Oscillations and Simple Harmonic Motion

Oscillations of a Spring

Oscillatory motion occurs when an object moves back and forth about an equilibrium position. A classic example is a mass attached to a spring.

• Hooke's Law: The restoring force is proportional to displacement:

• Simple Harmonic Motion (SHM): The motion where acceleration is proportional and opposite to displacement.

• Equation of Motion: , where is amplitude, is angular frequency, and is phase.

• Period of a Mass-Spring System:

• Frequency:

• Energy in SHM: Total mechanical energy

• Example: A 0.5 kg mass attached to a spring with N/m oscillates with a period s.

General Simple Harmonic Motion (Kinematics and Dynamics)

SHM can be described by kinematic equations and energy considerations.

• Acceleration:

• Maximum velocity:

• Maximum acceleration:

• Example: For m and rad/s, m/s.

Rotational Motion and Dynamics

Rotational Kinematics

Describes the motion of rotating bodies using angular quantities.

• Angular displacement: (radians)

• Angular velocity:

• Angular acceleration:

• Kinematic equations:

• Example: A wheel accelerates from rest at $2^2 s: rad/s.

Rotational Energy

Rotating objects possess kinetic energy due to their motion.

• Rotational kinetic energy:

• Moment of inertia (): Depends on mass distribution. For a solid disk:

• Example: A disk ( kg, m) spinning at rad/s has J.

Torque and Angular Momentum

Torque causes rotational acceleration, and angular momentum is conserved in isolated systems.

• Torque ():

• Relation to angular acceleration:

• Angular momentum ():

• Conservation of angular momentum: if

• Example: A figure skater pulls in her arms, reducing and increasing to conserve .

Rolling Motion

Rolling combines rotational and translational motion.

• Condition for rolling without slipping:

• Total kinetic energy:

• Example: A solid sphere rolling down a ramp has both translational and rotational energy.

Conservation Laws

Mechanical Energy Conservation

In the absence of non-conservative forces, total mechanical energy is conserved.

• Mechanical energy:

• Conservation:

• Example: A pendulum swings, converting potential energy to kinetic energy and back.

Linear and Angular Momentum Conservation

Momentum is conserved in isolated systems.

• Linear momentum:

• Conservation:

• Angular momentum:

• Example: Collisions (elastic and inelastic) conserve total momentum.

Additional Topics

Rotational Inertia

Rotational inertia quantifies resistance to changes in rotational motion.

• Depends on mass and its distribution relative to axis.

• Parallel Axis Theorem:

• Example: Calculating for a rod rotated about one end.

Newton's Law for Rotation

Analogous to Newton's second law for linear motion.

• Rotational form:

• Example: Applying torque to a wheel to find angular acceleration.

Work and Power in Rotational Motion

Work and power can be defined for rotational systems.

• Work:

• Power:

• Example: Calculating work done by a motor turning a shaft.

Forces in Rotational Systems

Forces cause torques, which produce rotational motion.

• Key forces: Gravity, friction, tension, normal force.

• Example: Analyzing forces on a rolling cylinder.

Summary Table: Key Equations and Concepts

Additional info: These notes synthesize the study guide topics, expanding on each with definitions, equations, and examples for exam preparation. Some sections (e.g., "excluded from exams") are omitted as per the original document's focus.