# 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%?

#### Questions & Answers

what is electromagnetism
It is the study of the electromagnetic force, one of the four fundamental forces of nature. ... It includes the electric force, which pushes all charged particles, and the magnetic force, which only pushes moving charges.
Energy
what is units?
units as in how
praise
What is th formular for force
F = m x a
Santos
State newton's second law of motion
can u tell me I cant remember
Indigo
force is equal to mass times acceleration
Santos
The acceleration of a system is directly proportional to the and in the same direction as the external force acting on the system and inversely proportional to its mass that is f=ma
David
The uniform seesaw shown below is balanced on a fulcrum located 3.0 m from the left end. The smaller boy on the right has a mass of 40 kg and the bigger boy on the left has a mass 80 kg. What is the mass of the board?
Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?
how can one calculate the value of a given quantity
means?
Manorama
To determine the exact value of a percent of a given quantity we need to express the given percent as fraction and multiply it by the given number.
AMIT
meaning
Winford
briefly discuss rocket in physics
ok let's discuss
Jay
What is physics
physics is the study of natural phenomena with concern with matter and energy and relationships between them
Ibrahim
a potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inductance of the inductor that should be connected to the capacitor for the circuit to oscillate at 1125Hza potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inducta
L= 0.002H
NNAEMEKA
how did you get it?
Favour
is the magnetic field of earth changing
what is thought to be the energy density of multiverse and is the space between universes really space
tibebeab
can you explain it
Guhan
Energy can not either created nor destroyed .therefore who created? and how did it come to existence?
this greatly depend on the kind of energy. for gravitational energy, it is result of the shattering effect violent collision of two black holes on the space-time which caused space time to be disturbed. this is according to recent study on gravitons and gravitational ripple. and many other studies
tibebeab
and not every thing have to pop into existence. and it could have always been there . and some scientists think that energy might have been the only entity in the euclidean(imaginary time T=it) which is time undergone wick rotation.
tibebeab
What is projectile?
An object that is launched from a device
Grant
2 dimensional motion under constant acceleration due to gravity
Awais
Not always 2D Awais
Grant
Awais
why not? a bullet is a projectile, so is a rock I throw
Grant
bullet travel in x and y comment same as rock which is 2 dimensional
Awais
components
Awais
no all pf you are wrong. projectile is any object propelled through space by excretion of a force which cease after launch
tibebeab
for awais, there is no such thing as constant acceleration due to gravity, because gravity change from place to place and from different height
tibebeab
it is the object not the motion or its components
tibebeab
where are body center of mass on present.
on the mid point
Suzana
is the magnetic field of the earth changing?
tibebeab
does shock waves come to effect when in earth's inner atmosphere or can it have an effect on the thermosphere or ionosphere?
tibebeab
and for the question from bal want do you mean human body or just any object in space
tibebeab
A stone is dropped into a well of 19.6m deep and the impact of sound heared after 2.056 second ,find the velocity of sound in air.
9.53 m/s ?
Kyla
In this case, the velocity of sound is 350 m/s.
Zahangir
why?
Kyla
some calculations is need. then you will get exact result.
Zahangir
i mean how? isn't it just a d over t?
Kyla
calculate the time it takes the stone to hit the ground then minus the stone's time to the total time... then divide the total distance by the difference of the time
Snuggly
awit lenard. Hahahah ari ga to!
Kyla