ENSC2004 - Lecture 20

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ENSC2004 - Lecture 19
Kinetic Energy (Rigid Bodies) - Slides

Kinetic Energy

• When we considered kinetic energy of a particle, there was no rotation
  $T=\frac{1}{2}m{v}^2$
• However, for a rigid body, we need to account for rotational velocity
  $T=\frac{1}{2}m(v_G{)}^2+\frac{1}{2}{I}_G{\omega }^2$
  Remember that $G$ means a vector is applied at the centre of mass

We can simplify this equation depending on the scenario:

1. Pure Translation $-$ body subject to only curvilinear or rectilinear translation (no rotational kinetic energy)
   $T=\frac{1}{2}m(v_G{)}^2$

2. Pure Rotation $-$ body rotates about fixed axis passing through $O$
   $T=\frac{1}{2}[I_G+m(r_G{)}^2]{\omega }^2=\frac{1}{2}{I}_O{\omega }^2$
   Note: $v_G=r_G\omega$

   • If rotation occurs around $G$ (i.e. $O=G$), $v_G=0$
     $\therefore T=\frac{1}{2}{I}_G{\omega }^2$

Work of a Force (Recap)

• Recall: $U_F=\int \text{F}\cdot dr={\int }_s(F\cos \theta )ds$
  • See Work of a Force
• When force is constant, equation reduces to:
  $U_{Fc}=(F_c\cos \theta )$
  $F_c\cos \theta$ is component of force acting in direction of displacement $s$
• Review: ENSC2004 - Lecture 14

Forces That Do No Work

• Reactions at fixed supports → no displacement at point of application
• Normal + friction forces acting on rolling body that does not slip → no instantaneous displacement of point of contact with ground
• Internal forces → act in equal and opposite pairs

Work of a Couple

• A couple only does work when the body undergoes rotation:
  $U_M={\int }_{{\theta }_1}^{{\theta }_2}Md\theta$
  $d\theta$ $-$ angular displacement
• If $M$ (couple moment) is constant, then:
  $U_M=M({\theta }_2-{\theta }_1)$

Work is positive is $M$ and (${\theta }_2-{\theta }_1$) are in the same direction

Principle of Work and Energy

• Recall: Sum of the initial kinetic energy and work done by all external forces and couple moments equals the body’s final kinetic energy
  $T_1+\sum U_{1-2}=T_2$

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Conservation of Energy (Rigid Bodies) - Slides

Conservation of Energy

• Conservation of energy easier to use than principle of work and energy

  • How do we tell if a force is conservative?
  • Recap: Conservative Force
  • Conservative → work done by force is independent of path

• If a rigid body is only subject to conservative forces:
  $T_1+V_1=T_2+V_2=\text{Constant}$
  i.e. as body moves from one position to another, potential energy is converted to kinetic energy and vice versa

  $T=\frac{1}{2}m(v_G{)}^2+\frac{1}{2}{I}_G{\omega }^2$
  $V=V_g+V_e$

Problems involving velocity, displacement and conservative force systems can be solved using the conservation of energy equation

Gravitational Potential Energy

• Potential energy is function of bodies’ height above/below a datum
  $V_g=W{y}_G$

  

Elastic Potential Energy

• Spring forces are also conservative
  $V_e=\frac{1}{2}k{s}^2$

  

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ENSC2004 - Lecture 21