Kinematics of Rigid Bodies

Why it matters

Understanding the kinematics of rigid bodies is crucial for designing and analyzing mechanical systems such as engines, machinery, and robotics. It helps engineers predict the motion of components, ensuring they function correctly and efficiently.

Key ideas

• Rigid Body: An idealization where the distance between any two points remains constant despite external forces.
• Types of Motion: Rigid bodies can undergo translational, rotational, or general plane motion.
  • Translational Motion: All points in the body move parallel to and in the same direction as every other point.
  • Fixed-axis rotation: Points away from the axis in the body move in circles about a single line, known as the axis of rotation.
  • General Plane Motion: A combination of translation and rotation.
• Instantaneous Center of Rotation (ICR): A point in a body or its extension where the velocity is zero at a particular instant.

Velocity and acceleration relations

For two points A and B fixed in the same rigid body, with r_B/A directed from A to B:

v_B = v_A + ω × r_B/A

a_B = a_A + α × r_B/A + ω × (ω × r_B/A).

Here ω is angular velocity in rad/s and α is angular acceleration in rad/s². The symbol × denotes a vector cross product. The last term is the centripetal acceleration; it must not be omitted even when angular speed is constant.

For rotation about a fixed axis, a point at perpendicular distance r has tangential speed v = ωr, tangential acceleration a_t = αr, and inward normal acceleration a_n = ω²r. Tangential and normal components are perpendicular.

For pure translation, ω = α = 0, so all points have the same instantaneous velocity and acceleration, even if their paths are curved.

Instantaneous centre in planar motion

When ω is nonzero, a planar rigid body's velocity field can be represented as instantaneous rotation about a point of zero velocity. This point may lie outside the material body. Pure translation has no finite instantaneous centre. The zero-velocity point generally has nonzero acceleration, so it cannot automatically be used as a fixed pivot for acceleration calculations.

For a wheel rolling without slip on a stationary horizontal surface, the contact point has zero instantaneous velocity. The top of the wheel has speed 2v_C, where v_C is the centre's speed. This statement does not mean that the contact point has zero acceleration.

Worked example

Given: A wheel with a radius of 0.5 m rotates about its fixed centre with an angular speed of 10 rad/s. Find the tangential speed of a point on the rim.

1. Identify the given values:
   • Radius, r = 0.5 m
   • Angular velocity, ω = 10 rad/s
2. Use the formula for linear velocity:
   • v = ω × r
3. Substitute the given values:
   • v = 10 rad/s × 0.5 m
4. Calculate the linear velocity:
   • v = 5 m/s

Final Answer: The point has tangential speed 5 m/s, perpendicular to its radius. If angular speed is constant, its acceleration is still nonzero: a_n = 10² × 0.5 = 50 m/s² toward the centre.

Common mistakes

• Confusing angular velocity with linear velocity.
• Forgetting to convert units, especially when dealing with angles (degrees vs. radians).
• Misidentifying the axis of rotation or the ICR.

For GATE ME

Questions often involve calculating velocities and accelerations of points on a rigid body, identifying the ICR, and analyzing combined translational and rotational motion. Practice problems involving different types of motion and ensure a strong grasp of vector mathematics.

Quick check

1. What is the tangential speed of a point 2 m from a fixed axis of rotation if the angular velocity is 3 rad/s?
2. Define the instantaneous center of rotation.
3. What type of motion involves both translation and rotation?

Answers: 1. 6 m/s 2. A point where the velocity is zero at a particular instant. 3. General plane motion.