Rotational Dynamics Chapter 8 Section 3

Torque Direction A net positive torque causes an object to rotate counterclockwise. A net negative torque causes an object to rotate clockwise. + -

Newton’s Second Law for Rotation Net Torque = (Moment of Inertia)(Angular Acceleration)

Translational vs. Rotational Newton’s 2nd Law Translational: F = ma Force = Mass x Acceleration Rotational: T = Iα Torque = Moment of Inertia x Angular Acceleration

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?

Example Problem #1 Answer T = Iα T = (½mr2)((ωf-ωi)/t) T = (½)(0.165kg)(0.135m)2 ((0rad/s – 30rad/s)/0.10s) T = -0.45 Nm

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.

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.

Angular Momentum Equation L = Iω Angular Momentum = (Moment of Inertia)(Angular Speed)

Angular Momentum The variable used for Angular Momentum. Capital letter “L” The SI units for angular momentum. Kgm2/s

Translational vs. Rotational Momentum Translational: p = mv Linear Momentum = mass x velocity Rotational: L = Iω Angular Momentum = moment of inertia x angular speed

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

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.

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?

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

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

Rotational Kinetic Energy Equation 𝐾𝐸 𝜔 = 1 2 𝐼 𝜔 2 Rotational Kinetic Energy = ½(Moment of Inertia) (Angular Speed)2

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. 𝐾𝐸 𝜔 = 1 2 𝐼 𝜔 2

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?

Example Problem #3 Diagram Vi = 0 m/s 0.89m h = ? d = 4.0m Vf = ?

Example Problem #3 Answer Remember that mechanical energy within a system must remain constant. 𝐼=𝑚 𝑟 2 𝜔 𝑓 = 𝑣 𝑓 𝑟