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Chapter 60: The Dynamics of Simple Harmonic Motion
The potential energy stored in the spring is equal to the work done to compress it.
The expression is (1/2) kx 2.
If the system is set into oscillation by stretching the spring by an amount, the maximum energy possessed by the system is given by U.
When it passes through x, the mass reaches its maximum speed.
The period of a system is not dependent on the amplitude.
If we want to treat cases in which the mass is between the two extremes of x and x, we note that the spring will still have some elastic potential energy.
The equilibrium point of a mass varies from position to position.
The speed of the mass should be ranked from fastest to slowest.
Justify your answer.
The mass is changing direction when the spring is fully compressed or extended.
The speed is the highest when the mass is traveling through the equilibrium point.
The direction of motion doesn't matter for positions I and III since the question is about speed.
Since position III is closer to equilibrium, it is slightly faster than position I.
A periodic motion in which the acceleration is proportional to the negative of the displacement is called a simple harmonic motion.
The period of oscillation for a mass attached to a spring is dependent on the mass and force constant k.
The period of oscillation for a simple pendulum is dependent on the length of the string and the value of the acceleration due to gravity.
The period of simple motion is not dependent on the amplitude.
The middle part of the motion has the greatest velocity.
At the end of the motion, the acceleration is the greatest.
The acceleration varies with the displacement, but in the opposite direction.
If the displacement angle is small, a simple pendulum approximates simple motion.
Under these small-angle circumstances, the period will be independent of mass and amplitude.
It's easier to use energy considerations since the equations don't have free-body diagrams to draw.
An astronauts uses a pendulum to determine the acceleration of gravity on a planet.
When a 0.05- kilogram mass is attached to a vertical spring, it is observed that it stretches 0.03 m.
There is a massless spring with a force constant of 50 N/m.
The system is on a horizontal surface.
There is a massless spring with a force constant of 20 N/m.
The system is on a surface with an amplitude of 4 cm.
A 2- kilo mass is attached to a spring.
10 J is the total energy of the system.
Two springs with force constants k 1 and k 2 are attached to a mass M. The mass can move over the surface.
There are two arrangements for the mass and springs.
A mass M is attached to two elastic strings made of the same material.
Equal tensions T exist in the two strings as shown below, because the mass is displaced vertically upward by a small displacement D y.
The mass starts to move up and down.
The tensions will not change if the displacement is small.
The pendula and springs can be used as timekeeping devices.
Compare the accuracy of these devices to the accuracy of timekeeping devices on the Moon, where gravity is 1/6 that of Earth's.
The values given in the question are 0.25 m.
We find that g is 17.5 m/s 2 using the values given in the question.
The period is given by the formula T=2pm/k.
We find that T is 0.35 s using the values given in the question.
E is the value given in the question.
We need to change the centimeters to meters to find a v of 0.28 m/s.
The total energy of the system will be affected if the spring and the amplitude of oscillations remain the same.
Newton's third law states that the forces must be equal and opposite.
There is a period of oscillation for arrangement.
The effective spring constant increases when the springs are connected in parallel.
The acceleration is proportional to the displacement and only for small angles is a pendulum a representation.
The object's mass can be determined using springs and horizontal oscillations, since the period of a horizontally oscillating mass on a spring is independent of the acceleration due to gravity.
The spring's period will be unaffected since the pendulum will have a longer period on the Moon.
The hanging mass-on-a-spring system will be stretched six times less on the Moon than it is on Earth.
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