Rotational Dynamics

1
Rotational Dynamics

• Chapter 9

2
Expectations

• After Chapter 9, students will
• calculate torques produced by forces
• recognize the condition of complete equilibrium
• calculate the location of the center of gravity
  of a collection of objects
• use the rotational form of Newtons second law of
  motion to analyze physical situations
• calculate moments of inertia

3
Expectations

• After Chapter 9, students will
• calculate the rotational work done by a torque
• calculate rotational kinetic energy
• calculate angular momentum
• apply the principle of the conservation of
  angular momentum in an isolated system

4
Preliminary Definitions

• Torque
• Complete Equilibrium
• Center of Gravity

5
Torque

• Torque the rotational analog to force
• Force produces changes in linear motion (linear
  acceleration). A force is a push or a pull.
• Torque produces changes in angular motion
  (angular acceleration). A torque is a twist.

6
Torque
length of lever arm

• Mathematical definition
• 
  The lever arm is the line
• 
  through the axis of
• 
  rotation, perpendicular to
• 
  the line of action of the
• 
  force.

SI units Nm
torque
force
7
Torque

• Torque is a vector quantity. It magnitude is
  given by
• and its direction by the right-hand rule
• 
  

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Torque

• For a given force, the torque depends on the
  location of the forces application to a rigid
  object, relative to the location of the axis of
  rotation.

more torque
less torque
9
Torque

• For a given force, the torque depends on the
  forces direction.

10
Complete Equilibrium

• A rigid object is in complete equilibrium if the
  sum of the forces exerted on it is zero, and the
  sum of the torques exerted on it is zero.
• An object in complete equilibrium has zero
  translational (linear) acceleration, and zero
  angular acceleration.

11
Center of Gravity

• In analyzing the equilibrium of an object, we see
  that where a force is applied to an object
  influences the torque produced by the force.
• In particular, we sometimes need to know the
  location at which an objects weight force acts
  on it.
• Think of the object as a collection of smaller
  pieces.

12
Center of Gravity

• In Chapter 7, we calculated the location of the
  center of mass of this system of pieces
• Multiply numerator and denominator by g

13
Center of Gravity

• But Substituting
• It is intuitive that the weight force acts at the
  effective location of the mass of an object.

14
Newtons Second Law Rotational

• Consider an object, mass m, in circular motion
  with a radius r. We apply a tangential force F
• The result is a
• tangential acceleration
• according to Newtons second law.

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Newtons Second Law Rotational

• The torque produced by the force is
• But the tangential acceleration
• is related to the angular
• acceleration
• Substituting

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Newtons Second Law Rotational

• This is an interesting result.
• If we define the quantity
• as the moment of inertia,
• we have
• the rotational form of Newtons second law.

17
Moment of Inertia

• The equation
• gives the moment of inertia of a particle
  (meaning an object whose dimensions are
  negligible compared with the distance r from the
  axis of rotation).
• Scalar quantity SI units of kgm2

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Moment of Inertia

• Not many real objects can reasonably be
  approximated as particles. But they can be
  treated as systems of particles

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Moment of Inertia

• The moment of inertia of an object depends on
• the objects total mass
• the objects shape
• the location of the axis of rotation

20
Rotational Work and Energy

• By analogy with the corresponding translational
  quantities
• Translational
  Rotational

SI units Nm J
SI units (kgm2) / s2 Nm J
21
Total Mechanical Energy

• We now add a term to our idea of the total
  mechanical energy of an object

total energy
gravitational potential energy
translational kinetic energy
rotational kinetic energy
22
Angular Momentum

• By analogy with linear momentum
• Angular momentum is a vector quantity. Its
  magnitude is given by
• and its direction is the same as the direction of
  w.
• w must be expressed in rad/s.

SI units kgm2 / s
23
Angular Momentum Conservation

• If a system is isolated (no external torque acts
  on it), its angular momentum remains constant.
• If a system is isolated (no external force acts
  on it), its linear momentum remains constant.