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<p><strong>Chapter 10: Fixed-Axis Rotation</strong></p>

<p><br />

<p><strong>Introduction:</strong></p>

<p>-Give an example of the use of rotaional motion in our Daily Life. <span
id="ans">Wind turbine to produce electricity </span></p>

<p>-What is rigid body? <span id="ans">A body which does not deform as it
moves</span>.</p>

<p></p>

<p><strong>1.1 Rotational Variables:</strong></p>

<p></p>

<p><strong>Angular Velocity:</strong></p>

<p>-What is the angular position of a particle in rotational motion? What are
its unit and dimension? <span id="ans">Angle swept by the particle about the
center of rotation. Radian (rad). Dimensionless</span></p>

<p>-How is angle related to the radius of the circle and arc length? <span
id="ans">&theta; = Arc length (s)/Radius (r).</span></p>

<p>-How are the degree and radian related? <span id="ans">2&pi; rad = 180
deg.</span></p>

<p>-How are the angle, angular position and arc length vectors related to each
other? <span id="ans">Equation 10.2</span></p>

<p>-What is angular veocity? What is its unit? <span id="ans">Time rate of
change of the angle. rad/sec.</span></p>

<p>-How is instantaneous angular velocity calculated? <span id="ans">Time
derivative of angle.</span></p>

<p>-How are the revolution and angle related? <span id="ans">1 revolution =
2&pi; rad </span></p>

<p>-How is the direction of angle determined? <span id="ans">Counterclockwise;
+ve and Clockwise;-ve</span></p>

<p>-What is tangential speed and how is it related to the angular velocity?
<span id="ans">Rate of change of arc-length. v<sub>t</sub>=r&omega;</span></p>

<p>-Notice how the tangential speed of the particles varies with the distance
from axis of rotation in Fig. 10.4.</p>

<p>-Look how the direction of the angular velocity is determined using the
right-hand rule in Fig. 10.5.</p>

<p>-How are the tanfential velocity, angular velocity and position vector
related?<span id="ans">Equation 10.5.</span></p>

<p></p>

<p><strong>Angular Acceleration:</strong></p>

<p>-What is instantaneous angular acceleration? <span id="ans">Derivative of
angular velocity with respect to time.</span></p>

<p>-What is the unit of angular acceleration? <span
id="ans">rad/s<sup>2</sup></span></p>

<p>-Look how the direction of the angular acceleration is along or opposite to
that of angular velocity in Fig. 10.7.</p>

<p>-Problem-Solving Strategy for Rotational kinematics. First, a rough sketch
is very helpful. Others is same as you do for translational motion. i.e. list
the known varibles and search for the correct equation to solve for unkown.
Don't forget to convert revolution to angle.</p>

<p>-Notice how the rpm (revolution per minute) is converted to rad/sec before
finding out the desired variables in Equation 10.2.</p>

<p>-Notice how the direction of the angular velocity of wind turbine is found
using Right-hand rule and how the direction of angular speed is determined with
the help of change in angular velocity in Example 10.3. Also, look how the
average angular acceleration and instantaneous velocity are found from the
angular velcoity function.</p>

<p></p>

<p></p>

<p><strong>10.2 Rotation with Constant Angular Acceleration:</strong></p>

<p><ins>-What are the variables used to analyze rotational motion for a rigid
body about a fixed axis under a constat angular acceleration? <span
id="ans">Angular dispalcement, Angular velocity and Angular
acceleration.</span></ins></p>

<p>-How is the averae angular velocity related to initial and final angular
velocity if the angular acceleartion is constant? <span id="ans">Half of sum of
inital and final values. Eqn. 10.9</span></p>

<p>-Notice how the final angular position at a time "<em>t</em>" is related to
initial angular position and angular velocity similar to in the translational
case in Equation 10.10.</p>

<p>-Look how the relation between the final angular velocity, intial angular
velocity, angular acceleration and time is obtained in Eqn 10.11 from the
defination of angular accleration.</p>

<p>-Look how the defination of angular velocity is used to find the expression
of angular position in terms of angular velocity and angular acceleration in
Eqn. 10.12.</p>

<p>-Kinematic equations for rotational motion with constant acceleration: Eqn.
10.9 to 10.13.</p>

<p>-Look how the general strategy of listing knowns and finding out the
unknowns from the kinematic equations (as we did for translational case) in
Example 10.4.</p>

<p>-In Example 10.6, an alternative approach is used to find the the angular
displacement by integrating the area under the curve of angular velocity vs.
time graph.</p>

<p></p>

<p></p>

<p><strong>10.3 Relating Angular and Translational Quantities:</strong></p>

<p><ins>-What are the rotational counter parts of position, velocity and
acceleration in translational motion? <span id="ans">Angluar position, angluar
velocity, angular acceleration.</span></ins></p>

<p>-How does the centripital acceleration arise?<span id="ans">Due to change in
direction of tangential velocity.</span></p>

<p>-How does the tangential acceleration arise? <span id="ans">Due to change in
magnitude of tangential velocity.</span></p>

<p>-Total linear acceleration vector is vector sum of tangential and
centripital acceleration.</p>

<p>-What is the angle between the tangential and centripital acceleration?
<span id="ans">Perpendicular. </span></p>

<p>-Look at the comparision of the rotional kinemetic equations with their
translational counter-part in Table 10.2.</p>

<p>-Look at the rotational (circular in this case) variables, their linear
counter parts and relationship between them in Table 10.3</p>

<p>-Look how the angle for the decelerating rotational motion comes out to be
negative in Example 10.7. This -ve sign indicates the total acceleation vector
is angled towards the opposite direction of the rotation.</p>

<p></p>

<p></p>

<p><strong>10.4 Moment of Inertial and Rotational Kinetic Energy:</strong></p>

<p></p>

<p><strong>Rotational Kinetic Energy:</strong></p>

<p>-Why is rotational kinetic energy expressed in terms of angular velocity but
not velocity? <span id="ans">Velocity is different for every point on rotating
body about axis.</span></p>

<p>-How is the rotatinal kinetic energy of a rigid body related to the
individual masses and their angular velocity? <span id="ans">Equation
10.16.</span></p>

<p></p>

<p><strong>Momentum of Inertia:</strong></p>

<p>-what is the rotatinal counter part of mass? <span id="ans">Moment of
inertia.</span></p>

<p>-How is the moment of inertial defined and what is its unit? <span
id="ans">Equation (10.17). kg.m<sup>2 </sup></span></p>

<p>-What is quantitative measure of the rotational inertia? <span
id="ans">Moment of inertia</span></p>

<p>-What does it signify?<span id="ans">Greater the moment of inertia, greater
is its resistance to change in angular velocity about a fixed axis of
rotaion.</span></p>

<p>-Why does the hollow cylinder has more rotational inertial than a solid
cylinder of same mass when rotating about an axis through center? <span
id="ans">Eqn. (10.17) , more the mass concentrated at greater distance from
axis of rotation, greater the moment of inertia.</span></p>

<p>-How do you express the rotational kinetic energy in terms of moment of
inertia and the angular velocity? <span id="ans">Eqn. (10.8).</span></p>

<p>-From Table 10.3, shows the rotational and translational analogue of the
inertia and kinetic energy.</p>

<p>-In Fig 10.20., Learn the moemtum of inertia for common shapes of
objects.</p>

<p></p>

<p><strong>Applying Rotational Kinetic Energy :</strong></p>

<p>-Problem-Solving Strategy for Rotatinal Energy. Find out the energy and work
involved. Use the conservation of mechanical energy (i.e. rotational KE +
translational KE + PE) if there is no losses of enrgy to due any
non-conservative forces (eg. friction) and solve for the desired unknown.</p>

<p>-Look at the Example 10.10, part b). Along with above stragtegy, the fact
that the horizontal component of velocity remains constat is used to find the
final height of Boomerang.</p>

<p></p>

<p></p>

<p><strong>10.5 Calculating Moments of Inertia</strong><strong>:</strong></p>

<p>-In Fig. 10.23, how would you conclude that it is twice as hard to rotate
barbell about the end than about its center? <span id="ans">The momentum
inertia is double in case of axis at end</span></p>

<p>-How can the moment of inertia for the not-point like masses calculated?
<span id="ans">Eqn. (10.19)</span></p>

<p>-Look how the moment of inertia of the uniform rod is calculated with axis
through center of mass and through end point.</p>

<p>-While finding the moment of inertia for the uniform thin rod, why do we
need to find a way to relate mass to spatial variable? <span id="ans">Since
intergration is over the mass distribution.</span></p>

<p>-Why is the moment of inertia of a uniform thin rod along end point is
greater than that along center of mass? <span id="ans">More mass is distributed
farther from the rotation axis.</span></p>

<p>-What is parallel axis theorem? <span id="ans">Eqn (10.20).</span></p>

<p>-Why is it needed? <span id="ans">To make it easy to find moment of inertia
through the axis parallel to axis through center of mass.</span></p>

<p>-Look how the moment of inertia of uniform thin disk about an axis through
the center is found.</p>

<p>-How can you calculate the moment of inertia of compound objects about a
common axis? <span id="ans">Sum of the each part of the object. Eqn.
(10.21)</span></p>

<p>-Look how the parallel-axis theorem is used to find the moment of inertia of
the compound objects in Example 10.12.</p>

<p>-Look how the conservation of mechanical energy is used to find the angular
velocity of the pendulum. Also, note that the mass thought to be concentrated
at center mass and the gravitaional potential energy is taken from that
reference.</p>

<p></p>

<p></p>

<p><strong>10.6 Torque:</strong></p>

<p>-What is required to describe the dynamics of the rotational motion? <span
id="ans">Torque.</span></p>

<p>-What is the rotational counter part of force? <span
id="ans">Torque.</span></p>

<p>-In Fig. (10.31), look how the magnitude &amp; direction of force and its
distance from the axis of rotation affects when you try to open the door.</p>

<p><span id="prob">-Note: The length of the red arrow represents the magnitude
of the Force in Fig. (10.31).</span></p>

<p>-How is torque related to the force applied and the position of force from
the axis of rotation? <span id="ans">Equation 10.22</span></p>

<p>-Look how the direction of torque is found using Right hand thumb rule in
the Fig. (10.32) .</p>

<p>-What is the SI unit of torque?<span id="ans">N.m</span></p>

<p>-What is lever arm? <span id="ans">perpendicular distance from axis of
rotation to line of action of force.</span></p>

<p>-How is direction of torque related to that of angular acceleration?<span
id="ans">Same. i.e. +ve for counterclockwise and -ve for clockwise.</span></p>

<p>-How is net torque is found from individual torque about a axis? <span
id="ans">Sum of individual torque. Equation 10.24.</span></p>

<p>-<span id="prob">Note: In Equation 10.24, the RHS has the sum of the
MAGNITUDE of the individual torques and the paragraph right above the equations
explains that the magnitude can be positive or -ve depending upon direction.
This might be confusing. Also, there is no sign of MAGNITUDE in LHS.</span></p>

<p>-Problem-Solving Strategy for Finding Net Torque. First, choose co-ordinate
system and find axis of rotation. Then find lever arm for individual force and
find out if they produce counter-clockwise or clockwise motion about the axis.
Add or substract the torque from the forces accordingly.</p>

<p>-Look how the above strategy is used to solve for net torque in Example
10.15. Especially, look how the torque due to force acting along the axis of
rotation vanishes since the lever arm is zero. Also, see how to calculate the
lever arm of a force acting not perpendicular to the axis of rotation.</p>

<p></p>

<p></p>

<p><strong>10.7 Newton's Second Law for Rotation:</strong></p>

<p>-What is the Newton's second law for rotation? <span id="ans">Total torque
is equal to the moment of inertia times angular acceleration. Eqn.
(10.25).</span></p>

<p>-What is equation for rotational dynamics? <span id="ans">Eqn.
(10.25)</span></p>

<p>-Look how the Newton's second law for rotation in vector form (Eqn. 10.26)
be derived from second law for translatinal motion.</p>

<p>-Problem-Solving Strategy for Rotational Dynamics. First, draw free body
diagram levelling all the external forces. Then, identify the axis of rotation
and apply the Newton's second law for rotation.</p>

<p>-Notice how the above strategy is used to find the angular acceleration of
the merry-go-round. Also, notice how the momentum inertia increases when there
is a child at a distance which inturn decreases the angular acceleration.</p>

<p></p>

<p></p>

<p><strong>10.8 Work and Power for Rotational Motion:</strong></p>

<p>-What is the work done on a rigid body rotating about a fixed axis? <span
id="ans">Sum of the toruques integrated over the angle through which the body
rotates. Eqn. (10.27) or (10.30).</span></p>

<p>-Problem-Solving strategy for work energy theorem for rotational motion.
First, find toruque due to the individual forces, calculate work done on body.
Then, use work-energy theorem.</p>

<p>-See how this strategy is used to find the angular velocity in the pullly in
Example 10.18. Also, see how the weight of the pully and normal force on the
bearing does not do any work since the they acts through the axis of
rotation.</p>

<p>-What is the power for the rotational motion? Torque time the angular
velocity. Eqn. (10.31)</p>

<p><span id="prob">-Error: While deriving the expression for the power, the
Eqn. (10.27) for work done is simplified NOT Eqn.(10.25).</span></p>

<p>-Look at the summary tables (Table 10.5, 10.6 and 10.7) showing rotational
and translatinal relations.</p>

<p></p>

<p></p>

<p><strong></strong></p>
</p>
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