# Modern Mechanics

Semester: Spring 2020

Course Code: PHYS218

Course Title: Modern Mechanics

Experiment #: TAP 3

Experiment Title: VARIABLE g PENDULUM

Date: ……………………….. Lab#…………………………..

Section: ……………………….

 Student Name Student ID

1. Introduction

This experiment explores the dependence of the period of a simple pendulum on the acceleration due to gravity. A simple rigid pendulum consists of a 35-cm long lightweight (28 g) aluminum tube with a 150-g mass at the end, mounted on a Rotary Motion Sensor. The pendulum is constrained to oscillate in a plane tilted at an angle from the vertical. This effectively reduces the acceleration due to gravity because the restoring force is decreased.

2. Objectives

· Measure the effective length of variable-g pendulum.

· Measure the period of a variable-g pendulum for different values of the tilt angle and verify the dependence of the function T versus .

· Measure moment of inertia

3. Experimental setup:

· Large rod base

· 45 cm stainless steel rod

· Angle indicator

· Rotary motion sensor

· Pendulum accessories

4. Theory

The period of a simple pendulum is given by:

(1)

Where is the acceleration due to gravity and the approximation becomes exact as the amplitude of the oscillation goes to zero. We will limit to angles less than 10° (0.17 rad) where assuming the equality in equation 1 holds produces an error of a fraction of a percent. Here it is understood that is a constant acceleration that acts in the plane of oscillation.

The pendulum we use is actually a physical pendulum (not a point mass) so equation 1 is replaced by the rotational analog:

(2)

where I is the moment of inertia of the system about the fixed axis, m is the mass of the brass masses (150 g) plus the rod (26 g), and r is the distance from the axis to the center of mass of the rod plus masses (~31 cm). Note that I, m, & are all constant and that I/mr must have the units of length so we may write:

(3)

where is the effective length of a simple pendulum that would behave the same as our physical pendulum. We may then re-write equation 2 in the form of equation 1:

(4)

We will determine by measuring the period when . Then we have:

(5)

In this experiment, the acceleration will be varied by tipping the plane of oscillation of the pendulum by an angle of θ from the vertical (figure 1). The component of g that is in the plane of oscillation is where:

(6) Figure 1: Components of g

Note that the component of g perpendicular to the plane of oscillation, , is cancelled by forces in the rod since no motion is allowed in this direction. Putting it all together gives:

(7)

Finally, combining equation (4) and (6) we have:

(8)

5. Pre-lab Preparation

Read section 11.2 (page 422). Also read the slides posted on Moodle corresponding to chapter 11.

6. Experimental Procedure

a) Adjust the an initial angle of 0° (figure 2)

b) In PASCO Capstone, click <Hardware Setup> in the Tools palette to open the Hardware Setup panel. Confirm that the Hardware Setup panel shows the Air Link interface you are using and the icon of the Rotary Motion Sensor (figure 3) Figure 2. Setup

Figure 3. Hardware Setup panel

c) Set up a data display. For example, drag the Graph icon from the Displays palette onto the workbook page, or double-click the icon to create a Graph display (figure 4)

d) Set up the Graph display to show Angle (rad) on the vertical axis. Click the “Select Measurement” menu button on the vertical axis and pick Angle (rad) from the menu. Time (s) automatically shows on the horizontal axis (figure 5)

e) Displace the pendulum from equilibrium (no more than 10 degrees [0.17 rad] amplitude) and let go   Figure 6

Figure 4

Figure 5

f) Click ‘Record’ in the lower left corner of the PASCO Capstone window to begin recording data. (The <Record> button changes to <Stop>.) (figure 6)

g) Let the timer run for 20 seconds and click STOP.

h) Read the period on the digits display

i) Change the value of the angle (given at table 1) and repeat steps (e), (f), (g) and (h).

j) Record your results and complete the table 1

7. Experimental Work (40 %)

Run the Interactive simulation of the link below (Click on the first button “Intro”) https://phet.colorado.edu/sims/html/pendulum-lab/latest/pendulum-lab_en.html Use the sliders in the right of the simulation to fix the pendulum parameters as follows:

 Length (m) Mass (kg) Gravity Friction 0.75 1.2 Earth None

· Left-click on the pendulum mass and drug it to an angle of 15° and release it.

· Untick the “Ruler” option and Tick the “Stopwatch” in order to measure the period of oscillations.

 Angles 15° 12° 10° 8° 5° Period T (s)

1. Fill the table below (5%)

2. Knowing that absolute error of time measurement using the stopwatch in this simulation is 0.05s, how does decreasing the angle change the Period T? (5%)

Use the sliders in the right of the simulation to fix the pendulum parameters as follows:

 Length (m) Mass (kg) Gravity Friction 0.75 Earth None

· Left-click on the pendulum mass and drug it to an angle of 15° and release it.

· Untick the “Ruler” option and Tick the “Stopwatch” in order to measure the period of oscillations.

3. Fill the table below (5%)

 Mass (Kg) 0.50 0.75 1.00 1.25 1.50 Period T (s)

4. Knowing that absolute error of time measurement using the stopwatch in this simulation is 0.05s, how does increasing the mass change the Period T? (5%)

Use the sliders in the right of the simulation to fix the pendulum parameters as follows:

 Length (m) Mass (kg) Gravity Friction 1.00 Earth None

· Left-click on the pendulum mass and drug it to an angle of 15° and release it.

· Untick the “Ruler” option and Tick the “Stopwatch” in order to measure the period of oscillations.

5. Fill the table below (10%)

 Length L (m) 0.30 0.40 0.50 0.75 1.00 Period T (s) T2 (s2)

6. Plot the Graph of the Function T2 versus L (10%)

Bonus Question : (10 %)

Knowing that theoretical formula of the simple pendulum period of oscillations in the small angle approximation (Angle less than 15°) is given by the formula:

Use the slope of the function in order to calculate the experimental value of the Earth Gravity Constant g.

8. Analysis/Report (60%)

1. Record your results and plot the graph Texp(s) versus θ(°) (15%(table)+5%(graph))

2. Explain how increasing of the angle effects the period (5%) ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

3. Plot the graphs Texp(s) versus (15%)

3. Determine graphically the slope of the function Texp(s) versus . (5%) ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

4. Using the result of the previous question and equation to calculate the effective length of the pendulum (10%) ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

5. Use the result of previous question and the formula (3) and given data and to calculate the moment of inertia I (5%) ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

T2 (s2) 0.3 0.4 0.5 0.75 1.0

0.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 75.0 80.0

θ(°)

Texp (s)

Texp (s)

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2 of 10