Chapter 10 Rotational Motion and Torque

10.1- Angular Position, Velocity and Acceleration For a rigid rotating object a point P will rotate in a circle of radius r away from the axis of rotation.

10.1 The location of point P can be described in polar coordinates (r, θ). The circular distance traveled is called the arc length according to When θ is measured in radians (1 radian is the angle swept by an arc length equal to the radius).

10.1 Angles measured in radians, degrees, revolutions 2π rad = 360o = 1 rev Angular displacement- the change in angular position

10.1 Angular Velocity- the rate of change in angular displacement – For constant rotations or averages – For Angular Position as function of time Measured in rad/s or rev/s

10.1 Quick Quizzes p 294 Angular Acceleration- the rate of change of angular velocity or

10.1 Angular Velocity/Acceleration Vector Directions- Right Hand Rule Generally CCW is positive, CW is negative Acceleration direction Points the same direction as ω, if ω is increasing, antiparallel if ω decreases

10.1 Quick Quiz p. 296

10.2 Rotational Kinematics Tracking the increasing and decreasing rotation can be done with the same relationships as increasing and decreasing linear motion. Remember Δx  Δθ v  ω a  α

10.2 Quick Quiz p. 297 Example 10.1

10.3 Angular and Linear Quantities When an object rotates on any axis, every particle in that object travels in a circle of constant radius (distance from axis) The motion of each point can be described linearly about the circular path Tangential Velocity-

10.3 Tangential Acceleration- We also know there is a centripetal acceleration

10.3 The resultant acceleration- Quick Quizzes p. 298 Examples 10.2

10.4 Rotational Kinetic Energy- the kinetic energy of a single particle in a rotating object is… The Total Kinetic Energy would be the sum of all K i Which can be rewritten...

10.4 This is a new term we will call Moment of Inertia Moment of Inertia has Dimensions ML 2 and units kg. m 2 (~ rotational counterpart to mass) Rotational Kinetic Energy-

10.4 Quick Quiz p. 301 Examples 10.3, 10.4

10.5 Calculating Moments of Inertia We can evaluate the moment of inertia of an extended object by adding up the M.o.I. for an infinite number of small particles.

10.5 Its generally easier to calculate based on the volume of elements rather than mass so using for small elements…. We have… If ρ is constant, the integral can be completed based on the geometric shape of the object.

10.5 Volumetric Density- ρ (mass per unit volume) Surface Mass Density- σ (mass per unit area) – (Sheet of uniform thickness (t) σ = ρt) Linear Mass Density- λ (mass per unit length) – (Rod of uniform cross sectional area (A) λ = M/L = ρA) See Board Diagrams

10.5 Example Common M.o.I. for high symmetry shapes (p. 304) Parallel Axis Theorem

10.5 Example 10.8

10.6 Torque Torque- the tendency of a force to cause rotation about an axis Where r is the distance from the axis of rotation and Fsinφ is the perpendicular component of the force Where F is the force and d is the “moment arm.”

10.6 Moment Arm- (lever arm) the perpendicular distance from the axis of rotation to the “line of action”

10.6 Torque is a vector has dimensions ML 2 T -2 which are the same as work, units will also be N. m Even though they have the same dimensions and units, they are two very different concepts. Work is a scalar product of two vectors Torque is a vector product of two vectors

10.6 The direction of the torque vector follows the right hand rule for rotation, and CCW torques will be considered positive, CW torques negative. Quick Quizzes p. 307 Example 10.9

10.7 Torque and Angular Acceleration Consider a tangential and radial force on a particle. The F t causes a tangential acceleration.

10.7 We can also look at the torque caused by the tangential force. And since…

10.7 Newton’s 2 nd Law (Rotational Analog) Quick Quiz p. 309 Review Examples

10.8 Work, Power, and Energy Rotational Analogs for Work, Power, and Energy – Work – Energy – Work-KE Theorem – Power

10.8 Quick Quiz p. 314 Examples 10.14, 10.15

10.9 Rolling Motion For an object rolling in a straight line path the translational motion of its center of mass can be related to its angular displacment, velocity and acceleration. Condition for Pure Rolling Motion- no slipping – If there is no slip, then every point on the outside of the wheel contacts the ground and following relationships hold.

10.9

Linear distance traveled (translational displacment)- CofM Velocity (trans. vel.)- CofM Accel. (trans. accel.)-

10.9

Total Kinetic Energy for a rolling object K tot = K r + K cm (using just translational speed) (using just angular speed)

10.9 Friction must be present to give the torque causing rotation, but does not cause a loss of energy because the point of contact does not slide on the surface. With zero friction the object would slide, not roll. Quick Quizzes p. 319 Example 10.16