Phys101Lec16B

Centre of Mass

• Overview: The Centre of Mass (CM) is a crucial concept in understanding the motion of objects in physics.

  • It can be thought of as the average position of all the mass in a system.

  • Represents translational, rotational, and vibrational motion.

Defining Center of Mass (CM)

• The general motion of an object:

  • Can be expressed as the sum of the translational motion of the CM plus other forms of motion (rotational, vibrational) around it.

Calculating Center of Mass

• For two particles:

  • The CM's position is affected by the mass of each particle.

  • Formula:

    • For masses mA and mB:

      • y_CM = (mA * yA + mB * yB) / (mA + mB)

      • x_CM = (mA * xA + mB * xB) / (mA + mB)

    • Indicates that the CM is closer to the massier particle.

• Example: If mA is greater than mB, then:

  • y_CM = (mA * yA + mB * yB) / (mA + mB)

  • x_CM = (mA * xA + mB * xB) / (mA + mB)

Practical Exercise

• Exercise: Determine the center of mass for three particles located at the corners of a right triangle:

  • Each particle has a mass of 2.50 kg, located at (0,0), (2,0), and (0,1.5).

• Solution involves using:

  • x_CM = (8/3) m

  • y_CM = (3/4) m

  • This results in CM calculated as:

    • Final coordinates are

      • (2.67 m, 0.75 m)

Example: L-Shaped Object

• Determine the CM for a uniform thin L-shaped object comprised of two rectangular parts:A and B.

  • Use respective centers of mass (xA, yA) and (xB, yB) calculated based on symmetry.

  • Incorporate the density and dimensions of the shapes to find:

    • Calculated positions of CM for A and B respectively.

Center of Mass and Translational Motion

• Total momentum of a particle system:

  • Given by the product of total mass (M) and velocity of CM (v):

    • P_Total = Mv_CM

• Total forces on the system relate to the acceleration of the CM:

  • F_net = M * a_CM

• Important concept: The system behaves like a single particle if subjected to net forces.

Fosbury Flop Example

• Notably, when executing the Fosbury flop in high jump,

  • The CM may not need to cross the bar, demonstrating the dynamics of CM in sports.

Conceptual Example: Two-Stage Rocket

• Overview of rocket motion:

  • When a rocket reaches its peak height, it separates into two parts after an explosion;

  • Motion dynamics are highlighted as part I falls and part II maintains trajectory.

• Analysis of height and horizontal landing point necessitates an understanding of CM and gravitational forces.

Explosion Dynamics

• Explosions are considered reverse inelastic collisions.

• Key point: Momentum conservation is maintained in explosions despite external forces like gravity influencing the components.