# 15.2 Energy in simple harmonic motion  (Page 5/8)

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If you are interested in this interaction, find the force between the molecules by taking the derivative of the potential energy function. You will see immediately that the force does not resemble a Hooke’s law force $\left(F=\text{−}kx\right)$ , but if you are familiar with the binomial theorem:

${\left(1+x\right)}^{n}=1+nx+\frac{n\left(n-1\right)}{2!}{x}^{2}+\frac{n\left(n-1\right)\left(n-2\right)}{3!}{x}^{3}+\cdots ,$

the force can be approximated by a Hooke’s law force.

## Velocity and energy conservation

Getting back to the system of a block and a spring in [link] , once the block is released from rest, it begins to move in the negative direction toward the equilibrium position. The potential energy decreases and the magnitude of the velocity and the kinetic energy increase. At time $t=T\text{/}4$ , the block reaches the equilibrium position $x=0.00\phantom{\rule{0.2em}{0ex}}\text{m,}$ where the force on the block and the potential energy are zero. At the equilibrium position, the block reaches a negative velocity with a magnitude equal to the maximum velocity $v=\text{−}A\omega$ . The kinetic energy is maximum and equal to $K=\frac{1}{2}m{v}^{2}=\frac{1}{2}m{A}^{2}{\omega }^{2}=\frac{1}{2}k{A}^{2}.$ At this point, the force on the block is zero, but momentum carries the block, and it continues in the negative direction toward $x=\text{−}A$ . As the block continues to move, the force on it acts in the positive direction and the magnitude of the velocity and kinetic energy decrease. The potential energy increases as the spring compresses. At time $t=T\text{/}2$ , the block reaches $x=\text{−}A$ . Here the velocity and kinetic energy are equal to zero. The force on the block is $F=+kA$ and the potential energy stored in the spring is $U=\frac{1}{2}k{A}^{2}$ . During the oscillations, the total energy is constant and equal to the sum of the potential energy and the kinetic energy of the system,

${E}_{\text{Total}}=\frac{1}{2}k{x}^{2}+\frac{1}{2}m{v}^{2}=\frac{1}{2}k{A}^{2}.$

The equation for the energy associated with SHM can be solved to find the magnitude of the velocity at any position:

$|v|=\sqrt{\frac{k}{m}\left({A}^{2}-{x}^{2}\right)}.$

The energy in a simple harmonic oscillator is proportional to the square of the amplitude. When considering many forms of oscillations, you will find the energy proportional to the amplitude squared.

Check Your Understanding Why would it hurt more if you snapped your hand 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.

Check Your Understanding Identify one way you could decrease the maximum velocity of a simple harmonic oscillator.

You could increase the mass of the object that is oscillating. Other options would be to reduce the amplitude, or use a less stiff spring.

## Summary

• The simplest type of oscillations are related to systems that can be described by Hooke’s law, F = − kx , where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system.
• Elastic potential energy U stored in the deformation of a system that can be described by Hooke’s law is given by $U=\frac{1}{2}k{x}^{2}.$
• Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
${E}_{\text{Total}}=\frac{1}{2}m{v}^{2}+\frac{1}{2}k{x}^{2}=\frac{1}{2}k{A}^{2}=\text{constant.}$
• The magnitude of the velocity as a function of position for the simple harmonic oscillator can be found by using
$|v|=\sqrt{\frac{k}{m}\left({A}^{2}-{x}^{2}\right)}.$

## Conceptual questions

Describe a system in which elastic potential energy is stored.

In a car, elastic potential energy is stored when the shock is extended or compressed. In some running shoes elastic potential energy is stored in the compression of the material of the soles of the running shoes. In pole vaulting, elastic potential energy is stored in the bending of the pole.

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.)

The temperature of the atmosphere oscillates from a maximum near noontime and a minimum near sunrise. Would you consider the atmosphere to be in stable or unstable equilibrium?

The overall system is stable. There may be times when the stability is interrupted by a storm, but the driving force provided by the sun bring the atmosphere back into a stable pattern.

## Problems

Fish are hung on a spring scale to determine their 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?

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 force 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?

a. $1.57\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{N/m}$ ; b. 77 kg, yes, he is eligible to play

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?

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?

a. $6.53\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{N/m}$ ; b. yes, when the man is at his lowest point in his hopping the spring will be compressed the most

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?

The length of nylon rope from which a mountain climber is suspended has an effective force constant of $1.40\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}\text{N/m}$ . (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. 1.99 Hz; b. 50.2 cm; c. 0.710 m

a particle projected from origin moving on x-y plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
definition of inertia
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
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Kushal
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Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
position, velocity and acceleration of vector
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
I need the solving for this question
philip
is the eye the same like the camera
I can't understand
Suraia
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
why classical mechanics is necessary for graduate students?
classical mechanics?
Victor
principle of superposition?
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
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Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
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Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
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what is the formula for pressure?
force/area
Kidus
force is newtom
Kidus
and area is meter squared
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so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus