Chapter 10 Rotation Rotation of a rigid body

Chapter 10 Rotation

Rotation of a rigid body • We consider rotational motion of a rigid body about a fixed axis • Rigid body rotates with all its parts locked together and without any change in its shape • Fixed axis: it does not move during the rotation • This axis is called axis of rotation • Reference line is introduced

Angular position • Reference line is fixed in the body, is perpendicular to the rotation axis, intersects the rotation axis, and rotates with the body • Angular position – the angle (in radians or degrees) of the reference line relative to a fixed direction (zero angular position)

Angular displacement • Angular displacement – the change in angular position. • Angular displacement is considered positive in the CCW direction and holds for the rigid body as a whole and every part within that body

Angular velocity • Average angular velocity • Instantaneous angular velocity – the rate of change in angular position

Angular acceleration • Average angular acceleration • Instantaneous angular acceleration – the rate of change in angular velocity

Rotation with constant angular acceleration • Similarly to Chapter 2 (case of 1 D motion with a constant acceleration) we can derive a set of formulas (Table 10 -1)

Chapter 10 Problem 6

Relating the linear and angular variables: position • For a point on a reference line at a distance r from the rotation axis: • θ is measured in radians

Relating the linear and angular variables: speed • ω is measured in rad/s • Period (recall Ch. 4)

Relating the linear and angular variables: acceleration • α is measured in rad/s 2 • Centripetal acceleration (Ch. 4)

Rotational kinetic energy • We consider a system of particles participating in rotational motion • Kinetic energy of this system is • Then

Moment of inertia • From the previous slide • Defining moment of inertia (rotational inertia) as • We obtain for rotational kinetic energy

Moment of inertia: rigid body • For a rigid body with volume V and density ρ(V) we generalize the definition of a rotational inertia: • This integral can be calculated for different shapes and density distributions • For a constant density and the rotation axis going through the center of mass the rotational inertia for 9 common body shapes is given in Table 10 -2 (next slide)

Moment of inertia: rigid body

Moment of inertia: rigid body • The rotational inertia of a rigid body depends on the position and orientation of the axis of rotation relative to the body • More information at: http: //scienceworld. wolfram. com/physics/Momentof. Inertia. html

Parallel-axis theorem • Rotational inertia of a rigid body with the rotation axis, which is perpendicular to the xy plane and going through point P: • Let us choose a reference frame, in which the center of mass coincides with the origin

Parallel-axis theorem

Parallel-axis theorem R

Chapter 10 Problem 39

Torque • We apply a force at point P to a rigid body that is free to rotate about an axis passing through O • Only the tangential component Ft = F sin φ of the force will be able to cause rotation

Torque • The ability to rotate will also depend on how far from the rotation axis the force is applied • Torque (turning action of a force): • SI unit: N*m (don’t confuse with J)

Torque • Torque: • Moment arm: r┴= r sinφ • Torque can be redefined as: force times moment arm τ = F r┴

Newton’s Second Law for rotation • Consider a particle rotating under the influence of a force • For tangential components • Similar derivation for rigid body