Conical Pendulum

Introduction

The goal of this laboratory is to study uniform circular motion

In this experiment, you will learn about uniform circular motion. Most people encounter difficulties with

circular motion problems. This laboratory should help you sort out some of the ideas

The physical case we will study is a type of pendulum. But unlike the simple pendulum you may be

familiar with, this pendulum moves in a horizontal circle

M

L

θ

m

Mg

FT

mg

FT

The tension in the string does two things. First, it supports the stationary mass M (see the diagram

above)

FT = M g .

Second, it supports mass m and keeps it running in a loop. To understand its influence on the moving

mass m, resolve the tension into components. For the vertical component we have FT cos θ = mg. For

the horizontal component we have FT sin θ = mv 2 / r, where r is the radius of the circle of motion, and

v is the linear speed of the object. Also, we know that

r = L sin θ ,

v = 2 π r

T ,

where T is the period.

1

· From these equations, find the relationship between the period and the length, and the masses m and M . That is, combine these equations to solve for T in terms of m, M , L and g

The answer you will get will look like this,

T = . . . m

M

L

g , (1)

except that this is an incomplete answer. You need to complete Equation (1) according to the answer that

you found. The expression will look similar to the equation for the period of a simple pendulum

When any object is in uniform circular motion, there must be a force directed toward the center of the

circle. We call this the centripetal force, to emphasize that it points towards the center. It is not a new

kind of force: the new name simply indicates its direction, but it is always one of the old familiar forces,

like gravity or the tension in a spring or rope. The centripetal force is always the result of action of other

forces, and is not a force by itself, strictly speaking. The centripetal forces causes a centripetal acceleration,

which is related to the angular velocity and the radius,

a = F

m =

v2

r = ω2 r (uniform circular motion).

We needed to use this expression for the acceleration to get the formula for the period given above. If

this expression were different, we would get different results for the period. So the formula for the period

gives us a way to test the ideas that went into the analysis. If the formula for the period checks out with

observations, we can take that as confirmation of our ideas about circular motion

Measurements

The idea of the laboratory is to measure the period T of the pendulum and compare it to the theoretical

prediction by measuring L, m, M and using Equation (1)

As always, we like to observe the behavior for as wide a range of parameters as possible (what principle is

this?). You should do a little investigation to determine the experimentally feasible ranges of parameters.

Can you make an argument that would reduce the parameter space from { L, m, M

} to { L, m/M

} ?

Also, as always, we want to work to make the measurement errors as small as possible. You already know

one trick — count as many periods as possible. Repeat this for 5 or 10 trials before changing parameter

values. A little dexterity is required to keep L constant. But with practice, you can learn to twirl it a

little faster or slower to keep L nearly constant

Planning ahead : we will be plotting T versus L for fixed m/M and T versus m/M for a fixed L

· first do one sequence of experiments with m/M constant, but varying L · then do a second sequence with L constant, but varying m/M (this may be a bit more challenging)

2

Your data tables for each combination should look similar to the following,

Configuration: m = M = L =

trial no. of periods time (s) period (s)

1

2

3

4

5

Average:

Your tables of analysis for each sequence should be some variation of the following,

Configuration: m = M =

L(m) Tobs(s) Ttheory(s) % error

(observed)

% error

(expected)

1

2

3

4

5

3

Graphic analysis

You will need to build three graphs for your analysis. Each graph should have Tobs on the vertical axis,

· plot T (L) for fixed m/M · plot T (m/M) for fixed L · plot Tobs as a function of Ttheory

If the theory was exact and the measurements were perfect, what would the last graph look like? What

conclusions can you draw from your graph?

Error analysis

For the last of the graphs we will put the error bars on. In earlier laboratories we learned how to estimate the

uncertainty in the average of a sequence of measurements. Apply those ideas to determine the uncertainty

in the mean period. Roughly speaking, if your timing is good to 0.2 s and you observed 20 periods, your

uncertainty in the mean should be 0.2/20 = 0.01 s. If you set these error bars on your observations, do

you get agreement between theory and observation? We can answer these questions by doing the following:

· each point should have a vertical error bar that is based on the number of trials, the number of cycles in a trial, and the timing error (not as a percentage)

· the length of the horizontal error bar is determined in the same manner as we have done the expected error in the past, with the exception that we need it in seconds rather than as a percentage

For the horizontal error bar, you need to derive an expression similar to the one we used in Laboratory 3.

The template for the answer is as follows,

∆Ttheory Ttheory

= . . .

∆L L

+ ∆g

g +

∆m

m +

∆M

M

 , (2) where the dots “. . .” represent a coefficient that you need to figure out by examining the appearance of

Equation (1). Our discussion of the relative uncertainty in the end of Laboratory 3 will help you. Now,

Equation (2) gives you the relative uncertainty, from which you can find the absolute uncertainty (the

horizontal error bar) by multiplying it by the respective value of Ttheory for each point on the graph. This

way you will get a different value of the horizontal error bar at each point

Determining the vertical error bar

Timing error:

Number of cycles per trial:

Number of trials:

Uncertainty in Tobs:

Determining the horizontal error bar

Uncertainty in L:

Uncertainty in m and M :

Uncertainty in g:

Relative uncertainty in Ttheory:

4

Laboratory evaluation

Please provide feedback on the following areas, comparing this laboratory to your previous labs. Please

assign each of the listed categories a value in 1 – 5, with 5 being the best, 1 the worst.

· how much fun you had completing this laboratory? 1 2 3 4 5 · how well the lab preparation period explained this laboratory? 1 2 3 4 5 · the amount of work required compared to the time allotted? 1 2 3 4 5 · your understanding of this laboratory? 1 2 3 4 5 · the difficulty of this laboratory? 1 2 3 4 5 · how well this laboratory tied in with the lecture? 1 2 3 4 5

Comments supporting or elaborating on your assessment can also be very helpful in improving the future

laboratories