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Department of Physics and Astronomy
2 nd^ Year Laboratory
G1 Kater’s Pendulum
Scientific aims and objectives
- To determine an accurate value for g in the lab here in Sheffield and compare it with an empirically calculated value.
- To compare the accuracy of g measured using three different pendulums o Simple pendulum o Compound pendulum o Kater’s pendulum
- To evaluate the validity of the small angle approximation made when deriving the equation of motion of all three pendulums.
Learning Outcomes
- To be able to read a vernier scale
- To fit complex equations by manipulating data to the form y = mx + c
- To be able to use χ^2 to fit an errors weighted straight line
- To apply compound errors formulae to complex equations
- To know the historical motivation behind the systematic improvements in
experimental methods used to measure g.
Apparatus
- Simple pendulum & o Stopwatch o Ruler
- Compound pendulum & o Stopwatch o Ruler
- Kater’s pendulum & o Electronic timing gate o Travelling microscope
Safety instructions
- Always replace the guard on the knife edge suspension point
- Take care when moving Kater’s pendulum around.
- Do not climb on the laboratory stools. Always demount the pendulum before adjusting the weight.
Task 1 - Pre-session questions You are required to complete the online questions found at the back of the script before starting your laboratory work. These questions will prepare you for the experiment by teaching you how to read a Vernier scale; manipulate complex equations to yield a simple straight line fit; and to use compound errors formula. The online questions will give you a context for the experiment by asking you to briefly research the history of the measurement of g using the internet.
Task 2 – The experiment
Three pendulums are provided.
The objective is to compare the value and accuracy of g using each pendulum in turn. Kater’s pendulum should yield a value precise to three significant figures. The other two pendulums will yield far less precise values and it is important to determine the errors for each pendulum to allow a discussion of the precision of the values of g.
In all cases the small angle approximation has been used to derive the equation of motion. The final part of the experiment is to measure the validity of this approximation by varying the amplitude of oscillation of Katers pendulum and recording the period. These results can be
checked against T (α )= T 0 ( 1 + 41 sin^2 α 2 ))(where α is the angle of oscillation), as derived from
the rigorous equation of motion.
Task 3 – Reporting
Using the Simple and Compound pendulums you should be able to report g to two significant figures and also calculate a value for the radius of gyration of the compound pendulum. Using Kater’s pendulum you should be able to report g to at least three significant figures and compare it directly with an empirically calculated value for g in Sheffield. You should discuss the methods that you used to analyse the data recorded from each pendulum, and discuss how you derived the values for errors in each case. You will be likely to need compound errors formula and you should show how you apply them.
3. The compound pendulum
A compound pendulum is a rigid body swinging in a vertical plane about any horizontal axis passing through the body. The resultant force acts through the centre of mass. The periodic time of the compound pendulum is related to the moment of inertia I about the point of suspension.
mgl
I
T = 2 π (A2)
I can be expressed in terms of the moment of inertia about the centre of mass ,
(^22) I = mk 0 + ml (A3)
giving,
( ) 2 2 1 0
2 (^2)
gl
l k
T π (A4)
where l is the distance between the suspension point and the centre of gravity of the pendulum and k 0 is the radius of gyration about a parallel axis through the centre of gravity. If different values of l are taken, and the corresponding values of T measured, both g and k 0 can be determined.
The radius of gyration k 0 is defined as the radius of a thin uniform hoop rotating about an axis through its centre and perpendicular to its plane, which has the same moment of inertia as the original object.
One way to determine k 0 and g is to make a series of measurements of T with the pivot point at different distances l from the centre of mass. Equation (A4) can then be manipulated into a straight line form and the intercept and gradient together will yield the unknown parameters.
You should determine the experimental error in T and l; ideally by repeat measurements, or alternatively by estimation, and using the appropriate compound errors formulae, determine the error in your X and Y variables for your straight line plot. Using the excel program “χ^2 “ you can then determine the error of g and k 0.
In addition you should compare your experimentally determined value for k 0 with a theoretical value found from the definition of the radius of gyration and the
integrated value for the moment of inertia of a metal bar 12
ml^2 I =.
4. Kater’s pendulum
An improvement in the precision of the measurement of g was developed in 1817 by Kater. He realised that by using a compound pendulum and suspending it from each end in turn the requirement to measure the distance from the centre of mass to the pivot could be removed. He made a very accurate measurement of g in London, a value that was used to define the metre for many years. The version of Kater's reversible pendulum used in this experiment has a knife-edge for suspension at
either end: thus there are two distances, l 1 and l 2 , and two periods T 1 and T 2. Using
equation (A2) it is possible to derive the following expression for g :
1 2
2 2
2 1 1 2
2 2
2 1
2
l l
T T
l l
T T
g −
(A5)
The distance between the knife-edges, the quantity L = l 1 + l 2 can be measured
directly, and accurately using the travelling microscope. On the other hand, it is much more difficult to measure the quantity l 1 - l 2 directly, however if the periods T 1 and T 2
are made to be the same then the term T 1^2 − T 22 goes to zero and the value of l 1 - l 2 is
not important.
The objective when using Kater’s pendulum is to equalise the period’s measured from the two pivots by adjusting the position of the weight on its thread. You can do this roughly by measuring the difference in the periods when the weight is positioned at its extreme limits and then estimating the required position using distance or the number of turns on the thread as a measure. However this methods will only yield an
approximate position of the weight for equality between T 1 and T 2. For a precise
measurement of g you should aim to take a series of measurements where T 1 and T 2
vary systematically. Equation (A5) can then be manipulated and approximated to a
straight line where X = T 12 − T 22 and Y = T 1 2 + T 22 assuming that l 1 (^) − l 2 is constant. The
intercept at X = 0 will yield g. Consider carefully the best conditions for this method to work and whether the approximation that l 1 (^) − l 2 is constant is valid. Alternative,
equally accurate methods are also possible.
You should determine the experimental error in T and L; ideally by repeat measurements, or alternatively by estimation, and using the appropriate compound errors formulae, determine the error in your X and Y variables of your straight line plot. Using the excel program “χ^2 “ you can then determine the error of g.
5. Compound errors
f (^) Absolute error Proportional error x^2 ∆ f^^ =^2 x ∆ x x
x f
f ∆
x n ∆ f = nxn^ −^1 ∆ x x
n x f
f ∆
x + y ( ∆ f )^2 =(∆ x )^2 +(∆ y )^2 2
2 2 2
2
( )
x y
x y f
f
xy (^) ( ∆ f )^2 = y^2 (∆ x )^2 + x^2 (∆ y )^222 2
( )^2
y
y x
x f
f
x / y 2 4
2 2
2 ( )^2 ( ) ( y ) y
x y
x f + ∆
2 2 2
( )^2
y
y x
x f
f
Pre-lab multiple choice questions
Q1 Read the following vernier scales
1.2 1.
Q2 Manipulate the following equations to yield Y = mX + C formula that could be fitted as straight- line graphs with Y = f 1 ( x , y )and X = f 2 ( x , y )to determine a and b.
A B C 2.1 (^) Y = y^2
X = x^2
Y = y^2 X = x −^2
Y = y^2 X = ax −^2
b
x y a
x y a
2 2 2 2 (^2) = + + − Y = x^2 + y^2 Y = x^2 − y^222
2 2
x y
x y Y −
2 2
2 X = x 1 + x
Y = x^2 + a^2 Y = x^2 − a^2
sin 2
x y a b
sin 2
x X
Y = y
x X
Y = y
x X
Y = y
Q3 Derive expressions for the proportional error ∆ f / f in the following equations.
A B C
y
x f x y
2 ( , )=
3.3 (^) f ( x , y )= x + 1 / x
Q4 Which of these statements are correct in the context of the historical measurement of g.
4.1 A tool for measuring the acceleration due to gravity is called a gravimeter 4.2 An accurate determination of g can be used to navigate nuclear submarines 4.3 In 1774 an accurate measurement of G was made using a pendulum and a isolated mountain in Scotland 4.4 Reversible pendulums like Kater’s were the standard way of measuring the acceleration due to gravity until 1950 4.5 In Kater’s original measurement in London he achieved a value of g = (9.81158 +/- 0.00001) m/s 2
b x a
y
(^2) +
2
f ( x , y )= x^2 y^22 (^4)
y
y x
x
2 2 (^2)
y
y x
x
y
y x
x 4
2 2 (^4)
y
y x
x 2 2 (^2)
y
y x
x
y
y x
x 4
2 2 2
4 2 2
( 1 )
x x
x x x 2
2 ( )^2
x
∆ x 2
( )^2 ( )^2
x
∆ x + ∆ x
