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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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