Notes on Simple Harmonic Motion

Class 1: Simple Harmonic Motion

Date: 17 March 2026
Time: 17:20

Definition of Periodic Motion

• Oscillatory Motion: Movement characterized by back-and-forth motion, exemplified by a child swinging on a swing.

• Periodic Function: Motion that repeats after a fixed interval of time. The mathematical representation is:

  • $f(t+nT) = f(t)$

• Time Period (T): The time taken to complete one full cycle of periodic motion. Example: Earth's motion around the Sun takes about 365 days and 4 hours.

Simple Harmonic Motion (SHM)

• SHM Definition: A type of periodic motion where an object oscillates back and forth along the same path.

  • The motion is driven by a restoring force that is directly proportional to the object's displacement from its equilibrium position and directed towards that position.

Everyday Examples of SHM

• Child swinging on a swing

• Ticking of a clock's pendulum

• A plucked guitar string

Restoring Force

• Restoring Force (F_R):

  • F= -Kx

  • Defined as the force acting to restore an object to its equilibrium position.

  • The restoring force is opposite to the displacement (x) and proportional in magnitude, where K is the spring constant.

Newton's Second Law in SHM

• The relationship between force, mass, and acceleration is given by:

  • $F = ma$

  • Thus, when considering SHM, we have:

  • $ma = -kx$

  • Rearranging gives:

  • $a = - rac{k}{m}x$

Characteristics of SHM

• The acceleration (a) of the object is directly proportional to its displacement (x) and acts in the opposite direction:

  • $a ext{ is proportional to } -x$

Systems Exhibiting SHM

• Different systems can exhibit SHM; notable examples include:

  • Object-Ideal Spring System

  • Pendulum

Conditions for Simple Harmonic Motion

• The presence of a restoring force that acts in the opposite direction of the object's displacement is necessary.

• The restoring force must be directly proportional to the object's displacement from equilibrium.

Pendulum Conditions

• A pendulum involves angular displacement, requiring a restoring torque:

  • The restoring torque must also be proportional to angular displacement but directed oppositely.

• Applicable only for small angular displacements.

Mathematical Relationships for Pendulum

• Angular relationships for small angles involve using:

  • $heta ext{ in radians}$

  • For small angles, $an( heta) ext{ and } heta ext{ are approximately equal}$

Kinematics and Dynamics in SHM

• For SHM, the period (T) can be expressed as:

  • $T = 2 rac{ ext{π}}{ω}$

  • The angular frequency (ω) relationships:

  • $ω^2 = rac{g}{L}$

  • where g is the acceleration due to gravity and L is the length of the pendulum.

Key Parameters in SHM

• Amplitude: Maximum distance from the equilibrium position.

• Equilibrium Position: The point where net force is zero.

• Restoring Force: Force acting to bring the object back to equilibrium.

• Period (T): Time to complete one cycle of motion.

• Frequency (f): Number of cycles per second, represented as:

  • $f = rac{1}{T}$

Kinematics Relationships

• For maximum velocity in an SHM, use:

  • $V_{max} = A imes ω$

Sample Problem A

Problem: Calculating the spring constant (k) of a spring stretched by a mass weight.

• Given: Mass (m) = 0.55 kg, stretch from equilibrium position (x) = 2.0 cm = 0.02 m.

  • $k = rac{mg}{x}$

  • Calculate gravitational force:

  • $mg = 0.55 imes 9.81 ext{ (which results in 5.3955 N)}$

  • Now using the earlier equation for k:

  • $k = rac{5.3955}{0.02} = 269.775 ext{ N/m}$

Energy Considerations

• Potential energy in the spring is calculated by:

  • $E_P.E = rac{1}{2} k A^2$

• At maximum displacement (amplitude), potential energy is maximized while kinetic energy (E_K) is zero.

• At equilibrium position, potential energy is zero and kinetic energy is maximized:

  • $E.P.E_{max} = rac{1}{2}kA^2$

  • Using conservation of energy, we can derive the maximum speed:

  • $V_{max} = rac{A ext{, k} ext{ and } ω}$

Final Kinematic Relationships

• For an oscillating mass on a spring, the expressions are as follows:

  • $x(t) = A ext{cos(}ωt)$

  • $V_{max} = ωA$

  • $ext{Where: } ω = ext{angular frequency} = rac{2 ext{π}}{T}$

Conclusion

• All learned principles apply to multiple systems, underscoring the universality of SHM across various physical contexts.

• The relationship between force, displacement, and acceleration in SHM provides foundational insights for more complex oscillatory systems.