6.1R Rotational Kinematics
Ellen Akers
Radians and Degrees
• In degrees, once around a circle is 360˚
• In radians, once around a circle is 2π
• A radian measures a distance around an arc equal to the
length of the arc’s radius
• ∆s = C = 2πr= 2π radians
• You can use this as a conversion factor to move back and forth
between degrees and radians
Example
• Convert 90 ̊ to radians
• Convert 6 radians to degrees
Revolutions
• Angles are also measured in terms of
revolutions (complete trips around a circle). A
complete single rotation is equal to 360˚;
therefore you can write the conversion factors
for rotational distances and displacements as
• 360˚ = 2π radians = 1 revolution
Example 2
• Convert 1.5 revolutions to both radians and
degrees
Rotational Kinematics
• Rotational kinematics is similar to translational
kinematics. All you have to do is learn the
rotational versions of the kinematic variables
and equations.
• In translational kinematics you learned that
velocity = ∆x
t
Linear vs. Angular Displacement
• Linear
position/displacement is
given by ∆x.
• Angular position or
displacement is given by
∆Θ.
• S=rΘ
• ∆s = r∆Θ
Linear vs. Angular Velocity
• Linear speed / velocity
is given by v in units of
meters per second
• Angular speed / velocity
is given by ω in units of
radians/sec
• v = ∆x ω = ∆Θ
t
∆t
Converting Linear to Angular Velocity
• V = ωr (linear or
translational velocity)
• ω = v/r
Example –Angular Velocity of the Earth
• Find the magnitude of Earth’s angular velocity
in radians per second.
Linear vs. Angular Acceleration
•
•
•
•
Linear acceleration is given by a
Angular acceleration is given by α
a = v/t
α = ω/t
Example 2
•
A frog rides a unicycle. If the unicycle wheel begins at rest, and accelerates
uniformly in a counter-clockwise motion to an angular velocity of 15 rpms in a
time of 6 seconds, find the angular acceleration of the unicycle wheel.
Comparing Kinematic Variables
Variable
Displacement
Velocity
Acceleration
Time
Translational
Angular
Putting them together
• It is quite straightforward to translate
between translational and angular variables as
well when you know the radius of the point of
interest on a rotating object and assume the
object is not slipping as it rotates.
Kinematic Variable Translations
Variable
Displacement
Velocity
Acceleration
Time
Translational
Angular
Example
• A knight swings a mace of radius 1 m in two
complete revolutions. What is the distance
traveled by the mace?
Kinematic Equations
• The parallel between translational and
rotational motion go even further. You
already know the linear kinematic equations
so you can develop a set of corresponding
equations for rotational motion.
• The rotational kinematic equations can be
used the same way. Once you know three of
the variables, you can solve for the other two.
Kinematic Equation Parallels
Translational
Rotational
Example 3
•
A mom spins her small child in a circle of radius 1 meter at a constant velocity of
60 rpm. What is the child’s linear velocity?
Example 4
•
A goat is being roasted on a spit, with the goat rotating at a radius of 0.3 meters
from the axis of rotation. What is the translational displacement of the goal as he
travels through 2 complete revolutions?
Example 5
•
On the playground, 4 children climb on the round-a-bout for a ride. If the round-a-bout
accelerates from rest to an angular velocity of 0.3 radians per second in a time of 10 seconds,
what is its angular acceleration? What is its linear acceleration at a raduis of 1.5 m from the
axis of rotation?
Example 6
•
A carpenter cuts a piece of wood with a high-powered circular saw. The saw blade
accelerates from rest with an angular acceleration of 14 rad/s2 to a maximum speed of
15,000 rpms. What is the maximum speed of the saw in radians per second? How long does
it take the saw to reach its maximum speed?
6.2R Torque
Ellen Akers
Torque
• Torque is a force that causes an object to turn
• Torque must be perpendicular to the displacement to
cause rotation
• The further away the force is applied from the point
of rotation, the more leverage you obtain.
• This distance is known as the lever arm (r).
Newton’s 2nd Law
• Translational vs. Rotational
• Fnet = ma
• τnet = Iα
Equilibrium
• Static equilibrium implies that the net force
and the net torque are zero, and the system is
at rest.
• Dynamic equilibrium implies that the net force
and the net torque are zero, and the system is
moving at constant translational and
rotational velocity
• Rotational equilibrium implies that the net
torque on an object is zero.
Example
•
A pirate captain takes the helm and turns the wheel of the ship by applying a force
of 20 N to a wheel spoke. If he applies the force at a radius of 0.2 meters from the
axis of rotation, at an angle of 80˚ to the line of action, what torque does he apply
to the wheel?
Example 2
•
A mechanic tightens the lugs on a tire by applying a torque of 100 newton meters at an angle
of 90 ̊to the line of action. What force is applied if the wrench is 0.4 meters long?
•
How long must the wrench be if the mechanic is only capable of applying a force of 200 N?
Example 3
•
A 3 kg café sign is hung from a 1 kg horizontal pole. A wire is attached to prevent the sign
from rotating. Find the tension in the wire.
Example 4
•
45 kg Sam sits on a teeter totter 1 meter from the fulcrum. Where must 75 kg Ben sit in
order to maintain static equilibrium? What is the force on the fulcrum?
6.3R Rotational Dynamics
Ellen Akers
Objectives
• Understand the moment of inertia (rotational
inertia) of an object or system depends upon
the distribution of mass within the object or
system.
• Determine the angular acceleration of an
object when an external torque or force is
applied.
Types of Inertia
• Inertial mass (translational inertia) is an object’s ability to
resist a linear acceleration.
• Moment of inertia (rotational inertia) is an objects resistance
to a rotational acceleration.
• Objects that have most of their mass near their axis of
rotation have smaller rotational inertias than objects with
more mass farther from their axis of rotation.
Moment of Inertia
Example
• Calculate the moment of inertia for a solid
sphere with a mass of 10 kg and a radius of
0.2m.
Example 1
•
Find the moment of inertia (I) of 2 5 kg masses joined by a meter long rod of negligible mass
when rotated about the center of the rod. Compare this to the moment of inertia when
rotated about one of the masses.
Newton’s 2nd Law for Rotation
• The acceleration of an object is equal to the
net force applied divided by the object’s
inertial mass.
• Fnet = ma
• The angular acceleration of an object is equal
to the net torque applied divided by the
object’s moment of inertia.
• Τnet = Iα
Example 2
•
Ian spins a top with a moment of inertia of 0.001 kg/m2 on a table by applying a torque of
0.01 Nm for 2 seconds. If the top starts from rest, find the final angular velocity of the top.
Example 3
•
What is the angular acceleration experienced by a uniform solid disc of mass 2 kg
and radius 0.1 m when a net torque of 10 Nm is applied? Assume the disc spins
about its center.
Example 5
•
A round-a-bout on a playground with moi of 100 kg m/s2 starts at rest and is accelerated by a
force of 150 N at a radius of 1 m from its center. If this force is applied at an angle of 90˚
from the line of action for a time of 0.5 seconds, what is the final rotational velocity of the
round-a-bout?
6.4R Angular Momentum
Ellen Akers
Angular Momentum
• Linear momentum, the product of an object’s inertial mass
and linear velocity p=mv is conserved in a closed system.
• Describes how difficult it is to stop a moving object.
• Angular momentum (L) the product of an objects moment of
inertia and its angular velocity about the center of mass, is
also conserved in a closed system with no external net
torques applied.
• Describes how difficult it is to stop a moving object.
Angular Momentum
Example
•
Anna spins on a rotating pedestal with an angular velocity of 8 radians per second. Bob
throws her an exercise ball which increases her moi from 2 kg m2 to 2.5 kg m2. What is
Anna’s angular velocity after catching the exercise ball? Neglect any external torque from the
ball.
Example 2
•
A disc with a moi of 1 kg m2 spins about an axle through its center of mass with angular
velocity of 10 rad/s. An identical disc which is not rotating is slid along the axle until it makes
contact with the first disc. If the two discs stick together, what is their combined angular
velocity?
6.5R Rotational Kinetic Energy
Ellen Akers
Types of Kinetic Energy
• Objects traveling with a translational velocity have
energy of motion, known as translational kinetic
energy.
• Objects traveling with angular (rotational) velocity
have rotational kinetic energy.
Translational vs. Rotational Equations
Example – KE of a Basketball
•
A 0.62 kg basketball flies through the air with a velocity of 8 m/s. What is its translational
kinetic energy?
•
The same basketball with radius 0.38m also spins about its axis with an angular velocity of 5
radians per second. Determine its moment of inertia and its rotational kinetic energy.
•
What is the total kinetic energy of the basketball?
Example 2 – Ice Skater
•
An ice skater spins with a specific angular velocity. She brings her arms and legs
closer to her body, reducing her moi to half its original value. What happens to
her angular velocity? What happens to her rotational kinetic energy?
Example 3- Bowling Ball
•
Gina rolls a bowling ball of mass 7 kg and radius 10.9 cm down a lane with a velocity of 6 m/s.
Find the rotational kinetic energy of the bowling ball, assuming it does not slip. What is its
total kinetic energy?