Physics Laboratory 9

A mass on a spring and a pendulum

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.


Experiment 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.

image

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.

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 T2 versus m.  The slope of the best fitting straight line equals 4π2/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.

Begin

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) T2 (s2)
1      
2      
4      
4      

Paste  your table and plot of T2 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?

 


Experiment 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.

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.

image

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

Begin

Assume θ(t) = θmaxcos(ωt + φ).   Find the most negative and the most positive angular displacements.  Average the absolute value of these numbers to find the amplitude θmax.  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) = θmaxcos(ωt + φ).  What value did you obtain for the amplitude θmax?


Convert your log into a lab report.

Name:
E-mail address:

Laboratory 9 Report

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