A 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,
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.
We use a table of values in order to draw a graph.
We draw a graph of vertical height versus periodic time squared.
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
FT∙sin(α) = mar (1).
In the y-direction, Fnet,y = 0 becomes
FT,y – mg = 0
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)
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
g/4π2 = 0.234
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%