Handout 4

• What is the main focus of the study material on kinematics of rigid bodies?


  The dynamics of rigid bodies that do not deform under applied forces
• What does it mean for a body to be considered rigid in mechanics?


  Distances between points within the body remain constant
• In what scenarios might we still assume a body is rigid despite some deformation?


  In cases like aircraft wings that deflect under loads
• What type of kinematics and dynamics does this unit focus on?


  Planar kinematics and dynamics of rigid bodies
• What are the two components of general plane motion for rigid bodies?

  Translation and rotation
• How is angular position represented in the study material?


  By the symbol θ with respect to a reference axis
• What do the time derivatives of angular position represent?


  Angular velocity ω and angular acceleration α
• How is angular velocity ω mathematically defined?


  ω = $\frac{d\theta}{dt} =$$\dot{\theta}$
• How is angular acceleration α mathematically defined?


  α = $\frac{d\omega}{dt} =$$\ddot{\theta}$
• What is the relationship between angular acceleration α and angular velocity ω?


  α = $\frac{d\omega}{d\theta} \cdot \frac{d\theta}{dt} =$$\frac{d\omega}{d\theta} \cdot \omega$
• What does the choice of reference axis affect in angular motion?


  It adds a constant offset β to the angular position but does not affect angular velocity and acceleration
• What is the formula for the velocity v of a point A on a rigid body rotating around a fixed point O?


  v = ωr
• What is the formula for the normal acceleration an of point A?


  an = ω²r
• What is the formula for the tangential acceleration at of point A?


  at = αr
• How can the equations for velocity and acceleration be expressed in vector form?


  Using the angular velocity vector ω and position vector r
• What is the vector expression for the velocity of point A?


  v = ω × r
• What is the significance of the order of the cross product in the velocity equation?


  It must be applied correctly as r × ω = −v
• What does the acceleration of point A consist of in vector form?


  a = ˙v = ω × ˙r + ˙ω × r
• How is the normal acceleration an expressed in vector form?


  an = ω × (ω × r)
• How is the tangential acceleration at expressed in vector form?


  at = α × r
• What is the general approach to describe the motion of a rigid body in absolute terms?

  Express the configuration in geometric relations and differentiate with respect to time
• For pure rolling of a wheel, what is the relationship between distance traveled s and rotation θ?


  s = θR
• How are the velocity and acceleration of a wheel found during pure rolling?


  By differentiating distance with respect to time
• What are the Cartesian coordinates of point A fixed to the wheel rim during rolling?


  x = R(θ - sin θ), y = R(1 - cos θ)
• How do you find the velocities of point A using the chain rule?


  ˙x = R˙θ(1 - cos θ), ˙y = R˙θsin θ
• How do you find the accelerations of point A?


  ¨x = R¨θ(1 - cos θ) + R˙θ²sin θ, ¨y = R¨θsin θ + R˙θ²cos θ
• What is the velocity of point A on the wheel expressed in terms of the center of the wheel?


  vA = vO + vA/O
• How is the position vector rA/O defined for point A on the wheel?


  rA/O = −R sin θi − R cos θj
• What is the significance of the point of zero velocity in the context of a rolling wheel?


  It is the contact point with the ground where the wheel has no velocity
• What happens to the velocity at the top of a purely rolling wheel?


  The velocity is v = 2ωR, which is double that of the center of the wheel
• What is the role of the slider-crank mechanism in mechanical systems?


  It converts rotational motion into linear reciprocating motion
• How is the point of zero velocity determined in a slider-crank mechanism?


  At the intersection of the normals to the velocity vectors of points A and B
• What is the relationship between the angular velocity ωAB and the velocities of points A and B?


  ωAB = $\frac{vA}{rA} =$$\frac{\omega R}{rA}$
• How is the velocity vB of the slider related to the angular velocity ωAB?


  vB = ωAB rB = $\frac{rB}{rA} \omega R$
• What is the significance of the instantaneous center of rotation in the slider-crank mechanism?


  It varies continuously as the mechanism moves
• What happens to the velocities when OA and AB are collinear in the slider-crank mechanism?


  The velocity vB can be determined at that instant
• When does the angular velocity ωAB change direction in the slider-crank mechanism?


  When the crank OA and connecting member AB are not aligned
• What are the key equations for planar kinematics of rigid bodies?


  • Angular position: θ
  • Angular velocity: ω = $\frac{d\theta}{dt}$
  • Angular acceleration: α = $\frac{d\omega}{dt}$
  • Velocity: v = ωr
  • Normal acceleration: an = ω²r
  • Tangential acceleration: at = αr
• What are the steps to analyze the motion of a rolling wheel?


  1. Define the relationship between distance and rotation: s = θR
  2. Differentiate to find velocity: v = ωR
  3. Differentiate again to find acceleration: a = αR
  4. Express Cartesian coordinates: x = R(θ - sin θ), y = R(1 - cos θ)
  5. Use chain rule for velocities and accelerations
• What are the components of the acceleration vector for point A on a rigid body?


  • Normal acceleration: an = ω × (ω × r)
  • Tangential acceleration: at = α × r