Swing Mechanics Slides Swing Mechanics
Isolating Pendulum Variables
LAB SESSION 01
The Swing Paradox
"If you sit on a swing with a friend, do you swing slower than when you are alone?"
A: Slower
B: Faster
C: The Same
Defining the System
Period (T)
The time it takes for one complete oscillation (back and forth).
Bob
The mass at the end of the pendulum string.
Length (L)
Measured from the pivot to the center of the bob.
Amplitude (θ)
The maximum angle from the equilibrium position.
Variable Isolation
To find the truth, we must change only one thing at a time.
Test 1: Mass
Change Bob Mass
Keep Length same
Keep Angle same
Test 2: Length
Change String Length
Keep Mass same
Keep Angle same
Test 3: Angle
Change Release Angle
Keep Length same
Keep Mass same
Make Your Hypotheses
1. Increasing mass will ______ the period.
2. Increasing length will ______ the period.
3. Increasing angle will ______ the period.
Start the Experiment
Variable Isolation Worksheet Lab Report: Variable Isolation
Physics 11 / Unit: SHM
Student Name:
Date:
01. Objective & Hypothesis
Identify which variables (Mass, Length, or Release Angle) affect the period ($T$) of a simple pendulum. For each variable, predict if increasing it will increase, decrease, or have no effect on the period.
Mass ($m$)
Length ($L$)
Angle ($\theta$)
02. Data Collection
Note: For each test, keep the other two variables constant.
Test 1: Effect of Mass (Constant L = 50cm, θ = 10°)
Mass (g) Time for 10 Swings (s) Period $T$ (s) 20g 50g 100g
Test 2: Effect of Length (Constant m = 50g, θ = 10°)
Length (cm) Time for 10 Swings (s) Period $T$ (s) 20cm 50cm 80cm
03. Synthesis & Analysis
1. Based on your data, which variable(s) had a significant effect on the period?
2. Did the mass of the bob change the period? Why might this be surprising given that heavier objects are pulled down with more force by gravity?
3. Describe the relationship between Length and Period. Is it linear (e.g., doubling length doubles the period)? Look closely at your data.
Challenge Question
If you want to build a clock that ticks exactly once every second (Period = 2.0s), what general adjustments would you make to your pendulum setup based on today's findings?
Angle Limits Slides ENGINEERING LIMITS
Small Angle Approximation
When does the simple pendulum formula actually stop working?
The Grandfather Clock Puzzle
Why do grandfather clocks only swing a tiny bit?
If a pendulum's period is based on length, shouldn't it keep perfect time no matter how wide it swings?
Ideal vs. Reality
The "Magic" of Radians
The Assumption
For very small angles:
sin(θ) ≈ θ
*Where θ is measured in radians.
Simple Harmonic Motion requires a restoring force directly proportional to displacement.
In a pendulum, the force is proportional to sin(θ), not θ.
Crucial Insight
The formula only works when the angle is small enough that sine and the angle are nearly equal.
Comparing Periods
Angle (θ) sin(θ) θ (rad) % Difference 5° 0.08716 0.08727 0.13% 15° 0.2588 0.2618 1.14% 30° 0.5000 0.5236 4.51% 60° 0.8660 1.0472 17.30%
At what point is the error too large for precision timekeeping?
Breaking the Model
Precision
Test angles from 5° up to 90° using the exact same pendulum setup.
Calculation
Calculate the "Ideal Period" using the formula and compare to your measured results.
The Threshold
Determine exactly where "Simple Harmonic Motion" stops and "Complex Motion" begins.
Angle Threshold Lab Worksheet Lab: The 15-Degree Threshold
EXP 02
Student:
Date:
01. Mathematical Prediction
Using your calculator (set to Radians ), compare the actual value of \(\sin(\theta)\) to the value of \(\theta\). Calculate the percent difference.
Angle (°) Angle (\(\theta\)) in Radians \(\sin(\theta)\) % Difference 5° 15° 45°
Percent Difference Formula: \(\left| \frac{\theta - \sin(\theta)}{\theta} \right| \times 100\)
02. Breaking the Formula
Measure the period of a 50cm pendulum at increasing release angles. We will see if the "Simple" formula remains accurate.
Calculated Ideal Period (\(L=0.5m, g=9.81\)): \( T_{ideal} = \) _________ s
Release Angle (°) Time for 10 Swings (s) Measured Period \( T \) (s) Error vs. \( T_{ideal} \) (%) 10° 30° 60° 90°
03. Analysis & Conclusions
1. Defining "Small":
At which angle did you first notice a percent error greater than 2%? Explain how this limits our use of the simple pendulum formula.
2. Restoring Force:
Simple Harmonic Motion requires a restoring force proportional to displacement (\(F = -kx\)). In a pendulum, the force is \(F = -mg \sin(\theta)\). Why does the motion stop being "simple" at large angles?
Grandfather Clocks
If a grandfather clock's battery/weight is low and the pendulum's swing amplitude starts to decrease (e.g., from 15° down to 5°), will the clock start running fast, running slow, or stay perfectly on time? Justify your answer using today's data.
Gravity Calculation Lab Reference: g = 9.806 m/s²
The Gravity Constant Lab
Experiment 03: Precision Calculation
Operator:
Station No:
I. Mathematical Foundation
The period of a simple pendulum is given by: \[ T = 2\pi \sqrt{\frac{L}{g}} \]
Task: Rearrange this formula to isolate \( g \). Show your algebraic steps below:
Workspace
Derived Formula: \( g = \) _______________________
II. Precise Measurements
Measure 3 different lengths. For each, time 20 full oscillations for maximum precision.
Length \( L \) (m) Time for 20 Swings (s) Period \( T \) (s) Calculated \( g \) (m/s²) Average Experimental Gravity (\( g_{exp} \)):
III. Error Analysis
1. Calculate your Percent Error:
\[ \% \text{ Error} = \left| \frac{9.81 - g_{exp}}{9.81} \right| \times 100 \]
Result: ________ %
2. Sources of Error:
Human reaction time (start/stop)
Air resistance (damping)
Measuring length to center of bob
Amplitude > 15 degrees (SA Approximation)
Question: Precision vs. Accuracy
If your trials were very close to each other (e.g., 9.54, 9.55, 9.54) but far from 9.81, are you being precise, accurate, or both? Explain.
The Lunar Challenge
Gravity on the Moon is \( 1.62 \text{ m/s}^2 \). If you took your favorite 1.0m pendulum from this lab to the Moon, what would happen to its period? Calculate the new period below.
Gravity Probe Slides Gravity Probe
Measuring 'g' with Precision
LATITUDE: 48.8566° N | ALTITUDE: 35M
The Primitive Tool
"Can we measure the gravity of Earth using just a string and a rock?"
Standard textbooks say gravity is 9.81 m/s².
But standard isn't precise. Gravity varies by location. Today, we measure it here.
Local Field Analysis
Isolating Gravity
Start Here:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
Solve for g:
\[ g = \frac{4\pi^2 L}{T^2} \]
Engineering Precision
L must be measured to the center of the mass.T is the average of 10-20 swings to reduce reaction time error.Units must be in Meters and Seconds for \( m/s^2 \).
How close were you?
\[ \% \text{ Error} = \left| \frac{\text{Actual} - \text{Experimental}}{\text{Actual}} \right| \times 100 \]
Ideal
0 - 1%
Research Grade
Solid
1 - 5%
Standard Lab
Check Tools
> 5%
Systemic Error
The Local Competition
We are competing for the lowest percent error. Check your measurements twice. Time your swings over the longest possible period.
Measure | Calculate | Prove
Rigid Body Slides Beyond the Point
Physical Pendulum Dynamics
SESSION 04
RIGID BODY MOTION
The Baseball Bat Paradox
Why does holding a bat at the barrel end change its swing?
It's the same mass. It's the same length. But the distribution of that mass has shifted.
In a simple pendulum, all mass is at a point. In the real world, mass is spread out.
Moment of Inertia Analysis
Simple vs. Physical
The Ideal
Simple Pendulum
Massless string
Point mass at the end
\( T = 2\pi \sqrt{\frac{L}{g}} \)
The Reality
Physical Pendulum
Rigid body (spread mass)
Pivot point can be anywhere
\( T = 2\pi \sqrt{\frac{I}{mgh}} \)
The Physical Formula
\[ T = 2\pi \sqrt{\frac{I}{mgh}} \]
I
Moment of Inertia
Resistance to rotational acceleration.
h
Pivot Distance
Distance from pivot to Center of Mass.
mg
Weight
The force causing the torque.
Key Insight: For physical pendulums, mass DOES NOT cancel out!
Today's Investigation
01
The Pivot Shift
Measure the period of a meter stick pivoted at 10cm, 30cm, and 50cm.
02
The Shape Showdown
Compare a disk vs. a hoop of the same mass and radius.
Does a solid object swing faster than a hollow one?
Physical Pendulum Lab Lab: The Rigid Body Swing
Physics 11 / Physical Pendulums
Student Name:
Partner:
01. Simple vs. Physical
A simple pendulum assumes a point mass at distance \( L \). A physical pendulum (like a meter stick) has its mass spread out. We use the formula: \[ T = 2\pi \sqrt{\frac{I}{mgh}} \]
Prediction
If you pivot a meter stick at its very end (100cm mark), will it swing faster or slower than a simple pendulum of the same 1.0m length? Why?
Moment of Inertia (\( I \))
For a rod of length \( L \) pivoted at its end: \[ I = \frac{1}{3}mL^2 \]
Notice that mass (\( m \)) appears in both \( I \) and the denominator (\( mgh \)). What does this mean for our final period calculation?
02. Data: The Pivot Shift
Measure the period of a meter stick (Total Length \( L = 1.0m \)) as you shift the pivot point further from the center of mass (CM).
Pivot Location (cm mark) Dist to CM (\( h \)) (m) Time for 10 Swings (s) Period \( T \) (s) 10 cm 0.40 m 25 cm 0.25 m 40 cm 0.10 m
03. Synthesis
1. The Relationship:
What happened to the period as the pivot point got closer to the center of mass (the 50cm mark)? Predict what would happen if you pivoted the stick exactly at the 50cm mark.
2. Comparison to Simple Model:
Calculate the period of a 0.40m simple pendulum. How does it compare to your measured period for the meter stick pivoted at the 10cm mark (\( h = 0.40m \))? Why is the meter stick period different?
The Baseball Bat Challenge
Standard baseball bats have more mass at the barrel end than the handle end. If you want a bat that swings with the fastest possible period (highest frequency), should you hold it by the thin handle or by the thick barrel? Explain using your observations about pivot distance and mass distribution.
Earth Motion Slides Evidence of Rotation
Earth in Motion
The Legacy of the Foucault Pendulum
The Museum Mystery
"How can a pendulum inside a building prove the Earth is spinning?"
If you set a giant weight swinging in a straight line, it will eventually start knocking over pins in a circle.
But why? Does the pendulum turn, or do we?
Inertia of the Plane
The plane of oscillation remains fixed in space while the Earth turns beneath it.
1851: Paris, France
The Specs
Length: 67 metersMass: 28 kg (Brass bob)Pivot: Frictionless jointEffect: The plane rotated 11.3° per hour.
"Vous êtes invités à venir voir tourner la Terre"
"You are invited to come and see the Earth turn."
Leon Foucault realized that a pendulum's swing path is fixed relative to the distant stars. If the path seems to turn relative to the floor, it's actually the floor that is moving.
The Sine of the Latitude
The rate of rotation of the pendulum depends on where you are on the globe.
Rotation Rate Formula
ω = 15°/hr × sin(φ)
Where φ is your latitude.
The North Pole
360° in 24 hours
Max Effect
The Equator
0° rotation
No Effect
Your Current Latitude
We will calculate this in today's case study.
Visualizing the Spin
Watch: The Foucault Pendulum
Observe how the pendulum's path seemingly shifts over time, knocking down the markers along the perimeter.
[Simulation / Video placeholder]
Prepare for the Case Study
Earth Spin Case Study The Foucault Study
Observational Proof of Planetary Rotation
Sequence Finale
LES. 05
Investigator Name:
Research Date:
I. The Physics of the Plane
"Newton's First Law states that an object in motion stays in motion in a straight line unless acted upon by an external force."
Based on your class discussion and the presentation, explain why a pendulum's plane of oscillation stays "fixed" while the building around it appears to rotate.
II. Mathematical Prediction
The rate of rotation (\( \omega \)) in degrees per hour is calculated by: \[ \omega = 15^\circ/\text{hr} \cdot \sin(\phi) \]
Scenario A: The North Pole
Latitude (\( \phi \)) = 90°
\( \omega = 15 \cdot \sin(90^\circ) = \)
_________ °/hr
How long does it take for a full 360° rotation?
Scenario B: Paris, France
Latitude (\( \phi \)) = 48.8°
\( \omega = 15 \cdot \sin(48.8^\circ) = \)
_________ °/hr
Compare this to Foucault's original data (11.3°/hr).
III. Observational Analysis
As you watch the simulation or video of the Foucault Pendulum, answer the following questions:
1. Direction of Shift:
In the Northern Hemisphere, does the plane rotate clockwise or counter-clockwise from the perspective of an observer on the ground? Why does this swap in the Southern Hemisphere?
2. Variable Impact:
Why must the pendulum be very long (like the one in Paris) for this experiment to be successful in a museum? (Hint: Think about energy loss and time.)
Thinking Like a Scientist
If you were standing on the surface of the Moon, would a Foucault Pendulum show any rotation? Explain your reasoning based on the Moon's rotation period and its gravitational field.
