Physics 102: Mechanics Lecture 12

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Physics 102 Mechanics Lecture 12

• Wenda Cao
• NJIT Physics Department

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Rotational Kinetic Energy and Angular Momentum

• Moment of Inertia
• Torque
• Newton 2nd Law for Rotational Motion Torque and
  angular acceleration
• Rotational Kinetic Energy
• Rotational Energy Conservation
• Angular Momentum
• Angular Momentum Conservation

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Moment of Inertia of Point Mass

• For a composite particle, the definition of
  moment of inertia is
• mi is the mass of the ith single particle
• ri is the rotational radius of ith particle
• SI units of moment of inertia are kg.m2
• Moment of inertia and mass of an object are
  different quantities
• It depends on both the quantity of matter and its
  distribution (through the r2 term)

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Moment of Inertia of Extended Objects

• Divided the extended objects into many small
  volume elements, each of mass Dmi
• We can rewrite the expression for I in terms of
  Dm
• With the small volume segment assumption,
• If r is constant, the integral can be evaluated
  with known geometry, otherwise its variation with
  position must be known

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Moment of Inertia for some other common shapes
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General Definition of Torque

• Let F be a force acting on an object, and let r
  be a position vector from a rotational center to
  the point of application of the force. The
  magnitude of the torque is given by
• ? 0 or ? 180
• torque are equal to zero
• ? 90 or ? 270 magnitude of torque attain
  to the maximum

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Torque Units and Direction

• The SI units of torque are N.m
• Torque is a vector quantity
• Torque magnitude is given by
• Torque will have direction
• If the turning tendency of the force is
  counterclockwise, the torque will be positive
• If the turning tendency is clockwise, the torque
  will be negative

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Net Torque

• The force will tend to cause a
  counterclockwise rotation about O
• The force will tend to cause a clockwise
  rotation about O
• St t1 t2 F1d1 F2d2
• If St ? 0, starts rotating
• If St 0, rotation rate does not change

• Rate of rotation of an object does not change,
  unless the object is acted on by a net torque

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Newtons Second Law for a Rotating Object

• When a rigid object is subject to a net torque
  (?0), it undergoes an angular acceleration
• The angular acceleration is directly proportional
  to the net torque
• The angular acceleration is inversely
  proportional to the moment of inertia of the
  object
• The relationship is analogous to

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Rotational Kinetic Energy

• An object rotating about z axis with an angular
  speed, ?, has rotational kinetic energy
• Each particle has a kinetic energy of
• Ki ½ mivi2
• Since the tangential velocity depends on the
  distance, r, from the axis of rotation, we can
  substitute
• vi wri

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Rotational Kinetic Energy, cont

• The total rotational kinetic energy of the rigid
  object is the sum of the energies of all its
  particles
• Where I is called the moment of inertia

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Rotational Kinetic Energy, final

• There is an analogy between the kinetic energies
  associated with linear motion (K ½ mv 2) and
  the kinetic energy associated with rotational
  motion (KR ½ Iw2)
• Rotational kinetic energy is not a new type of
  energy, the form is different because it is
  applied to a rotating object
• Units of rotational kinetic energy are Joules (J)

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Extended Work-Energy Theorem

• The work-energy theorem tells us
• When Wnc 0,
• The total mechanical energy is conserved and
  remains the same at all times
• Remember, this is for conservative forces, no
  dissipative forces such as friction can be present

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Total Energy of a System

• A ball is rolling down a ramp
• Described by three types of energy
• Gravitational potential energy
• Translational kinetic energy
• Rotational kinetic energy
• Total energy of a system

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Conservation of Mechanical Energy

• Conservation of Mechanical Energy
• Remember, this is for conservative forces, no
  dissipative forces such as friction can be
  present

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Work-Energy in a Rotating System

• In the case where there are dissipative forces
  such as friction, use the generalized Work-Energy
  Theorem instead of Conservation of Energy
• If we only consider friction,

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Problem Solving Hints

• Choose two points of interest
• One where all the necessary information is given
• The other where information is desired
• Identify the conservative and non-conservative
  forces
• Write the general equation for the Work-Energy
  theorem if there are non-conservative forces
• Use Conservation of Energy if there are no
  non-conservative forces
• Use v w to combine terms
• Solve for the unknown

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A Ball Rolling Down an Incline

• A ball of mass M and radius R starts from rest at
  a height of 2.0 m and rolls down a 30? slope,
  what is the linear speed of the bass when it
  leaves the incline? Assume that the ball rolls
  without slipping.

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Blocks and Pulley

• Two blocks with masses m1 5 kg and m2 7 kg
  are attached by a string over a pulley with mass
  M 2kg. The pulley, which truns on a
  frictionless axle, is a hollow cylinder with
  radius 0.05 m over which the string moves without
  slipping. The horizontal surface has coefficient
  of kinetic friction 0.35. Find the speed of the
  system when the block of mass m2 has dropped 2 m.

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General Problem Solving Hints

• The same basic techniques that were used in
  linear motion can be applied to rotational
  motion.
• F becomes ?
• m becomes I
• a becomes ?
• v becomes ?
• x becomes ?

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Angular Momentum

• Similarly to the relationship between force and
  momentum in a linear system, we can show the
  relationship between torque and angular momentum
• Linear momentum is defined as
• Angular momentum is defined as

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Angular Momentum and Torque

• Net torque acting on an object is equal to the
  time rate of change of the objects angular
  momentum
• Angular momentum is defined as
• Analog in impulse

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Angular Momentum Conservation

• If the net torque is zero, the angular momentum
  remains constant
• Conservation of Angular Momentum states The
  angular momentum of a system is conserved when
  the net external torque acting on the systems is
  zero.
• That is, when
• then

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The Spinning Stool

• A student sits on a pivoted stool while holding a
  pair of weights. The stool is free to rotate
  about a vertical axis with negligible friction.
  The moment of inertia is 2.25 kgm2. The student
  is set in rotation with arms outstretched, making
  one complete turn every 1.26 s, arms
  outstretched. (a) What is the initial angular
  speed of the system? (b) As he rotates, he pulls
  the weights inward so that the new moment of
  inertia becomes 1.8 kgm2. What is the new angular
  speed of the system?