Rotational Motion: Dynamics

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Rotational MotionDynamics

• Torque
• Moment of inertia
• Newtons second law

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Reading Question
The most important relationship in this chapter
is This relationship can be derived from

• 1. This is a new law.
• 2. Conservation of energy.
• Newtons second law.
• Impulse-Momentum theorem.
• None of the above.

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Reading Question
The most important relationship in this chapter
is This relationship can be derived from

• 1. This is a new law.
• 2. Conservation of energy.
• Newtons second law.
• Impulse-Momentum theorem.
• None of the above.

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Reading Question
A new way of multiplying two vectors is
introduced in this chapter. What is it called?
1. Dot Product 2. Scalar Product 3. Tensor
Product 4. Cross Product 5. Angular Product
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Reading Question
A new way of multiplying two vectors is
introduced in this chapter. What is it called?
1. Dot Product 2. Scalar Product 3. Tensor
Product 4. Cross Product 5. Angular Product
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Reading Question
Moment of inertia is
1. the rotational equivalent of mass. 2. the
point at which all forces appear to act. 3. the
time at which inertia occurs. 4. an alternative
term for moment arm.
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Reading Question
Moment of inertia is
1. the rotational equivalent of mass. 2. the
point at which all forces appear to act. 3. the
time at which inertia occurs. 4. an alternative
term for moment arm.
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Reading Question
A rigid body is in equilibrium if
1. 2. 3. neither 1 nor 2. 4. either 1 or 2. 5.
both 1 and 2.
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Reading Question
A rigid body is in equilibrium if
1. 2. 3. neither 1 nor 2. 4. either 1 or 2. 5.
both 1 and 2.
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Rotational Motion Dynamics
Important concept
Linear motion and the motion about the CM
(rotation) are independent. This means we can
solve each alone with out concern about the other
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Rotational Motion Dynamics
Torque - a measure of the effectiveness of a
force to produce a change in rotational motion
If you want to tighten a nut, where do you place
your hand?
What angle does your hand make to the wrench?
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Rotational Motion Dynamics
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Rotational Motion Dynamics
Vector cross product
The cross product of two vectors A, B is written
and is a vector with magnitude
Q is the angle between A and B, and the direction
is given by the right hand rule.
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Rotational Motion Dynamics
Torque as a vector cross product
Magnitude
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Rotational Motion Dynamics
Right-hand rule
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Rotational Motion Dynamics
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Rotational Motion Dynamics
--

0
--

0
t5 gt t1 t2 gt t3 t4 gt t6
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Rotational Motion Dynamics
You are tightening your cars wheel nut after
changing a flat tire. The instructions specify a
tighten torque of 95 N-m to ensure the nut want
come loose. If your 45-cm-long wrench makes a 67
degrees angle with the horizontal, with what
force must you pull horizontally to produce the
required torque?
Right-hand rule for direction
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Class Question
Rank in order, from largest to smallest, the five
torques The rods all have the same length and
are pivoted at the dot.
1. 2. 3. 4. 5.
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Class Question
Rank in order, from largest to smallest, the five
torques The rods all have the same length and
are pivoted at the dot.
1. 2. 3. 4. 5.
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Rotational Motion Dynamics

• Moment of inertia

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Rotational Motion Dynamics
The moment of inertia depends on how the mass is
distributed.
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Rotational Motion Dynamics

• Moment of inertia for a continuous body

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Rotational Motion Dynamics

• Moment of inertia for a continuous body

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Rotational Motion Dynamics

• Moment of inertia for a continuous body

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Rotational Motion Dynamics
Examples Moments of Inertia
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Rotational Motion Dynamics
negative
Gravity or not?
The angular velocity is constant if the net
torque on the ball is zero. The torque due to
the rod is zero. There is a if in gravity and
thus the angular velocity is increasing.
Zero or negative
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Rotational Motion Dynamics
4 gt 3 2 gt 1
4 3 gt 2 1
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Class Question
Four Ts are made from two identical rods of equal
mass and length. Rank in order, from largest to
smallest, the moments of inertia Ia to Id for
rotation about the dotted line.
1. Ic gt Ib gt Id gt Ia 2. Ic Id gt Ia Ib 3.
Ia Ib gt Ic Id 4. Ia gt Id gt Ib gt Ic 5. Ia gt
Ib gt Id gt Ic
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Class Question
Four Ts are made from two identical rods of equal
mass and length. Rank in order, from largest to
smallest, the moments of inertia Ia to Id for
rotation about the dotted line.
1. Ic gt Ib gt Id gt Ia 2. Ic Id gt Ia Ib 3.
Ia Ib gt Ic Id 4. Ia gt Id gt Ib gt Ic 5. Ia gt
Ib gt Id gt Ic
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Rotational Motion Dynamics
Important concept
Linear motion and motion about the CM (rotation)
are independent. This means we can solve each
alone with out concern about the other
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Rotational Motion Dynamics
Example 13.13 Lowering a block A 2.0 kg block
is attached to a massless string that is wrapped
around a 1.0 kg, 4.0-cm-diameter cylinder as
shown. The cylinder rotates on an axle through
the center. The block is released from rest 1.0
m above the floor. How long does it take to
reach the floor?
Model
Dont forget to use the Dynamical Worksheets in
your Workbook
Visualize Free-body diagram
Solve
Three unknowns and two equations
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Rotational Motion Dynamics
Example 13.13 Lowering a block A 2.0 kg block
is attached to a massless string that is wrapped
around a 1.0 kg, 4.0-cm-diameter cylinder as
shown. The cylinder rotates on an axle through
the center. The block is released from rest 1.0
m above the floor. How long does it take to
reach the floor?
Model
Visualize Free-body diagram
Solve