you have completed exercise 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. |

**Part 2:**

In part 2 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.

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). Then copy the
data table into Excel and add a column displaying θ(rad) = tan^{-1}(x/y).
(In cell D2 type =ATAN(B2/C2).) 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.

Open Microsoft Word and prepare a report using the template shown below.

E-mail address:

In one or two sentences state the goal of this lab.

Part 1:

Show your spreadsheet containing your plot of T^{2} versus mass
and your value for the slope of the best fitting straight line. | |

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? |

Part 2:

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. | |

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

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}? |

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