Simple Harmonic Motion (SHM) is a fundamental concept in physics that describes periodic motion. It plays a crucial role in various systems, including pendulums, springs, and even quantum mechanics. Two key aspects of SHM are velocity and acceleration, which determine how an object moves over time.
This topic explains velocity and acceleration in SHM, their equations, graphical representations, and real-world applications.
Understanding Simple Harmonic Motion
What is Simple Harmonic Motion?
SHM is a type of oscillatory motion where an object moves back and forth around an equilibrium position under a restoring force. This force is always directed towards the equilibrium position and is proportional to the displacement.
Mathematically, SHM is defined by:
where:
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F is the restoring force,
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k is the force constant,
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x is the displacement from the equilibrium position.
Examples of SHM include:
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A mass attached to a spring oscillating back and forth.
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A pendulum swinging at small angles.
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Vibrations of atoms in a crystal lattice.
Velocity in Simple Harmonic Motion
Definition of Velocity in SHM
Velocity in SHM refers to the rate of change of displacement with respect to time. It determines how fast an object moves at different points in its motion.
The velocity equation in SHM is:
where:
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v is the velocity,
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omega is the angular frequency,
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A is the amplitude,
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t is time,
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phi is the phase constant.
Velocity at Different Positions
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At the equilibrium position ( x = 0 ):
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The velocity is maximum:
v_{text{max}} = omega A -
The object moves fastest at the center.
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At the extreme positions ( x = pm A ):
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The velocity is zero.
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The object momentarily stops before reversing direction.
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Graphical Representation of Velocity
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The velocity graph is a cosine function when displacement follows a sine function.
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It is 90° out of phase with displacement.
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When displacement is maximum, velocity is zero and vice versa.
Acceleration in Simple Harmonic Motion
Definition of Acceleration in SHM
Acceleration in SHM refers to the rate of change of velocity with respect to time. It determines how quickly an object’s speed changes.
The acceleration equation in SHM is:
where:
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a is the acceleration,
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omega is the angular frequency,
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x is the displacement.
Acceleration at Different Positions
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At the equilibrium position ( x = 0 ):
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Acceleration is zero.
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The object moves fastest here, but without any force pulling it further.
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At the extreme positions ( x = pm A ):
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Acceleration is maximum:
a_{text{max}} = omega^2 A -
The object experiences the greatest force pulling it back towards equilibrium.
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Graphical Representation of Acceleration
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The acceleration graph is a sine function, but in the opposite phase to displacement.
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When displacement is positive, acceleration is negative, and vice versa.
Phase Relationship Between Displacement, Velocity, and Acceleration
In SHM, displacement, velocity, and acceleration are all interconnected through phase differences:
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Displacement ( x ) follows a sine function:
x = A sin(omega t + phi) -
Velocity ( v ) is a cosine function, leading displacement by 90°:
v = A omega cos(omega t + phi) -
Acceleration ( a ) is a negative sine function, meaning it is 180° out of phase with displacement:
a = -A omega^2 sin(omega t + phi)
This phase relationship means:
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When displacement is maximum, velocity is zero, and acceleration is at its maximum negative value.
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When displacement is zero, velocity is maximum, and acceleration is zero.
Energy Considerations in SHM
Although velocity and acceleration govern motion, energy provides additional insight into how an object oscillates.
Kinetic Energy ( KE ) in SHM
Kinetic energy depends on velocity and is given by:
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Maximum at the equilibrium position.
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Zero at the extreme positions.
Potential Energy ( PE ) in SHM
Potential energy depends on displacement and is given by:
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Maximum at the extreme positions.
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Zero at the equilibrium position.
Total Energy in SHM
The total mechanical energy remains constant and is given by:
This conservation of energy ensures that energy transfers smoothly between kinetic and potential forms during motion.
Real-World Applications of Velocity and Acceleration in SHM
1. Pendulums
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The velocity of a pendulum bob is maximum at the lowest point and zero at the highest points.
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Acceleration is maximum at the endpoints and zero at the center.
2. Mass-Spring Systems
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Springs follow SHM when stretched or compressed.
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Velocity is highest when passing through equilibrium.
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Acceleration is highest when the spring is most compressed or stretched.
3. Vibrations in Engineering
- Machines and buildings are designed considering SHM principles to avoid dangerous resonance effects.
4. Sound Waves
- Ptopics in sound waves undergo SHM, moving back and forth as waves propagate.
5. Quantum Mechanics
- The motion of subatomic ptopics in quantum harmonic oscillators follows SHM equations.
Velocity and acceleration play essential roles in Simple Harmonic Motion, determining how an object moves in periodic systems. Key takeaways include:
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Velocity in SHM follows a cosine function and reaches maximum at equilibrium.
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Acceleration in SHM follows a sine function and is maximum at extreme positions.
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Velocity and acceleration are out of phase, meaning when velocity is highest, acceleration is zero.
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Energy remains conserved in SHM, transferring between kinetic and potential forms.
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Real-world examples include pendulums, springs, sound waves, and even quantum systems.
Understanding these concepts helps in fields ranging from engineering to quantum physics, making SHM one of the most important topics in mechanics.