Download center of gravity

Today: (Ch. 8)
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Rotational Motion
Center of Gravity
• For the purposes of calculating the torque due to
the gravitational force, you can assume all the
force acts at a single location
• The location is called the center of gravity of the
object
– The center of gravity and the center of mass of an
object are usually the same point
Rotational Equilibrium
• Equilibrium may include rotational equilibrium
• An object can be in equilibrium with regard to both
its translation and its rotational motion
• Its linear acceleration must be zero and its angular
acceleration must be zero
• The total force being zero is not sufficient to ensure
both accelerations are zero
Equilibrium Example
• The applied forces are equal in
magnitude, but opposite in direction
• Therefore, ΣF = 0
• However, the object is not in equilibrium
• The forces produce a net torque on
the object
• There will be an angular acceleration
in the clockwise direction
• For an object to be in complete equilibrium, the angular
acceleration is required to be zero
• Στ = 0
• This is a necessary condition for rotational equilibrium
– All the torques will be considered to refer to a single axis of rotation
– The same ideas can also be applied to multiple axes
Rotational Equilibrium, Lever
• Use rotational equilibrium to
find the force needed to just
lift the rock
– We can assume that the
acceleration is zero
– Also ignore the mass of the
lever
• The force exerted by the
person can be less than the
weight of the rock
• The lever will amplify the
force exerted by the person
– If Lperson > Lrock
Tipping a Crate
• We can calculate the force
that will just cause the crate
to tip
• When on the verge of
tipping, static equilibrium
applies
• If the person can exert
about half the weight of the
crate, it will tip
Moment of Inertia
• The moment of inertia composed of many pieces of mass is
I   mi ri 2
i
• The moment of inertia of an object depends on its mass
and on how this mass is distributed with respect to the
rotation axis
• The definition can be applied to find the moment of inertia
of various objects for any rotational axis
• Units of moment of inertia are kg · m2
Various Moments of Inertia
Rotational Dynamics
• Newton’s Second Law for a rotating system states
Στ = Iα
• Once the total torque and moment of inertia are found,
the angular acceleration can be calculated
• Then rotational motion equations can be applied
• For constant angular acceleration:
  o   t
1 2
  o  o t   t
2
 2  o2  2    o 
  Final Angle
 0  Initial Angle
  Final Angular Velocity
0  Initial Angular Velocity
  Angular Accelerati on
t  Time
Kinematic Relationships
Real Pulley with Mass, Example
• Up to now, we have assumed a massless
pulley
• Using rotational dynamics, we can deal with
real pulleys
• The torque on the pulley is due to the
tension in the rope
• Apply Newton’s Second Laws for
translational motion and for rotational motion
• The crate undergoes translational motion
• The pulley undergoes rotational motion
• For the pulley:
• The tension in the rope supplies the torque
• The pulley rotates around its center, so that is a logical axis of
rotation
Motion of a Crate, Example, cont
• Pulley equation
– Στ = - T R = Ipulley α
– The pulley is a disc, so I = ½ mpulley R²pulley
• For the crate
– Take the +y direction as +
– Equation: ΣF = T – mcrate g = mcrate a
• Relating the accelerations
– a = α Rpulley
• Combine the equations and solve
Example
If the mass of a wheel is increased by a factor of 17 and the
radius is increased by a factor of 9, by what factor is the
moment of inertia increased?
A factor of
Example
If the mass and height of the object is increased by twice,
what would be the increase/decrease of final potential
energy?
Example
If final velocity is increased by 3 times, what would be
increase/decrease in final kinetic energy?
Tomorrow: (Ch. 9)
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Energy and Momentum of Rotational Motion