TORQUE

1
TORQUE

• Torque, due to a force, is a measure of the
  effectiveness of the force in producing a
  rotation, a turning about some axis.
• Torque radial distance from point of force x
  the force x sin0
• Angle0 is between the radial line and the force.
• Units of torques are Nm
• Torque has conventional direction
• A) CCW is positive
• B) CW rotation is negative

2
EQUILIBRIUM

• The condition implies no net forces acting, no
  net torques acting.
• The condition says there is no acceleration, no
  change in behavior.
• The object, or system, may be sitting still OR it
  may be moving with some constant velocity.
• The vector sum of all forces acting zero
• The vector sum of all torques acting zero

3
CENTER OF GRAVITY

• It is the point where all the weight of an object
  may be considered to be concentrated
• The point of balance.
• It is the same as center of mass so long as the
  gravitational field is considered constant.

4
POSITION OF AXIS IS ARBITRARY

• If the sum of the torques is zero about one axis,
  it will be zero about ALL other axes that are
  parallel to the first.

5
THE MOMENT OF INERTIA

• Symbol I kgmm (kg times meters squared)
• Just as in inertia of an object, the moment of
  inertia is a measure of rotational inertia.
• High I means object is difficult to set into
  rotational motion. Small I, easy to rotate.
• The I is calculated by mrran object is
  really broken into all the bits of mass in that
  object and each bits distance(r squared) from
  the axis of rotation Isum of all the bits of
  mass times radius squared
• (radius is distance from axis of rotation)

6
DEFINING RADIUS OF GYRATION

• Symbol k
• k is the distance a point mass M must be from
  the axis if the point mass is to have the same
  I as the object.
• It is the radial distance from an axis at which
  the mass of an object could be concentrated
  without altering the rotational inertia of the
  object.

7
TORQUE AND ANGULAR ACCELERATION

• TORQUE I_at_ which is the TORQUE acting on an
  object that has some I value. The Torque
  produces some angular acceleration (_at_).
• Torque rFsin0 I_at_
• Torque Nm I kgmm _at_rad/s/s

8
THE KE OF ROTATION

• The KE of a rotating body is
• KE ½ I w w
• A mass with some moment of inertia(I) about a
  fixed axis that rotates with some angular
  velocity,(w)
• KE Joules, J
• KE ½ x kgmm x rad/sec x rad/sec

9
COMBINED ROTATION AND TRANSLATION

• The KE of a rolling object, like a ball, is
  really the SUM of the rotational KE plus the
  translational KE..that is rolling plus linear.
• So KE ½ I w squared 1/2Mvv.
• I is the moment of inertia about an axis
  through its mass center.

10
WORK

• Symbol W
• Measuring work done ON a rotating body during an
  angular displacement (0) by a constant torque.
• W Torque x angle 0
• Work is in Joules and 0 in radians. Torque is in
  Nm.

11
POWER

• Symbol is P
• We are measuring power transmitted TO a body by a
  torque is
• P torque x angular speed (w)
• P Nm x rad/sec
• Radian measure is used for w

12
ANGULAR IMPULSE

• An angular impulse causes a change in angular
  momentum.
• Impulse torque applied to an object for some
  time.
• Torque x time Nms
• Torque x time delta I w
• Different shapes have different Moments of
  Inertia
• See Walker page 288.

13
Compare the symbols used for linear measures and
for nonlinear measures

• Linear displacement
• Linear speed
• Linear acceleration
• Inertial mass
• Force
• Linear momentum
• Linear impulse

• Angular displacement
• Angular speed
• Angular acceleration
• Moment of inertia
• Torque
• Angular momentum
• Angular impulse

14
Write the appropriate equations for both linear
and nonlinear measures.

• LINEAR
• A) Force
• B)Work
• C)KE
• D)Power
• Impulse
• Momentum
• Velocity
• Displacement
• Acceleration

• NONLINEAR
• Force
• Work
• KE
• Power
• Impulse
• Momentum
• Velocity
• Displacement
• Acceleration

15
Moment of Inertia for some common objects

• Write out the moment of Inertia for the following
  common objects
• Hoop or hollow cylinder
• Uniform disk or cylinder
• Uniform rod
• Uniform sphere
• INCLUDE ALL YOUR SOURCES IN YOUR ANSWERS!!!!!

• Write a short, general explanation which clearly
  and ONLY answers these two questions
• 1. Why do different objects have different
  moments of inertia.
• 2. Explain the derivation of ONE of the formulas
  for ONE of the shapes.
• Ex If I 1/12MLL
• How did they arrive at that?