To determine the acceleration due to gravity by means of a conical pendulum

ExpConPenA student conducted an experiment to measure the acceleration of gravity. He used a conical pendulum. The conical pendulum is a string with a bob (weight) that revolves around an axis through its point of suspension.

The following table summarizes results of the experiment,

h(m)  0.17  0.15  0.09  0.05  0.04
 T(s)  0.85  0.78  0.63  0.46  0.4

where T is a periodic time of revolution and h is a vertical height.

a) Draw a graph of vertical height versus periodic time squared.
b) Find the acceleration of gravity by using the graph.

Solution.

We use a table of values in order to draw a graph.

h(m)  0.17  0.15  0.09  0.05  0.04
 T(s)  0.85  0.78  0.63  0.46   0.4
 T2(s2)  0.7225  0.6804  0.3969  0.2116  0.16

We draw a graph of vertical height versus periodic time squared.

GraphConicalP

According to Newton’s Second Law for uniform circular motion, the net force acting on the bob equals mar.

Fnet = mar

The expression Fnet = mar is a vector equation, so we can write it as two component equations: Fnet,x = mar and Fnet,y = 0, because the bob does not accelerate along y-axis.

In the x-direction, there is only FT,x. Thus,

FT,x = mar

or

FT∙sin(α) = mar    (1).

In the y-direction, Fnet,y = 0 becomes

FT,y – mg = 0

or

FT∙cos(α) = mg    (2).

We know that ar can be presented as 4π2r/T2.

ar = 4π2r/T2     (3)

We substitute the equation (3) into the equation (1) and we get

FT∙sin(α) = m∙4π2r/T2    (4)

Now, we divide the equation (4) by equation (2) and obtain

tan(α) = 4π2r/(g∙T2).

Using the fact that tan(α) = r/h, we can rewrite the last equation

r/h = 4π2r/(g∙T2)

or

h = (g/4π2)∙T2    (5)

The equation (5) is the equation of the graph that we drew, where

g/4π2 is the slope of the graph.

We calculate the slope of the graph by dividing the change in h by the change in T2. We use two widely separated points on the trend line.

(0.16-0.05)/(0.67-0.2) ≈ 0.234

Hence,

g/4π2 = 0.234

and finally,

g ≈ 9.24 m/s2.

 
Let’s calculate the percent error

The formula for calculating percent error is:

percent error = (experimental value – accepted value)∙100/accepted value

percent error = (9.24 – 9.81)∙100/9.81 ≈ 5.81%

 
 

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