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

 Page 2 / 2

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

A soccer player kicked off a ball at velocity of 62 ft/s at angle 45°. A goal keeper is 43 yard away from the direction in which ball kicked off. At what minimum velocity he runs to meet the ball?
A soccer player kicked off the ball at the velocity of 62 ft/s at 45° with horizontal.A goal keeper is 43 yard away from the ball kicked position.At what minimum velocity he runs to meet the ball?
Ram
what is torque
The turning effect of force is called torque.
Uzair
What is the effect of static electricity
Ruth
what there factors affect the surface tension of a liquid
formula for impedance
ehat is central forces
what is distance?
What does mean ohms law imply
ohms law state that the electricity passing through a metallic conductor is directly proportional to the potential difference across its end
muyiwa
what is matter
Anything that occupies space
Kevin
Any thing that has weight and occupies space
Victoria
Anything which we can feel by any of our 5 sense organs
Suraj
Right
Roben
thanks
Suraj
what is a sulphate
Alo
Alo
the time rate of increase in velocity is called
acceleration
Emma
What is uniform velocity
Victoria
Greetings,users of that wonderful app.
how to solve pressure?
how do we calculate weight and eara eg an elefant that weight 2000kg has four fits or legs search of surface eara is 0.1m2(1metre square) incontact with the ground=10m2(g =10m2)
Cruz
P=F/A
Mira
can someone derive the formula a little bit deeper?
Bern
what is coplanar force?
forces acting and lying on d same plane
Promise
what is accuracy and precision
How does a current follow?
follow?
akif
which one dc or ac current.
akif
how does a current following?
Vineeta
?
akif
AC current
Vineeta
AC current follows due to changing electric field and magnetic field.
akif
Abubakar
ok bro thanks
akif
flows
Abubakar
but i wanted to understand him/her in his own language
akif
but I think the statement is written in English not any other language
Abubakar
my mean that in which form he/she written this,will understand better in this form, i write.
akif
ok
Abubakar
ok thanks bro. my mistake
Vineeta
u are welcome
Abubakar
what is a semiconductor
substances having lower forbidden gap between valence band and conduction band
akif
what is a conductor?
Vineeta
replace lower by higher only
akif
convert 56°c to kelvin
Abubakar
How does a current follow?
Vineeta
A semiconductor is any material whose conduction lies between that of a conductor and an insulator.
AKOWUAH