# 16.4 The simple pendulum  (Page 2/3)

 Page 2 / 3

## Measuring acceleration due to gravity: the period of a pendulum

What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s?

Strategy

We are asked to find $g$ given the period $T$ and the length $L$ of a pendulum. We can solve $T=2\pi \sqrt{\frac{L}{g}}$ for $g$ , assuming only that the angle of deflection is less than $\text{15º}$ .

Solution

1. Square $T=2\pi \sqrt{\frac{L}{g}}$ and solve for $g$ :
$g={4\pi }^{2}\frac{L}{{T}^{2}}.$
2. Substitute known values into the new equation:
$g={4\pi }^{2}\frac{0\text{.}\text{75000}\phantom{\rule{0.25em}{0ex}}\text{m}}{{\left(1\text{.}\text{7357 s}\right)}^{2}}.$
3. Calculate to find $g$ :
$g=9\text{.}\text{8281}\phantom{\rule{0.25em}{0ex}}\text{m}/{\text{s}}^{2}.$

Discussion

This method for determining $g$ can be very accurate. This is why length and period are given to five digits in this example. For the precision of the approximation $\text{sin θ}\approx \theta$ to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about $\text{0.5º}$ .

## Making career connections

Knowing $g$ can be important in geological exploration; for example, a map of $g$ over large geographical regions aids the study of plate tectonics and helps in the search for oil fields and large mineral deposits.

## Take home experiment: determining $g$

Use a simple pendulum to determine the acceleration due to gravity $g$ in your own locale. Cut a piece of a string or dental floss so that it is about 1 m long. Attach a small object of high density to the end of the string (for example, a metal nut or a car key). Starting at an angle of less than $\text{10º}$ , allow the pendulum to swing and measure the pendulum’s period for 10 oscillations using a stopwatch. Calculate $g$ . How accurate is this measurement? How might it be improved?

An engineer builds two simple pendula. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of $\text{10}\phantom{\rule{0.25em}{0ex}}\text{kg}$ . Pendulum 2 has a bob with a mass of $\text{100 kg}$ . Describe how the motion of the pendula will differ if the bobs are both displaced by $\text{12º}$ .

The movement of the pendula will not differ at all because the mass of the bob has no effect on the motion of a simple pendulum. The pendula are only affected by the period (which is related to the pendulum’s length) and by the acceleration due to gravity.

## Phet explorations: pendulum lab

Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It’s easy to measure the period using the photogate timer. You can vary friction and the strength of gravity. Use the pendulum to find the value of $g$ on planet X. Notice the anharmonic behavior at large amplitude.

## Section summary

• A mass $m$ suspended by a wire of length $L$ is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about $\text{15º}.$

The period of a simple pendulum is

$T=2\pi \sqrt{\frac{L}{g}},$

where $L$ is the length of the string and $g$ is the acceleration due to gravity.

## Conceptual questions

Pendulum clocks are made to run at the correct rate by adjusting the pendulum’s length. Suppose you move from one city to another where the acceleration due to gravity is slightly greater, taking your pendulum clock with you, will you have to lengthen or shorten the pendulum to keep the correct time, other factors remaining constant? Explain your answer.

## Problems&Exercises

As usual, the acceleration due to gravity in these problems is taken to be $g=9.80\phantom{\rule{0.25em}{0ex}}\text{m}/{\text{s}}^{2}$ , unless otherwise specified.

What is the length of a pendulum that has a period of 0.500 s?

6.21 cm

Some people think a pendulum with a period of 1.00 s can be driven with “mental energy” or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?

What is the period of a 1.00-m-long pendulum?

2.01 s

How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot?

The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency?

2.23 Hz

Two parakeets sit on a swing with their combined center of mass 10.0 cm below the pivot. At what frequency do they swing?

(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is $9\text{.}\text{79}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ is moved to a location where it the acceleration due to gravity is $9\text{.}\text{82}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ . What is its new period? (b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity.

(a) 2.99541 s

(b) Since the period is related to the square root of the acceleration of gravity, when the acceleration changes by 1% the period changes by $\left(0\text{.}\text{01}{\right)}^{2}=0\text{.}\text{01%}\text{}$ so it is necessary to have at least 4 digits after the decimal to see the changes.

A pendulum with a period of 2.00000 s in one location $\left(g=9\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)$ is moved to a new location where the period is now 1.99796 s. What is the acceleration due to gravity at its new location?

(a) What is the effect on the period of a pendulum if you double its length?

(b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?

(a) Period increases by a factor of 1.41 ( $\sqrt{2}$ )

(b) Period decreases to 97.5% of old period

Find the ratio of the new/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is $1\text{.}\text{63}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ .

At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is $1\text{.}\text{63}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}$ , if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock’s hour hand to make one revolution on the Moon.

Slow by a factor of 2.45

Suppose the length of a clock’s pendulum is changed by 1.000%, exactly at noon one day. What time will it read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.

If a pendulum-driven clock gains 5.00 s/day, what fractional change in pendulum length must be made for it to keep perfect time?

length must increase by 0.0116%.

#### Questions & Answers

Why is the sky blue...?
Star Reply
It's filtered light from the 2 forms of radiation emitted from the sun. It's mainly filtered UV rays. There's a theory titled Scatter Theory that covers this topic
Mike
A heating coil of resistance 30π is connected to a 240v supply for 5min to boil a quantity of water in a vessel of heat capacity 200jk. If the initial temperature of water is 20°c and it specific heat capacity is 4200jkgk calculate the mass of water in a vessel
fasawe Reply
A thin equi convex lens is placed on a horizontal plane mirror and a pin held 20 cm vertically above the lens concise in position with its own image the space between the undersurface of d lens and the mirror is filled with water (refractive index =1•33)and then to concise with d image d pin has to
Azummiri Reply
Be raised until its distance from d lens is 27cm find d radius of curvature
Azummiri
what happens when a nuclear bomb and atom bomb bomb explode add the same time near each other
FlAsH Reply
A monkey throws a coconut straight upwards from a coconut tree with a velocity of 10 ms-1. The coconut tree is 30 m high. Calculate the maximum height of the coconut from the top of the coconut tree? Can someone answer my question
Fatinizzah Reply
v2 =u2 - 2gh 02 =10x10 - 2x9.8xh h = 100 ÷ 19.6 answer = 30 - h.
Ramonyai
why is the north side is always referring to n side of magnetic
sam Reply
who is a nurse
Chilekwa Reply
A nurse is a person who takes care of the sick
Bukola
a nurse is also like an assistant to the doctor
Gadjawa
explain me wheatstone bridge
Malik Reply
good app
samuel
Wheatstone bridge is an instrument used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component.
MUHD
Rockwell Software is Rockwell Automation’s "Retro Encabulator". Now, basically the only new principle involved is that instead of power being generated by the relative motion of conductors and fluxes, it’s produced by the modial interaction of magneto-reluctance and capacitive diractance. The origin
Chip
what refractive index
Adjah Reply
write a comprehensive note on primary colours
Harrison Reply
relationship between refractive index, angle of minimum deviation and angle of prism
Harrison
Who knows the formula for binding energy,and what each variable or notation stands for?
Agina Reply
1. A black thermocouple measures the temperature in the chamber with black walls.if the air around the thermocouple is 200 C,the walls are at 1000 C,and the heat transfer constant is 15.compute the temperature gradient
Tikiso Reply
what is the relationship between G and g
Olaiya Reply
G is the u. constant, as g stands for grav, accelerate at a discreet point
Mark
Is that all about it?
Olaiya
pls explain in details
Olaiya
G is a universal constant
Mark
g stands for the gravitational acceleration point. hope this helps you.
Mark
balloon TD is at a gravitational acceleration at a specific point
Mark
I'm sorry this doesn't take dictation very well.
Mark
Can anyone explain the Hooke's law of elasticity?
Olaiya Reply
extension of a spring is proportional to the force applied so long as the force applied does not exceed the springs capacity according to my textbook
Amber
does this help?
Amber
Yes, thanks
Olaiya
so any solid can be compressed how compressed is dependent upon how much force is applied F=deltaL
Amber
sorry, the equation is F=KdeltaL delta is the triangle symbol and L is length so the change in length is proportional to amount of Force applied I believe that is what Hookes law means. anyone catch any mistakes here please correct me :)
Amber
I think it is used only for solids and not liquids, isn't it?
Olaiya
basically as long as you dont exceed the elastic limit the object should return to it original form but if you exceed this limit the object will not return to original shape as it will break
Amber
Thanks for the explanation
Olaiya
yh, liquids don't apply here, that should be viscosity
Chiamaka
hope it helps 😅
Amber
also, an object doesnt have to break necessarily, but it will have a new form :)
Amber
Yes
Olaiya
yeah, I think it is for solids but maybe there is a variation for liquids? that I am not sure of
Amber
ok
Olaiya
good luck!
Amber
Same
Olaiya
aplease i need a help on spcific latent heat of vibrations
Bilgate
specific latent heat of vaporisation
Bilgate
how many kilometers makes a mile
Margaret Reply
about 1.6 kilometres.
Faizyab
near about 1.67 kilometers
Aakash
equal to 1.609344 kilometers.
MUHD

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