PS6 1

Chapter 6: Systems of Particles and Rotational Motion

Rigid Body

• A rigid body is defined as an object with a perfectly definite and unchanging shape.

• Distances between any two points within the body remain constant.

Motion of a Rigid Body

• Non-pivoted or fixed bodies: Movement can either be pure translation or a combination of translation and rotation.

  • 1) Pure Translational Motion:

    • All particles of the body have the same velocity at any instant.

    • Example: A block moving down an inclined plane where all points move at the same velocity.

  • 2) Pure Rotational Motion:

    • Every point in a rigid body has the same angular velocity, but different linear velocities.

    • Examples:

      • Rotation about a fixed axis (e.g., ceiling fan, potter's wheel): points move in circles in a plane perpendicular to the axis.

      • Rotation about a non-fixed axis (e.g., spinning top).

  • 3) Rolling Motion:

    • A combination of translational and rotational motion.

    • Example: A solid cylinder moving down an inclined plane, with varying velocities across different points but zero velocity at the point of contact when rolling without slipping.

Centre of Mass

• Defined as a hypothetical point where the entire mass of a system can be assumed to be concentrated.

• For a two-particle system:

  • Coordinate formula:

    • X = (m1x1 + m2x2) / (m1 + m2)

• Example: For three particles at the vertices of an equilateral triangle with masses 100g, 150g, and 200g, each side length equals 0.5m.

  • To find the centre of mass, use coordinates to calculate the average based on their weights and distances.

Motion of Centre of Mass

• Position: R = (m1r1 + m2r2 + ...) / (m1 + m2 + ...)

• Velocity: V = (m1v1 + m2v2 + ...) / M

• Acceleration: A similar differentiation leads to the relationship of acceleration with external forces acting on the center of mass.

Law of Conservation of Momentum for Systems of Particles

• If the sum of external forces (F_ext) acting on a system of particles is zero:

  • dP/dt = 0, which means momentum (P) remains constant.

Vector Product or Cross Product of Vectors

• Defined as A x B = |A| |B| sin(θ) n, where:

  • A and B are magnitudes of the vectors.

  • θ is the angle between them.

  • n is the unit vector perpendicular to the plane containing A and B.

• Notable Properties:

  • Non-commutative (A x B ≠ B x A)

  • Follows distributive law.

  • Self-cross products yield zero (e.g., A x A = 0).

Angular Velocity & Acceleration

• Angular velocity (ω): A vector quantity that indicates how fast an object rotates.

  • Relation: v = rω, where v is the linear velocity at a distance r from the axis.

• Angular acceleration (α): Rate of change of angular velocity.

Torque or Moment of Force

• Defined as τ = rFsin(θ), where:

  • r = distance from the point of rotation to the line of action of the force.

  • Torque unit: Newton-metre (Nm).

• Angular Momentum (L): L = r x p, where p is linear momentum.

Conservation of Angular Momentum

• If the total external torque (T_ext) on a system is zero, then angular momentum remains constant.

Example

• Calculate the torque from a given force vector acting on a particle with a specific position vector using the cross product.

Equilibrium of a Rigid Body

• A rigid body is in mechanical equilibrium if both linear and angular motion are constant.

  • Translational Equilibrium: Sum of all forces is zero.

  • Rotational Equilibrium: Sum of all torques is zero.

  • Partial Equilibrium: Only one type of equilibrium exists (e.g., translational, not rotational).

Centre of Gravity and Moment of Inertia

• Centre of Gravity: Point where gravitational torque is zero, typically coincides with the centre of mass unless gravity varies.

• Moment of Inertia (I): Measure of rotational inertia about an axis, calculated as I = mr² for particles, depends on mass distribution and axis of rotation.

Specific Moments of Inertia

• Various rigid body shapes have specific moments of inertia about certain axes (e.g., thin ring, solid disc).

Rotational Kinetic Energy & Radius of Gyration

• Kinetic energy for rotational motion is K.E. = (1/2)Iω².

• Radius of Gyration (k): The distance from the axis of rotation that gives the same moment of inertia as the actual distribution of mass.

  • Relation: I = Mk².

Flywheel

• Machines utilizing a flywheel (e.g., engines) benefit from a large moment of inertia, ensuring smooth speed variations during operation.

Kinematics of Rotational Motion about a Fixed Axis

• Kinematic equations for uniform angular acceleration mirror those of linear motion, showing relations between linear displacement and angular displacement.

Dynamics of Rotational Motion

• Comparing linear and rotational definitions helps in transitioning principles from linear to rotational mechanics (e.g., displacement to angular displacement).