Rotational Dynamics Chapter 8 Section 3. Torque Direction A net positive torque causes an object to...

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Rotational Dynamics Chapter 8 Section 3

Transcript of Rotational Dynamics Chapter 8 Section 3. Torque Direction A net positive torque causes an object to...

Page 1: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Rotational DynamicsChapter 8 Section 3

Page 2: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Torque Direction A net positive torque causes an object

to rotate counterclockwise.

A net negative torque causes an object to rotate clockwise.

+

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Page 3: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Newton’s Second Law for Rotation

ΣT = IαNet Torque = (Moment of Inertia)(Angular Acceleration)

Page 4: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Translational vs. RotationalNewton’s 2nd Law

Translational: F = ma

Force = Mass x Acceleration

Rotational: T = Iα

Torque = Moment of Inertia x Angular Acceleration

Page 5: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Example Problem #1 A toy flying disk with a mass of 165.0

grams and a radius of 13.5 cm that is spinning at 30 rad/s can be stopped by a hand in 0.10 sec. What is the average torque exerted on the disk by the hand?

Page 6: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Example Problem #1 AnswerT = IαT = (½mr2)((ωf-ωi)/t)T = (½)(0.165kg)(0.135m)2 ((0rad/s – 30rad/s)/0.10s)

T = -0.45 Nm

Page 7: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Resistance to Change Swinging a sledge hammer, or a

similarly heavy object, takes some effort to start rotating the object.

The same can be said about stopping a heavy object that is rotating.

Page 8: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Momentum Translational Momentum – A vector

quantity defined as the product of an object’s mass and velocity. Also known as, “Inertia In Motion”

Angular Momentum – The product of a rotating object’s moment of inertia and angular speed about the same axis.

Page 9: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Angular Momentum Equation

L = Iω

Angular Momentum = (Moment of Inertia)(Angular Speed)

Page 10: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Angular Momentum The variable used for Angular

Momentum. Capital letter “L”

The SI units for angular momentum. Kgm2/s

Page 11: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Translational vs. RotationalMomentum

Translational: p = mv

Linear Momentum = mass x velocity

Rotational: L = Iω

Angular Momentum = moment of inertia x angular speed

Page 12: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Conservation of Angular Momentum The Law of Conservation of Angular

Momentum - When the net external torque acting on an object is zero, the angular momentum of the object does not change.

Li = Lf

Iωi = Iωf

Page 13: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Angular Momentum Example

Angular momentum is conserved as a skater pulls his arms towards their body, assuming the ice they are skating on is frictionless.

During an ice skaters spin, they will bring their hands and feet closer to the body which will in turn decrease the moment of inertia and as a result increase the angular speed.

Page 14: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Example Problem #2 A 0.11kg mouse rides on the edge of a

rotating disk that has a mass of 1.3 kg and a radius of 0.25m. If the rotating disk begins with an initial angular speed of 3.0 rad/s, what is its angular speed after the mouse walks from the edge to a point 0.15m from the center? What is the tangential speed of the disk at the outer edge?

Page 15: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Example Problem #2 Answer ω = 3.2 rad/s v = 0.8 m/s

Page 16: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Kinetic Energy Rotational Kinetic Energy – Energy of an

object due to its rotational motion.

Greater angular speeds and greater moment of Inertia, yields greater rotational kinetic energy

Page 17: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Rotational Kinetic Energy Equation

Rotational Kinetic Energy = ½(Moment of Inertia)(Angular Speed)2

Page 18: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Momentum vs. Energy Unlike Angular momentum, rotational

energy increases when the moment of inertia deceases when no external torques are introduced.

A greater angular speed will increase rotational kinetic energy because of the square term in the equation.

Page 19: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Example Problem #3 A car tire has a diameter of 0.89m and

may be approximated as a hoop. How fast will it be going starting from rest to roll without slipping 4.0m down an incline that makes an angle of 35 degrees with the horizontal?

Page 20: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Example Problem #3 Diagram

d = 4.0m

h = ?

0.89m

Vi = 0 m/s

Vf = ?

Page 21: Rotational Dynamics Chapter 8 Section 3. Torque Direction  A net positive torque causes an object to rotate counterclockwise.  A net negative torque.

Example Problem #3 Answer Remember that mechanical energy

within a system must remain constant.

𝜔 𝑓=𝑣 𝑓

𝑟

𝐼=𝑚𝑟2