# 16.5 Energy and the simple harmonic oscillator  (Page 2/2)

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A similar calculation for the simple pendulum produces a similar result, namely:

${\omega }_{\text{max}}=\sqrt{\frac{g}{L}}{\theta }_{\text{max}}.$

## Determine the maximum speed of an oscillating system: a bumpy road

Suppose that a car is 900 kg and has a suspension system that has a force constant $k=6\text{.}\text{53}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{N/m}$ . The car hits a bump and bounces with an amplitude of 0.100 m. What is its maximum vertical velocity if you assume no damping occurs?

Strategy

We can use the expression for ${v}_{\text{max}}$ given in ${v}_{\text{max}}=\sqrt{\frac{k}{m}}X$ to determine the maximum vertical velocity. The variables $m$ and $k$ are given in the problem statement, and the maximum displacement $X$ is 0.100 m.

Solution

1. Identify known.
2. Substitute known values into ${v}_{\text{max}}=\sqrt{\frac{k}{m}}X$ :
${v}_{\text{max}}=\sqrt{\frac{6\text{.}\text{53}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{N/m}}{\text{900}\phantom{\rule{0.25em}{0ex}}\text{kg}}}\left(0\text{.}\text{100}\phantom{\rule{0.25em}{0ex}}\text{m)}.$
3. Calculate to find ${v}_{\text{max}}\text{= 0.852 m/s}.$

Discussion

This answer seems reasonable for a bouncing car. There are other ways to use conservation of energy to find ${v}_{\text{max}}$ . We could use it directly, as was done in the example featured in Hooke’s Law: Stress and Strain Revisited .

The small vertical displacement $y$ of an oscillating simple pendulum, starting from its equilibrium position, is given as

$y\left(t\right)=a\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\mathrm{\omega t},$

where $a$ is the amplitude, $\omega$ is the angular velocity and $t$ is the time taken. Substituting $\omega =\frac{2\pi }{T}$ , we have

$yt=a\phantom{\rule{0.25em}{0ex}}\text{sin}\left(\frac{2\pi t}{T}\right).$

Thus, the displacement of pendulum is a function of time as shown above.

Also the velocity of the pendulum is given by

$v\left(t\right)=\frac{2\mathrm{a\pi }}{T}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\left(\frac{2\pi t}{T}\right),$

so the motion of the pendulum is a function of time.

Why does it hurt more if your hand is snapped with a ruler than with a loose spring, even if the displacement of each system is equal?

The ruler is a stiffer system, which carries greater force for the same amount of displacement. The ruler snaps your hand with greater force, which hurts more.

You are observing a simple harmonic oscillator. Identify one way you could decrease the maximum velocity of the system.

You could increase the mass of the object that is oscillating.

## Section summary

• Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
$\frac{1}{2}{\text{mv}}^{2}+\frac{1}{2}{\text{kx}}^{2}=\text{constant.}$
• Maximum velocity depends on three factors: it is directly proportional to amplitude, it is greater for stiffer systems, and it is smaller for objects that have larger masses:
${v}_{\text{max}}=\sqrt{\frac{k}{m}}X.$

## Conceptual questions

Explain in terms of energy how dissipative forces such as friction reduce the amplitude of a harmonic oscillator. Also explain how a driving mechanism can compensate. (A pendulum clock is such a system.)

## Problems&Exercises

The length of nylon rope from which a mountain climber is suspended has a force constant of $1\text{.}\text{40}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{N/m}$ .

(a) What is the frequency at which he bounces, given his mass plus and the mass of his equipment are 90.0 kg?

(b) How much would this rope stretch to break the climber’s fall if he free-falls 2.00 m before the rope runs out of slack? Hint: Use conservation of energy.

(c) Repeat both parts of this problem in the situation where twice this length of nylon rope is used.

(a) $\text{1.99 Hz}$

(b) 50.2 cm

(c) 1.41 Hz, 0.710 m

Engineering Application

Near the top of the Citigroup Center building in New York City, there is an object with mass of $4\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{kg}$ on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being driven—the driving force is transferred to the object, which oscillates instead of the entire building. (a) What effective force constant should the springs have to make the object oscillate with a period of 2.00 s? (b) What energy is stored in the springs for a 2.00-m displacement from equilibrium?

(a) $3\text{.}\text{95}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{N/m}$

(b) $7\text{.}\text{90}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{J}$

Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.
hour glass, pendulum clock, atomic clock?
S.M
tnks
David
how did they solve for "t" after getting 67.6=.5(Voy + 0)t
Find the following for path D in [link] : (a) The distance traveled. (b) The magnitude of the displacement from start to finish. (c) The displacement from start to finish.
the topic is kinematics
David
can i get notes of solid state physics
Lohitha
just check the chpt. 13 kinetic theory of matter it's there
David
is acceleration a fundamental unit.
no it is derived
Abdul
no
Nisha
K thanks
David
no it's not its derived
Emmanuel
hi
Hello
hi guys can you teach me how to solve a logarithm?
how about a conceptual framework can you simplify for me? needed please
Villaflor
Hello what happens when electrone stops its rotation around its nucleus if it possible how
Afzal
I think they are constantly moving
Villaflor
yep what is problem you are stuck into context?
S.M
not possible to fix electron position in space,
S.M
Physics
Beatriz
yes of course Villa flor
David
equations of kinematics for constant acceleration
A bottle full of water weighs 45g when full of mercury,it weighs 360g.if the empty bottle weighs 20g.calculate the relative density of mercury and the density of mercury....pls I need help
well You know the density of water is 1000kg/m^3.And formula for density is density=mass/volume Then we must calculate volume of bottle and mass of mercury: Volume of bottle is (45-20)/1000000=1/40000 mass of mercury is:(360-20)/1000 kg density of mercury:(340/1000):1/50000=(340•40000):1000=13600
Sobirjon
the latter is true
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture...take density of water as 1g/cm3 and density of liquid 1.2g/cm3
Lila
plz hu can explain Heisenberg's uncertainty principle
who can help me with my problem about acceleration?
ok
Nicholas
how to solve this... a car is heading north then smoothly made a westward turn during the travel the speed of the car remains constant at 1.5km/h what is the acceleration of the car? the total travel time of the car as it smoothly changed its direction is 15 minutes
Vann
i think the acceleration is 0 since the car does not change its speed unless there are other conditions
Ben
yes I have to agree, the key phrase is, "the speed of the car remains constant...," all other information is not needed to conclude that acceleration remains at 0 during the entire time
Luis
who can help me with a relative density question
Lila
1cm3 sample of tin lead alloy has mass 8.5g.the relative density of tin is 7.3 and that of lead is 11.3.calculate the percentage by weight of tin in the alloy. assuming that there is no change of volume when the metals formed the alloy
Lila
morning, what will happen to the volume of an ice block when heat is added from -200°c to 0°c... Will it volume increase or decrease?
no
Emmanuel
hi what is physical education?
Kate
BPED..is my course.
Kate
No
Emmanuel
I think it is neither decreases nor increases ,it remains in the same volume because of its crystal structure
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture. take density of water as 1g/cm3 and density of liquid as 1.2g/cm3
Lila
Sorry what does it means"no changes in volume occured"?
Sobirjon
volume can be the amount of space occupied by an object. But when an object does not change in shape it will still occupy the same space. Thats why the volume will still remain the same
Ben
Most soilds expand when heated but if it changes state at 0C it will have less volume. Ice floats because it is less dense ie a larger mass per unit volume.
Richard
how to calculate velocity
v=d/t
Emeka
Villaflor
Villaflor
v=d/t
Nisha
hello bro hw is life with you
Mine is good. How about you?
Chase
Hi room of engineers
yes,hi sir
Okwethu
hello
akinmeji
Hello
Mishael
hello
Jerry
hi
Sakhi
hi
H.C
so, what is going on here
akinmeji
Ajayi
good morning ppl
ABDUL
If someone has not studied Mathematics enough yet, should theu study it first then study Phusics or Study Basics of Physics whilst srudying Math as well?
whether u studied maths or not, it is advisable to start from d basics cuz it is essential to know dem
Nuru
yea you are right
wow, you got this w/o knowing math
Thomas
I guess that's it
Thomas
later people
Thomas
mathematics is everywhere
Anand
thanks but dat doesn't mean it is good without maths @Riaz....... Maths is essential in sciences particularly wen it comes to PHYSICS but PHYSICS must be started from the basic which may also help in ur mathematical ability
Nuru
A hydrometer of mass 0.15kg and uniform cross sectional area of 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
Lila
16.66 meters?
Darshik
16.71m2
aways
,i have a question of let me give answer
aways
the mass is stretched a distance of 8cm and held what is the potential energy? quick answer
aways
oscillation is a to and fro movement, it can also be referred to as vibration. e.g loaded string, loaded test tube or an hinged door