Lecture 6 Physics I Chapter 4 Rotational Motion Course website: http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Today we are going to discuss: Chapter 4: Uniform Circular Motion: Section 4.4 Nonuniform Circular Motion: Section 4.6 PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics In addition to translation, objects can rotate There is rotation everywhere you look in the universe, from the nuclei of atoms to spiral galaxies Rotational Motion Spiraling Galaxies Need to develop a vocabulary for describing rotational motion PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics In order to describe rotation, we need to define How to measure angles? You know degrees, but in the scientific world RADIANS are more popular. Let’s introduce them. PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Angular Position in polar coordinates Consider a pure rotational motion: an object moves around a fixed axis. y Instead of using x and y cartesian coordinates, we will define object’s position with: r, s arclength r s x θ in radians! r Its definition as a ratio of two length makes it a pure number without dimensions. So, the radian is dimensionless and there is no need to mention it in calculations (Thus the unit of angle (radians) is really just a name to remind us that we are dealing with an angle) . Use Radians to get an arc length If angle is given in radians, we can get an arclength spanning angle θ s 60 r s r s r 3 PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Examples: angles in radians 2 r s arc length r / 2 y 4 s r / 2 r Apply radians! r r 2 x y s 2r s 2r 2 rad r r r x 360 2 rad 1 rad 360/ 2 57.3 PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Now we need to introduce rotational kinematic quantities like we did for translational motion. Angular displacement, Angular velocity Angular acceleration for rotational kinematic equations PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Angular displacement and velocity Angular displacement: 2 1 The average angular velocity is defined as the total angular displacement divided by time: t The instantaneous angular velocity: d lim ; ; ; t0 t dt Angular velocity is the rate at which particle’s angular position is changing. For both points, ∆θ and ∆t are the same so is the same for all points of a rotating object. That is why we can say that Earth’s angular velocity is 7.2x10-5 rad/sec without connecting to any point on the Earth. All points have the same . So is like an intrinsic property of a solid rotating object. PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Sign of Angular Velocity When is the Angular velocity positive/negative? As shown in the figure, ω can be positive or negative, and this follows from our definition of θ. (Definition of θ: An angle θ is measured (convention) from the positive x-axis in a counterclockwise direction.) Example: For a clock hand, the angular velocity is negative PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics ConcepTest Bonnie and Klyde Bonnie sits on the outer rim of a A) same as Bonnie’s merry-go-round, and Klyde sits midway between the center and the B) twice Bonnie’s rim. The merry-go-round makes C) half of Bonnie’s one complete revolution every D) one-quarter of Bonnie’s 2 seconds. E) four times Bonnie’s Klyde’s angular velocity is: 1rev 2 rad is the same for both rad t 2sec 2sec sec rabbits The angular velocity of any point on a solid object rotating about a fixed axis is the same. Both Bonnie and Klyde go around one Bonnie revolution (2 radians) every 2 sec. Klyde A particle moves with uniform circular motion if its angular velocity is constant. Now, if a rotation is not uniform (angular velocity is not constant), we can introduce angular acceleration ≠ const Angular acceleration PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Angular Acceleration The angular acceleration is the rate at which the angular velocity changes with time: 2 1 t Average angular acceleration: t2 1 2 1 Units rad/s2 t t2 t1 d lim Instantaneous angular acceleration: t0 t dt The units of angular acceleration are rad/s2 Since is the same for all points of a rotating object, angular acceleration also will be the same for all points. Thus, and α are properties of a rotating object PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics The Sign of Angular Acceleration, α Initial Initial Initial Final Final Final α is positive if |ω| is Positive Positive Positive Positive decreasing and ω is clockwise. and larger and smaller and smaller and larger Initial f i f i 0 0 t f ti t f ti So, α is positive if |ω| is α is negative if |ω| is increasing and is decreasing and ω is Final counter-clockwise. counterclockwise. α is negative if |ω| is increasing and ω is clockwise. PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics ConcepTest The signs of and α The fan blade is slowing down. A) ω is positive and α is positive. What are the signs of ω and α? B) ω is positive and α is negative. C) ω is negative and α is positive D) ω is negative and α is negative E) ω is positive and α is zero 1) is negative (rotation CW) 2) is slowing down (f< i) Less Negative More Negative For example f i 2 (5) 0 3rad / s2 3 2 t f ti Case 3 (the previous slide) Now, since we have introduced all angular quantities, we can write down Rotational Kinematic Equations For motion with constant angular acceleration PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Rotational kinematic equations The equations of motion for translational and rotational motion (for constant acceleration) are identical Translational kinematic Rotational kinematic equations equations Analogs v v vo at o t 1 1 x x v t at 2 x t t 2 o o 2 o o 2 2 2 2 2 v vo 2a(x xo ) a o 2 o PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics For a rotating object we can also introduce a linear velocity which is called the Tangential velocity Now we need to introduce a useful expression relating linear velocity and angular velocity PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Relation between tangential and angular velocities The tangential velocity component vt is the rate ds/dt at which the particle moves around the circle, where s is the arc length. By definition, linear velocity: vt In the 1st slide, ds we defined: ds d v = r = r d t dt dt r Remember it!!!! vt r You will use it often!!! Relation between linear and angular velocities ( in rad/sec) Each point on a rotating rigid body has the same angular displacement, velocity, and acceleration! The corresponding linear (or tangential) variables depend on the radius and the linear velocity is greater for points farther from the axis. PHYS.1410 Lecture 6 Danylov The end of the class Department of Physics and Applied Physics ConcepTest Bonnie and Klyde II Bonnie sits on the outer rim of a A) Klyde merry-go-round, and Klyde sits midway between the center and the B) Bonnie rim. The merry-go-round makes one C) both the same revolution every 2 seconds. Who has D) linear velocity is zero the larger linear (tangential) velocity? for both of them We already know that all points of a rotating body have the same angular velocity . But their linear speeds v will be different because v t r and Klyde Bonnie is located farther out (larger Bonnie 1 radius R) than Klyde.V V Klyde 2 Bonnie For a rotating object we can also introduce the Acceleration Tangential acceleration Centripetal acceleration Now we need to introduce a useful expression relating linear acceleration and angular acceleration PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Tangential acceleration The particle in the figure is moving along a circle and is speeding up. (Definition) Tangential acceleration is the rate at vtan which the tangential velocity changes, at = dvt /dt. at dv d a t = r r a r t dt t dt atotal There is a tangential acceleration a , which is always t tangent to the circle. aR There is also the centripetal acceleration is 2 r ar = vt /r, where vt is the tangential speed (next slide). Finally, any object that is undergoing circular motion experiences two accelerations: centripetal and tangential. Let’s get a total acceleration: 2 2 atotal at ar atotal at ar PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics Centripetal acceleration • In uniform circular motion (=const), although the speed is constant, there is an acceleration because the vtan direction of the velocity vector is always changing. r • The acceleration of uniform circular motion is called a aR centripetal acceleration. R • The direction of the centripetal acceleration is toward a the center of the circle. R v vtan tan • The magnitude of the centripetal acceleration is constant for uniform circular motion: v 2 t (toward center of circle) ar r (without derivation) Centripetal acceleration can be rewritten in term of angular velocity, 2 vt 2 ar r vtan r r PHYS.1410 Lecture 6 Danylov Department of Physics and Applied Physics ConcepTest Car on a curve A car is traveling around a A) Yes curve at a steady 45 mph.