PHYS 1441 Section 002 Lecture #19

Transcription

1 PHYS 1441 Section 00 Lecture #19 Monday, April 8, 013 Fundamentals o the Rotational Motion Rotational Kinematics Equations o Rotational Kinematics Relationship Between Angular and Linear Quantities Rolling Motion o a Rigid Body Today s homework is homework #10, due 11pm, Monday, Apr. 15!!

2 Announcements Second non-comp term exam Date and time: 4:00pm, Wednesday, April 17 in class Coverage: CH6.1 through what we inish Monday, April 15 This exam could replace the irst term exam i better Special colloquium or 15 point extra credit Wednesday, April 4, University Hall RM116 Class will be substituted by this colloquium Dr. Ketevi Assamagan rom Brookhaven National Laboratory on Higgs Discovery in ATLAS Please mark your calendars!!

3 Rotational Motion and Angular Displacement In the simplest kind o rotation, points on a rigid object move on circular paths around an axis o rotation. The angle swept out by the line passing through any point on the body and intersecting the axis o rotation perpendicularly is called the angular displacement. Δ θ =θ θ o It s a vector!! So there must be a direction How do we deine directions? +:i counter-clockwise -:i clockwise The direction vector points gets determined based on the right-hand rule. These are just conventions!! 3

4 SI Unit o the Angular Displacement θ (in radians) = Arc length Radius = s r θ = π r r For one ull revolution: Since the circumerence o a circle is πr = π rad Dimension? None π rad = 360 o One radian is an angle subtended by an arc o the same length as the radius!

5 Unit o the Angular Displacement How many degrees are in one radian? 1 radian is 1 rad= 360 πrad How radians is one degree? And one degrees is π 1 o = 1 o 360 1rad = 180 π π = o o o o How many radians are in 10.5 revolutions? 10.5rev = 10.5rev π rad rev o o rad 1π rad = ( ) Very important: In solving angular problems, all units, degrees or revolutions, must be converted to radians. 5

6 Example 8- A particular bird s eyes can just distinguish objects that subtend an angle no smaller than about 3x10-4 rad. (a) How many degrees is this? (b) How small an object can the bird just distinguish when lying at a height o 100m? (a) One radian is 360 o /p. Thus ( rad ) ( 360 o π rad ) rad l = = = (b) Since l=rθ and or small angle arc length is approximately the same as the chord length. rθ = 100m rad = 3 10 m= 3 cm 6 o

7 Ex. Adjacent Synchronous Satellites Synchronous satellites are put into an orbit whose radius is m. I the angular separation o the two satellites is.00 degrees, ind the arc length that separates them. What do we need to ind out? The Arc length!!! Convert degrees to radians s = θ (in radians) = Arc length Radius = s r π rad.00deg = rad 360 deg rθ = ( m)( rad) 6 = m (90 miles) 7

8 Ex. A Total Eclipse o the Sun The diameter o the sun is about 400 times greater than that o the moon. By coincidence, the sun is also about 400 times arther rom the earth than is the moon. For an observer on the earth, compare the angle subtended by the moon to the angle subtended by the sun and explain why this result leads to a total solar eclipse. θ (in radians) = Arc length Radius = s r I can even cover the entire sun with my thumb!! Why? Because the distance (r) rom my eyes to my thumb is ar shorter than that to the sun. 8

9 Angular Displacement, Velocity, and Acceleration Angular displacement is deined as Δθ = How about the average angular velocity, the rate o change o angular displacement? Unit? rad/s θ θ i By the same token, the average angular acceleration, rate o change o the angular velocity, is deined as Unit? rad/s Dimension? [T -1 ] Dimension? [T - ] ω α θ θ i θ t ω t θ t i i ω t i i Δθ = Δt Δω = Δt When rotating about a ixed axis, every particle on a rigid object rotates through the same angle and has the same angular speed and angular acceleration. 9

10 Ex. Gymnast on a High Bar A gymnast on a high bar swings through two revolutions in a time o 1.90 s. Find the average angular velocity o the gymnast. What is the angular displacement? π rad Δ θ =.00 rev 1 rev = 1.6 rad Why negative? Because he is rotating clockwise!! ω = 1.6 rad 1.90 s = 6.63rad s 10

11 Ex. A Jet Revving Its Engines As seen rom the ront o the engine, the an blades are rotating with an angular speed o -110 rad/s. As the plane takes o, the angular velocity o the blades reaches -330 rad/s in a time o 14 s. Find the angular acceleration, assuming it to be constant. α = ω t ω t i i = Δω Δt ( 330rad s) ( 110rad s) = = 16rad s 14 s 11

12 Rotational Kinematics The irst type o motion we have learned in linear kinematics was under a constant acceleration. We will learn about the rotational motion under constant angular acceleration (α), because these are the simplest motions in both cases. Just like the case in linear motion, one can obtain Angular velocity under constant ω angular acceleration: v = vo + at Angular displacement under θ = constant angular acceleration: 1 x = x0 + vot + at One can also obtain ω v + a x x Linear kinematics Linear kinematics Linear kinematics v = ( ) o i = ω + αt = 0 0 θ + ω 0 t + 1 αt ( ) ω α θ θ 1

13 Problem Solving Strategy Visualize the problem by drawing a picture. Write down the values that are given or any o the ive kinematic variables and convert them to SI units. Remember that the unit o the angle must be in radians!! Veriy that the inormation contains values or at least three o the ive kinematic variables. Select the appropriate equation. When the motion is divided into segments, remember that the inal angular velocity o one segment is the initial velocity or the next. Keep in mind that there may be two possible answers to a kinematics problem. 13