Topic 6: Kinetics of Rigid Bodies

linear acceleration

what causes the forces in translation

angular acceleration

what causes the moments for a body in rotation

0 (moving in straight line and not rotating)

what is the angular acceleration for a particle

• ∑F = ma_G

  • a_G - acceleration of centre of mass

what is the force equation of motion for a rigid body

• ∑M_G = I_Gα

  • M_G - resultant moment of all external forces and torques acting about the c.o.m

  • I_G - mass moment of inertia about c.o.m

  • α - angular acceleration of the body

what is the moment equation of motion

1. Identify provided information and what is required

2. Draw a FBD

   • forces, moments, lines of action

3. choose a coordinate system

   • include direction of positive rotation

4. state any kinematic constraints

   • any relationship between translation and rotational motion

   • e.g. wheel rolling without slip

5. write equations of motion

   • force-acceleration: ∑F = ma_G

   • use chosen coordinate system and resolve into components

   • moment-angular acceleration:

     • ∑M_G = I_Gα (about c.o.m)

     • ∑M₀ = I₀α (only about a fixed axis)

6. solve for the required information

7. check that your answers make sense

what are the steps involved in solving kinetics of RB problems

1. isolate the body whose forces or motion you are interested in

2. draw the body’s outline (small details not required). include centre of mass (c.o.m) and important dimensions if necessary

3. identify all forces on the body, add these vectors (correct scale not necessary)

   • types of forces include:

     • externally applied forces

     • gravity

     • normal reactions

     • friction reactions

   • for rigid bodies, the line of action is important

how to draw FBDs

0 (as the body does not rotate and all points on the body have the same acceleration)

what is the angular velocity and angular acceleration of a rigid body in translation

• rectilinear translation

  • ∑F_x = ma_x

  • ∑F_y = ma_y

  • ∑M_G = 0

• curvilinear translation

  • ∑F_n = ma_n

  • ∑F_t = ma_t

  • ∑M_G = 0

what are the equations of motion for a body in rectilinear and curvilinear translation

• a_n = rω²

• a_t = rα

  • r is the distance between the point around rotation is occurring and the centre of mass

what equations are used to calculate the components of acceleration for a rigid body in fixed axis rotation

• ∑M_G = I_Gα

  or:

• ∑M₀ = I₀α

what are the moment equations of motion that can be used for a rigid body in fixed axis rotation

how difficult is to rotate a rigid body about an axis - this is the rotational analogy to mass

what does mass moment of inertia represent

• I_P = I_G + md²

  • I_P - mass moment of inertia about a point

  • I_G - mass moment of inertia about c.o.m

  • m - mass of object

  • d - distance between two lines of axis

what is the equation for the parallel axis theorem

• I_P = mk²

  • where k is the radius of gyration

what is the equation for the mass moment of inertia in terms of the radius of gyration