Circular & Rotational Motion + SHM

• Notes on dynamics of circular motion, rotational dynamics, and simple harmonic motion

Dynamics of Circular Motion

• Describes the motion of an object along a circular path.
• Uniform Circular Motion (UCM) features constant speed but changing velocity due to directional changes.
• Non-Uniform Circular Motion involves changes in both speed and direction.
• Centripetal Acceleration (𝑎𝑐): Acceleration directed toward the center of the circular path; defined as 𝑎𝑐 = 𝑣2/𝑟 = 𝜔2𝑟.
• Centripetal Force (𝐹𝑐): Force required to maintain an object's circular motion; calculated as 𝐹𝑐 = 𝑚𝑎𝑐 = 𝑚𝑣2/𝑟 = 𝑚𝜔2𝑟.
• Angular Velocity (𝜔): Rate of rotation, given by 𝜔 = 𝑣/𝑟 = 2𝜋/𝑇, where 𝑇 is the period of one revolution.
• Common problems involve finding centripetal force on objects like cars on circular tracks or determining tension in strings for objects moving in horizontal circles.

Rotational Dynamics

• Focuses on motion around a fixed axis, analogous to linear motion but uses angular quantities.
• Torque (𝜏): Rotational equivalent of force, defined as 𝜏 = 𝑟𝐹sin𝜃.
• Moment of Inertia (𝐼): Rotational equivalent of mass, calculated as 𝐼 = ∑𝑚𝑟2.
• Rotational Newton’s Second Law: States that 𝜏 = 𝐼𝛼, where 𝛼 is angular acceleration.
• Angular Momentum (𝐿): Given by 𝐿 = 𝐼𝜔.
• Conservation of Angular Momentum: Expressed as 𝐼1𝜔1 = 𝐼2𝜔2.
• Example problems include calculating torque on a door or determining the final angular velocity of a rotating system after mass redistribution.

Simple Harmonic Motion (SHM)

• Oscillatory motion where acceleration is proportional to displacement and directed towards equilibrium.
• General Equation: Describes displacement as a function of time: 𝑥 = 𝐴cos(𝜔𝑡 + 𝜙).
• Velocity: Changes with time, 𝑣 = −𝐴𝜔sin(𝜔𝑡 + 𝜙).
• Acceleration: Related to displacement, 𝑎 = −𝜔2𝑥.
• Time Period (𝑇) and Frequency (𝑓): Defined as 𝑇 = 2𝜋/𝜔 and 𝑓 = 1/𝑇.
• Restoring Force (Hooke’s Law): 𝐹 = −𝑘𝑥, where 𝑘 is the force constant.
• Mass-Spring System: The period is 𝑇 = 2𝜋√(𝑚/𝑘).
• Simple Pendulum: The period is 𝑇 = 2𝜋√(𝑙/𝑔).

Relationship Between SHM and Circular Motion

• SHM can be visualized as the projection of uniform circular motion onto a straight line.
• Displacement, velocity, and acceleration in SHM correspond to the horizontal components of a point undergoing circular motion.
• Equivalence: Defines SHM parameters as 𝑥 = 𝐴cos(𝜔𝑡), 𝑣 = −𝐴𝜔sin(𝜔𝑡), and 𝑎 = −𝐴𝜔2cos(𝜔𝑡).

Center of Gravity

• The point where the entire weight of an object is considered to act.
• For discrete masses: The center of gravity is found using 𝑥𝑐𝑔 = ∑𝑚𝑖𝑥𝑖 / ∑𝑚𝑖 and 𝑦𝑐𝑔 = ∑𝑚𝑖𝑦𝑖 / ∑𝑚𝑖.
• For a uniform object: the center of gravity lies at its geometric center.
• Common problems involve finding the center of gravity for systems of masses or composite bodies.

Final Tips for Exam Success

• Crucial to memorize key formulas and understand how to apply them.
• Practice with past questions is highly recommended, focusing on problem-solving techniques.
• Dimensional analysis is useful to ensure the units are correct in answers.
• Regular revision of concepts and the use of flashcards are effective for quick review.