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# Conceptual Questions

Exercise $$\PageIndex{1}$$

Describe a system in which elastic potential energy is stored.

# Problems & Exercises

Exercise $$\PageIndex{1}$$

Fish are hung on a spring scale to determine their mass (most fishermen feel no obligation to truthfully report the mass).

(a) What is the force constant of the spring in such a scale if it the spring stretches 8.00 cm for a 10.0 kg load?

(b) What is the mass of a fish that stretches the spring 5.50 cm?

(c) How far apart are the half-kilogram marks on the scale?

Solution:

(a) $$1.23 \times 10^3 \space N/m$$

(b) $$6.88 \space kg$$

(c) $$4.00 \space mm$$

Exercise $$\PageIndex{2}$$

It is weigh-in time for the local under-85-kg rugby team. The bathroom scale used to assess eligibility can be described by Hooke’s law and is depressed 0.75 cm by its maximum load of 120 kg.

(a) What is the spring’s effective spring constant?

(b) A player stands on the scales and depresses it by 0.48 cm. Is he eligible to play on this under-85 kg team?

Exercise $$\PageIndex{3}$$

One type of BB gun uses a spring-driven plunger to blow the BB from its barrel. (a) Calculate the force constant of its plunger’s spring if you must compress it 0.150 m to drive the 0.0500-kg plunger to a top speed of 20.0 m/s. (b) What force must be exerted to compress the spring?

Solution:

(a) 889 N/m

(b) 133 N

Exercise $$\PageIndex{4}$$

(a) The springs of a pickup truck act like a single spring with a force constant of $$1.30 \times 10^5 \space N/m$$.  By how much will the truck be depressed by its maximum load of 1000 kg?

(b) If the pickup truck has four identical springs, what is the force constant of each?

Exercise $$\PageIndex{5}$$

When an 80.0-kg man stands on a pogo stick, the spring is compressed 0.120 m.

(a) What is the force constant of the spring?

(b) Will the spring be compressed more when he hops down the road?

Solution:

(a) $$6.53 \times 10^3 \space N/m$$

(b) Yes

Exercise $$\PageIndex{6}$$

A spring has a length of 0.200 m when a 0.300-kg mass hangs from it, and a length of 0.750 m when a 1.95-kg mass hangs from it.

(a) What is the force constant of the spring?

(b) What is the unloaded length of the spring?

# Conceptual Questions

Exercise $$\PageIndex{1}$$

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

Exercise $$\PageIndex{1}$$

The length of nylon rope from which a mountain climber is suspended has a force constant of $$1.40 \times 10^4 \space N/m$$.

(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.

Solution:

(a) $$1.99 \space Hz$$

(b) 50.2 cm

(c) 1.41 Hz, 0.710 m

Exercise $$\PageIndex{2}$$: Engineering Application

Near the top of the Citigroup Center building in New York City, there is an object with mass of $$4.00 \times 10^5 \space kg$$ 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?

Solution:

(a) $$3.95 \times 10^6 \space N/m$$

(b) $$7.90 \times 10^6 \space J$$

# Problems & Exercises

Exercise $$\PageIndex{1}$$

(a)What is the maximum velocity of an 85.0-kg person bouncing on a bathroom scale having a force constant of $$1.50 \times 10^{6} N/m$$, if the amplitude of the bounce is 0.200 cm?

(b)What is the maximum energy stored in the spring?

Solution:

(a) 0.266 m/s

(b) 3.00 J

Exercise $$\PageIndex{2}$$

A novelty clock has a 0.0100-kg mass object bouncing on a spring that has a force constant of 1.25 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position? (b) How many joules of kinetic energy does the object have at its maximum velocity?

Exercise $$\PageIndex{3}$$

At what positions is the speed of a simple harmonic oscillator half its maximum? That is, what values of $$x/X$$ give $$v = \pm v_{max}/2$$, where $$X$$ is the amplitude of the motion?

Solution:

$$\pm \frac{\sqrt{3}}{2}$$

Exercise $$\PageIndex{4}$$

A ladybug sits 12.0 cm from the center of a Beatles music album spinning at 33.33 rpm. What is the maximum velocity of its shadow on the wall behind the turntable, if illuminated parallel to the record by the parallel rays of the setting Sun?