8-1 Torque. Torque A quantity that measures the ability of a force to rotate an object around some...
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8-1 Torque
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Torque A quantity that measures the ability of a force to rotate an object around some axis.
Greek letter ‘tau’
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Net Torque produces rotation.Just like a net force
causes motion/acceleration.
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Torque depends upon a force and the length of the
lever arm.
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How easily an object rotates depends not only on ‘how much’ force is applied, but also where the force is applied.
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Lever ArmThe perpendicular distance from the axis of rotation to a line drawn along the direction of the force.
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Torque also depends upon the angle
between a force and the lever arm.
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l
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= Fd(sin)Metric Units of torque are Newton-meters (N-m)
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Torque, like displacement and force,
is a vector quantity.Since torque is a rotational motion it has two directions:
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1.Counterclockwise (CCW), which is positive.
2.Clockwise (CW), which is negative.
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Ex 1: Sean has a flat and is changing his tire, to remove
the lug nuts he applies a force of 25 N on the tire iron at an
angle of 87o. What is the torque produced if the tire iron
is 0.6 m long?
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G: d = 0.6 m, F= 25 N, = 87o
U: = ?
E: = Fd(sin)S: =
S: =
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Ex 2: What angle produces a torque of 400 N-m, if the force applied is 505 N at a distance from the axis of rotation of 0.82 m?
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G: = 400 N-m, F = 505 N, d =0.82
m
U: = ?
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G: = 400 N-m, F = 505 N, d =0.82
m
U: = ?
E:
Fd
sin
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G: = 400 N-m, F = 505 N, d =0.82
m
U: = ?
E:
Fd
1sin
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)82.0)(505(
400sin : 1S
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)82.0)(505(
400sin : 1S
:S
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8-2 Rotation and Inertia
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Point massWhere all the mass is assumed to be located in one point.
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Center of MassThe point at which all the mass of a body can be considered to be concentrated when analyzing translational motion.
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Rotational and translational motion can
be combined.We use the the center of mass, as a reference, to analyze its translation
motion.
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Center of GravityThe position at which the gravitational force acts on an extended object as if it were a point mass.
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Toppling• If the Center of Gravity (CG) is
above the support area, then the object will remain upright.
• If the CG extends outside the support area, the object will topple.
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Unstable EquilibriumIs when any movement/
displacement of a balanced object lowers the CG.
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Stable Equilibrium
•Is when any motion/displacement of a balanced object raises its CG.
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Neutral Equilibrium•Is when any motion/displacement of a balanced object neither raises nor lowers its CG.
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Moment of Inertia (MOI)
The tendency of a body to rotate freely about a fixed axis to resist a change in rotational motion.
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**MOI is the rotational analog of mass.
**Very similar to mass, but MOI is not an intrinsic property of an object.
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**It depends upon the object’s mass and
the distribution of mass around the axis
of rotation.
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The farther the mass is, on average from the axis of rotation, the greater the
the object’s MOI and the more difficult it is to
rotate the object.
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Calculating the MOIPg 285: Table 8-1 has equations/formulas for a few common shapes.
M – mass in kilogramsR – radius in metersl – length in meters
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Units for MOI are kg-m2
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For an object to be in complete
equilibrium, requires zero net
force and zero net torque.
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If the net force is zero the object is in translational
equilibrium. (1st Condition of Equilibrium)
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If the net torque is zero, than it is in
rotational equilibrium. (2nd Condition for
Equilibrium)
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8-3 Rotational Dynamics
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Newton’s 2nd Law for Rotation
F = ma
We know in a rotational system, torque is a function of force, also:
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1. MOI replaces the mass.
2. Acceleration is replaced with angular acceleration.
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Net torque = MOI x Angular acel.
Inet
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Remember that if the net torque is zero, that the object can
still be rotating, just at a constant velocity.
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Ex 4: Simon decides to ride the ‘Gravitron’, it has a radius of 3 m, if his mass is 79.4 kg. What is the net torque produced when his angular acceleration is 7 rads/sec2?
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G: R = 3 m, M= 79.4 kg, = 7 rads/sec2
U: net = ?
E: net = I
We are going to assume he is a point mass.
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MOI, I = MR2
S:net = MR2
S:net =
S:net =
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Ex 5: In, 1995, a fully functional pencil with a mass of 24 kg and a length of 2.74 m was made. Suppose this pencil is suspended at its midpoint and a force of 1.8 N is applied perpendicular to its end, causing it to rotate. What is the angular acceleration of the pencil?
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G: M = 24kg, F = 1.8 N, l = 2.74 m
U: = ?
E: = I = / IWe need to find the values for Torque and MOI.
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Since the midpoint is the pivot point the lever arm is ½ l. Also, the force acts perpendicular, so the angle is 90 degrees.
= Fd(sin = ½Fl (sin)
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= ½(1.8)(2.74)sin90
=
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We use the formula for MOI on pg 285 for a thin rod rotating around its midpoint.
I = 1/12 Ml 2 I =I =
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S: =
S: =
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Angular Momentum (L)
The product of a rotating object’s moment of
inertia and the angular speed about the same
axis.
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L = IL = angular momentum
I = Moment of Inertia
= angular speed
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Angular Momentum Units
Kg-m2/s
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Law of Conservation of Momentum
When the net external torque acting on an object or objects
is zero, the angular momentum does not change.
Li = Lf
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Ex 6: A figure skater jumps into the air to perform a twisting maneuver. When she first jumps her moment of inertia is 86 kg-m2. While she’s in the air she brings her arms in and decreases her moment of inertia to 77 kg-m2. If her initial angular speed was 2 rads/sec what is her final angular speed, if momentum is conserved?
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G: i = 2 rads/sec, Ii = 86 kg-m2 , If = 77 kg-m2
U: f = ?Since momentum is conserved:
E: Li = Lf
Iii = Iff
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f = Iii / If
S: f =
S: f =
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Rotational Kinetic Energy (KERot)
Energy of an object due to its rotational motion.
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Calculating Rotational Kinetic Energy
KErot = ½ I2
KErot = rotational KE I = moment of inertia = angular speed
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EX 8: What is the MOI for a 0.75 kg top
spinning at an angular speed of 1.75 rads/sec, if the Rotational KE is
12 Joules?
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G: M = 0.75 kg, KErot = 12 J, = 1.75 rads/sec
U: I = ?
E: KErot = ½ I2
I = 2 KErot /2
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S: I =
S: I =
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Remember Mechanical Energy may be conserved. ME is the sum of all types of KE and PE.
ME = KEtrans + KErot + PE
ME = ½ mv2 + ½ I2 + mgh
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If ME is conserved:
MEinitial = MEfinal
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If ME is conserved:
MEinitial = MEfinal
ffrotftrans
iirotitrans
PEKEKE
PEKEKE
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