# 15.4 Pendulums  (Page 4/7)

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## Measuring the torsion constant of a string

A rod has a length of $l=0.30\phantom{\rule{0.2em}{0ex}}\text{m}$ and a mass of 4.00 kg. A string is attached to the CM of the rod and the system is hung from the ceiling ( [link] ). The rod is displaced 10 degrees from the equilibrium position and released from rest. The rod oscillates with a period of 0.5 s. What is the torsion constant $\kappa$ ?

## Strategy

We are asked to find the torsion constant of the string. We first need to find the moment of inertia.

## Solution

1. Find the moment of inertia for the CM:
${I}_{\text{CM}}=\int {x}^{2}dm={\int }_{\text{−}L\text{/}2}^{+L\text{/}2}{x}^{2}\lambda dx=\lambda {\left[\frac{{x}^{3}}{3}\right]}_{\text{−}L\text{/}2}^{+L\text{/}2}=\lambda \frac{2{L}^{3}}{24}=\left(\frac{M}{L}\right)\frac{2{L}^{3}}{24}=\frac{1}{12}M{L}^{2}.$
2. Calculate the torsion constant using the equation for the period:
$\begin{array}{ccc}\hfill T& =\hfill & 2\pi \sqrt{\frac{I}{\kappa }};\hfill \\ \hfill \kappa & =\hfill & I{\left(\frac{2\pi }{T}\right)}^{2}=\left(\frac{1}{12}M{L}^{2}\right){\left(\frac{2\pi }{T}\right)}^{2};\hfill \\ & =\hfill & \left(\frac{1}{12}\left(4.00\phantom{\rule{0.2em}{0ex}}\text{kg}\right){\left(0.30\phantom{\rule{0.2em}{0ex}}\text{m}\right)}^{2}\right){\left(\frac{2\pi }{0.50\phantom{\rule{0.2em}{0ex}}\text{s}}\right)}^{2}=4.73\phantom{\rule{0.2em}{0ex}}\text{N}·\text{m}\text{.}\hfill \end{array}$

## Significance

Like the force constant of the system of a block and a spring, the larger the torsion constant, the shorter the period.

## Summary

• A mass m suspended by a wire of length L and negligible mass is a simple pendulum and undergoes SHM for amplitudes less than about $15\text{°}$ . 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.
• The period of a physical pendulum $T=2\pi \sqrt{\frac{I}{mgL}}$ can be found if the moment of inertia is known. The length between the point of rotation and the center of mass is L .
• The period of a torsional pendulum $T=2\pi \sqrt{\frac{I}{\kappa }}$ can be found if the moment of inertia and torsion constant are known.

## 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.

A pendulum clock works by measuring the period of a pendulum. In the springtime the clock runs with perfect time, but in the summer and winter the length of the pendulum changes. When most materials are heated, they expand. Does the clock run too fast or too slow in the summer? What about the winter?

The period of the pendulum is $T=2\pi \sqrt{L\text{/}g}.$ In summer, the length increases, and the period increases. If the period should be one second, but period is longer than one second in the summer, it will oscillate fewer than 60 times a minute and clock will run slow. In the winter it will run fast.

With the use of a phase shift, the position of an object may be modeled as a cosine or sine function. If given the option, which function would you choose? Assuming that the phase shift is zero, what are the initial conditions of function; that is, the initial position, velocity, and acceleration, when using a sine function? How about when a cosine function is used?

## Problems

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

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?

24.8 cm

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

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?

4.01 s

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

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

1.58 s

(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is $9.79\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$ is moved to a location where the acceleration due to gravity is $9.82\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$ . 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 pendulum with a period of 2.00000 s in one location ( $g=9.80{\text{m/s}}^{2}$ ) 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?

$9.82002\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$

(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%?

what's the working difference between a dynamo and a pump?
a dynamo is basically a dc generator while pump is usually equipped with a motor
vedanth
a dynamo converts mechanical energy to electrical while a pump is opposite to that
vedanth
nice
Piyali
why sea water looks bluish?
cuz the sky is blue...
Mehdi
you see the reflexion of the "blue" sky in the water
Mehdi
somewhere sea water turns green why?
Piyali
never seen bro... are u sure ?
Mehdi
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
Piyali
waaaww... you re awesome
Mehdi
i need solutions of unuversity pbysucs volume 1
what is three dimensional coordinate system?
considering the change of vectors in all three dimensions of space
vedanth
direction co-sign of vector questions
opposite
Sonu
opposite mean
Vipin
what is the different between action and reaction?
action is external force. Reaction exists because of the external action. Reaction is mostly an internal force
Yoblaze
The difference is said in the word itself. There is no existance of reaction without an application of action. So action occur first then reaction. Reaction may produce in body itself or to other body i.e., it may be internal or external.
Amalesh
what is total energy
kenetic energy + potential energy + rotational energy
Tony
any body or object can have it ..even in the molecular level
Tony
As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk of diameter 8.50 ± 0.02 cm and thickness 0.050 ± 0.005 cm. Find the average volume of a cookie and the uncertainty in the volume.
how can proportionality sign can be replaced by = and propotionality constant aznd why?🙄🙄
because proportionality means linear relationship between the variables involved
Ranjan
When we say to quantities are proportional to each other, what we mean is that the variables maintain a constant ratio to each other.
Ranjan
let us consider an equation y= mx here y and x are variables and m is constant so whenever we divide y by x it maintains constant ratio called m , if m = 1 y=x if m doesn't equal 1 it may be other but for a closed function it is always same I.e the ratio remains constant
krishna
this applying for closed function proportionality sign can be replaced by = followed by a constant multiplier ratio
krishna
considering the propagation of a wave along the x direction on a vibrating string,how can one derive the one dimensional wave equation for the vibrating sting and the speed expression
show whether or not the expression v^2= u^2 sin^2 d- 2gs is dimensionally constant
the period T of a pendulum depends on its mass m, length l and acceleration due to gravity g. using dimensional analysis, derive for T.
Oyetayo
what is physics
Physics is the tool humans use to understand the properties characteristics and interactions of where they live - the universe. Thus making laws and theories about the universe in a mathematical way derived from empirical results yielded in tons of experiments.
Jomari
This tool, the physics, also enhances their way of thinking. Evolving integrating and enhancing their critical logical rational and philosophical thinking since the greeks fired the first neurons of physics.
Jomari
nice
Satyabrata
Physics is also under the category of Physical Science which deals with the behavior and properties of physical quantities around us.
Angelo
Physical Science is under the category of Physics*... I prefer the most is Theoretical Physics where it deals with the philosophical view of our world.
Jomari
what is unit
Metric unit
Arzoodan
A unit is what comes after a number that gives a precise detail on what the number means. For example, 10 kilograms, 10 is the number while "kilogram" is the unit.
Angelo
there are also different types of units, but metric is the most widely used. It is called the SI system. Please research this on google.
Angelo
Unit? Bahay yon
Jomari
How did you get the value as Dcd=0.2Dab
Why as Dcd=0.2Dab? where are you got this formula?...
Arzoodan
since the distance Dcd=1.2 and the distance Dab=6.0 the ratio 1.2/6.0 gives the equation Dcd=0.2Dab
sunday
Well done.
Arzoodan
how do we add or deduct zero errors from result gotten using vernier calliper?
how can i understand if the function are odd or even or neither odd or even
hamzaani
I don't get... do you mean positive or negative@hamzaani
Aina
Verner calliper is an old calculator
Antonio
Function is even if f(-x) =f(x)
Antonio
Function is odd if f(-x) = - f(x)
Antonio