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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
Feedback/Comments:
Grade: …….. /100
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
· Air link PASPORT interface
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, & r 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.htmlUse 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 graphTexp(s) versus θ(°)
(15%(table)+5%(graph))
2. Explain how increasing of the angle effects the period (5%)
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_______________
3. Plot the graphsTexp(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)
1 of 10
2 of 10
Use the below formulae to fill Table 1
( Not sure should we give it to them or not )
θ(°)
%Error
0
1,152
15
1,172
20
1,187
25
1,21
30
1,236
35
1,269
40
1,311
45
1,374
50
1,444
55
1,517
60
1,631
65
1,774
70
2,001
75
2,284
80
2,731

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Semester Spring 2020Course Code PHYS218Course Title.docx

  • 1. 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 Feedback/Comments: Grade: …….. /100 1. Introduction
  • 2. 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 · Air link PASPORT interface 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:
  • 3. (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, & r 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)
  • 4. 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
  • 5. 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).
  • 6. 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.htmlUse 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
  • 7. 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%)
  • 8. 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)
  • 9. 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 graphTexp(s) versus θ(°) (15%(table)+5%(graph))
  • 10. 2. Explain how increasing of the angle effects the period (5%) _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _______________ 3. Plot the graphsTexp(s) versus (15%) 3. Determine graphically the slope of the function Texp(s) versus . (5%) _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________
  • 11. _____________________________________________________ _____________________________________________________ _______________ 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
  • 12. 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) 1 of 10 2 of 10 Use the below formulae to fill Table 1 ( Not sure should we give it to them or not ) θ(°) %Error 0 1,152 15 1,172 20