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 VariablesVariable Translational Angular

Displacement

Velocity

Acceleration

Time

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 TranslationsVariable Translational Angular

Displacement

Velocity

Acceleration

Time

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 ParallelsTranslational 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?