DSE ROTATIONAL KINEMATICS OF A RIGID BODY

DSE ROTATIONAL KINEMATICS OF A RIGID BODY

Lecture Details

• Date: 20/03/2026

• Presented by: Dr. Godwin K. Aboagye

• Department: Master of Education, Department of Science Education, UCC

Rigid Body Definition

• Definition: A rigid body is defined as a body that can undergo both translational and rotational motions without being deformed.

• Example: Consider a solid uniform disk free to rotate about a fixed axis through its center.

Angular Motion Concepts

Average Angular Velocity

• When the angular position increases by $riangle heta$ over a time interval $riangle t$, the average angular speed or velocity ar{w} is given by:
  ar{w} = rac{ riangle heta}{ riangle t} ag{2}

• In the limit as $riangle t <br>ightarrow 0$, we define the instantaneous angular speed, $w$, as:
  $w = rac{d heta}{dt} ag{3}$

Instantaneous Angular Speed & Acceleration

• The unit of instantaneous angular speed is rad/s.

• If angular speed changes by $riangle w$ over a time $riangle t$, the average angular acceleration ar{eta} is proportional:
  ar{eta} = rac{ riangle w}{ riangle t}

• In the limit as $riangle t <br>ightarrow 0$, we define the instantaneous angular acceleration as:
  eta = rac{deta}{dt} = rac{dw}{dt} ag{4}

Relationship between Angular and Tangential Motion

• There exists a close relationship between angular quantities and the tangential motion of a particle on the rim of a disk:
  $v_t = rw$

• Differentiating with respect to time (with r constant):
  a_t = reta ag{5}

• The tangential acceleration is present only if angular speed $w$ is changing.

• A particle that moves in a circle always experiences centripetal acceleration towards the center defined as:
  $a<em>c = rac{v</em>t^2}{r} = r rac{v_t^2}{r^2} = r rac{w^2}{r} = w^2 r ag{6}$

Total Acceleration

• The total acceleration ($a$) of a particle can be determined by combining tangential and centripetal components:
  $a = ext{√(a<em>t^2 + a</em>c^2)}$

Kinematic Equations for Rotational Motion

• The angular quantities (eta) describe the rotation of the disk, while the tangential quantities describe the motion of a particle on the rim.

• If a thin paper is wound around the disk, its linear motion ($s$) is related to the disk's rotational motion ($d$) as follows:

  • Writing the kinematic equations:

  • $s = s_0 + vt + rac{1}{2} a t^2$

  • Hence:

    • heta = heta_0 + eta t + rac{1}{2} eta t^2

• More specifically, we have: heta = heta_0 + wt + rac{1}{2}eta t^2 ag{7} $s = r heta$

  • With the kinematic equations for rotational motion now established.

Moment of Inertia

Definition and Calculation

• Definition of Moment of Inertia (I): The sum of the mass $m_i$ of particles multiplied by the square of their perpendicular distance from the axis of rotation.

• For a rigid body rotating about a fixed axis, we consider a particle at a distance $r<em>i$, with resultant kinetic energy (KE): $ext{K.E.} = rac{1}{2} m</em>i v<em>i^2 = rac{1}{2} m</em>i (r_i w)^2$

• The total kinetic energy of the entire rotating body is represented as:
  $ext{K.E.} = rac{1}{2} I w^2$

• The moment of inertia for continuous mass distribution is given by the integral form:
  $I = rac{1}{2} imes extstyle{ ext{Mass}} imes r^2 ag{8}$

Parallel Axis Theorem

• To find the moment of inertia $I$ about any axis parallel to an axis through the center of mass (centroid):

  • $I = I_{cm} + Mh^2$
    where $h$ is the perpendicular distance between the axes.

Standard Moments of Inertia for Symmetrical Bodies

• The moments of inertia for various symmetrical bodies include:

  • Solid Cylinder or Disk about its axis:
    $I = rac{1}{2} M R^2$

  • Hoop about its axis:
    $I = M R^2$

  • Solid Sphere:
    $I = rac{2}{5} M R^2$

  • Thin Rod about the center:
    $I = rac{1}{12} M L^2$

Radius of Gyration

• Definition: The radius of gyration $K$ from any given axis where the mass of the body could be concentrated without altering the moment of inertia.

• If the mass $M$ is concentrated at a distance $K$ from an axis, the moment of inertia can be defined as:
  $I = K^2 M ag{9}$

Torque

Definition and Calculation

• Definition of Torque ($au$): The tendency of a force to rotate an object about an axis.

• Torque is given by: $au = F imes d imes ext{sin}( heta)$ where:

  • $F$ = applied force,

  • $d$ = perpendicular distance from the line of action to the axis of rotation.

• The torque expression highlights that if multiple forces act on a rigid body, each creates rotation about the pivot point. Thus,
  $au<em>{net} = au</em>1 + au<em>2 = au</em>{2} - au_{1}$

Torque as a Vector

• Torque can be represented as a vector using the cross product:
  $au = r imes F ag{10}$

Angular Momentum

• Definition of Angular Momentum ($L$): Given by the moment of linear momentum about an axis.
  $L = r imes p = r imes mv ag{11}$

• The relationship between torque and angular momentum is established as:
  $au = rac{dL}{dt}$

Conservation of Angular Momentum

• Law: The total momentum of a system remains constant if the net external torque acting on it is zero.

• Written as:
  $ext{If} au_{net} = 0, ext{then } L = ext{constant}$

Rotational Work and Energy

• The work done $W$ by a constant torque when turning an object through an angle is given as:
  $W = au heta$

• Total mechanical energy includes translational and rotational components:
  $E = rac{1}{2} mv^2 + rac{1}{2} I w^2$

Application Problem

• Given: A motor in an electric saw applies torque related to the blade's angular velocity.

• Challenge: Calculate the torque required to reach the rated angular velocity using the known moment of inertia.