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,

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 |

T^{2}(s^{2}) |
0.7225 | 0.6804 | 0.3969 | 0.2116 | 0.16 |

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 ma_{r}.

F_{net} = ma_{r}

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

In the x-direction, there is only F_{T,x}. Thus,

F_{T,x} = ma_{r}

or

F_{T}∙sin(α) = ma_{r} (1).

In the y-direction, F_{net,y} = 0 becomes

F_{T,y} – mg = 0

or

F_{T}∙cos(α) = mg (2).

We know that a_{r} can be presented as 4π^{2}r/T^{2}.

a_{r} = 4π^{2}r/T^{2} (3)

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

F_{T}∙sin(α) = m∙4π^{2}r/T^{2} (4)

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

tan(α) = 4π^{2}r/(g∙T^{2}).

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

r/h = 4π^{2}r/(g∙T^{2})

or

h = (g/4π^{2})∙T^{2} (5)

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

g/4π^{2} is the slope of the graph.

We calculate the slope of the graph by dividing the change in h by the change in T^{2}. We use two widely separated points on the trend line.

(0.16-0.05)/(0.67-0.2) ≈ 0.234

Hence,

g/4π^{2} = 0.234

and finally,

g ≈ 9.24 m/s^{2}.

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%