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

ω max = g L θ max . size 12{ω rSub { size 8{"max"} } = sqrt { { {g} over {L} } } θ rSub { size 8{"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 . 53 × 10 4 N/m size 12{k=6 "." "53" times "10" rSup { size 8{4} } `"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 max size 12{v rSub { size 8{"max"} } } {} given in v max = k m X size 12{v size 8{"max"}= sqrt { { {k} over {m} } } X} {} to determine the maximum vertical velocity. The variables m size 12{m} {} and k size 12{k} {} are given in the problem statement, and the maximum displacement X size 12{X} {} is 0.100 m.

Solution

  1. Identify known.
  2. Substitute known values into v max = k m X size 12{v size 8{"max"}= sqrt { { {k} over {m} } } X} {} :
    v max = 6 . 53 × 10 4 N/m 900 kg (0 . 100 m) . size 12{v size 8{"max"}= sqrt { { {6 "." "53" times "10" rSup { size 8{4} } "N/m"} over {"900"" kg"} } } 0 "." "100"" m"} {}
  3. Calculate to find v max = 0.852 m/s . size 12{v rSub { size 8{"max"} } } {}

Discussion

This answer seems reasonable for a bouncing car. There are other ways to use conservation of energy to find v max size 12{v rSub { size 8{"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 size 12{v rSub { size 8{"max"} } } {} of an oscillating simple pendulum, starting from its equilibrium position, is given as

y ( t ) = a sin ωt , size 12{y \( t \) =a"sin"ωt} {}

where a size 12{a} {} is the amplitude, ω size 12{ω} {} is the angular velocity and t size 12{t} {} is the time taken. Substituting ω = T size 12{ω= { {2π} over {T} } } {} , we have

y t = a sin t T . size 12{y left (t right )=a"sin" left ( { {2πt} over {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 ( t ) = 2 T cos t T , size 12{v \( t \) = { {2aπ} over {T} } "cos" left ( { {2πt} over {T} } right )} {}

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

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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.

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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.

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Section summary

  • Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
    1 2 mv 2 + 1 2 kx 2 = constant. size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } + { {1} over {2} } ital "kx" rSup { size 8{2} } =" 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 max = k m X . size 12{v rSub { size 8{"max"} } = sqrt { { {k} over {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.)

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Problems&Exercises

The length of nylon rope from which a mountain climber is suspended has a force constant of 1 . 40 × 10 4 N/m size 12{1 "." "40" times "10" rSup { size 8{4} } "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) 1.99 Hz size 12{ "1.99 Hz" } {}

(b) 50.2 cm

(c) 1.41 Hz, 0.710 m

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Engineering Application

Near the top of the Citigroup Center building in New York City, there is an object with mass of 4 . 00 × 10 5 kg size 12{4 "." "00" times "10" rSup { size 8{5} } "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 . 95 × 10 6 N/m size 12{3 "." "95" times "10" rSup { size 8{6} } "N/m"} {}

(b) 7 . 90 × 10 6 J size 12{7 "." "90" times "10" rSup { size 8{6} } "J"} {}

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Questions & Answers

what is heat
John Reply
Heat is a form of energy where molecules move
saran
Can you please help me with some questions
Janet Reply
topic-- question
Salman
I know this is unrelated to physics, but how do I get the MCQs and essay to work. they arent clickable.
Jake Reply
20cm3 of 1mol/dm3 solution of a monobasic acid HA and 20cm3 of 1mol/dm3 solution of NaOH are mixed in a calorimeter and a temperature rise of 274K is observed. If the heat capacity of the calorimeter is 160J/K, calculate the enthalpy of neutralization of the acid.(SHCw=4.2J/g/K) Formula. (ms*cs+C)*T
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why is a body moving at a constant speed able to accelerate
Lilian Reply
20cm3 of 1mol/dm3 solution of a monobasic acid HA and 20cm3 of 1mol/dm3 solution of NaOH are mixed in a calorimeter and a temperature rise of 274K is observed. If the heat capacity of the calorimeter is 160J/K, calculate the enthalpy of neutralization of the acid.(SHCw=4.2J/g/K) Formula. (ms*cs+C)*T
Lilian
because it changes only direction and the speed is kept constant
Justice
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
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v2 =u2 - 2gh 02 =10x10 - 2x9.8xh h = 100 ÷ 19.6 answer = 30 - h.
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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
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what refractive index
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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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