Angular Velocity. Midterm on Thursday at 7:30pm! Old exams available on website. Chapters 6 9 are covered. Go to same room as last time.

Transcription

1 Angular Velocity Announcements: Midterm on Thursday at 7:30pm! Old exams available on website. Chapters 6 9 are covered. Go to same room as last time. You are allowed one calculator and one doublesided sheet of paper with hand written notes 14 Multiple choice plus two long answer questions Test time from 7:30pm 9:00pm. If you have a conflict, need extra time, etc., then contact Professor Daniel Dessau. Web page:

2 Clicker Score Update Clickers are posted on D2L up to March 9. 05C628EB 0E8D199A 25D B9D AD801F 32D3D B08 33B BC9817 These clickers are being used in recently recorded responses, but they are not registered to any students in the class. Please check your clicker score on D2L!

3 Exam Rooms

4 Clicker question 1 Set frequency to BA Steve (m=50. kg) is skating east with velocity v=1.0 m/s. Scott (m=70. kg) is skating north with velocity v=1.0 m/s. They happen to collide at the center of the rink, and hold tight (i.e. they stick together, oomph!) What is the final speed of the pair, after the collision? A) 1.4 m/s B) 1.0 m/s C) 0.5 m/s D) 0.0 m/s E) 0.72 m/s

5 Clicker question 1 Set frequency to BA Steve (m=50. kg) is skating east with velocity v=1.0 m/s. Scott (m=70. kg) is skating north with velocity v=1.0 m/s. They happen to collide at the center of the rink, and hold tight (i.e. they stick together, oomph!) What is the final speed of the pair, after the collision? A) 1.4 m/s B) 1.0 m/s C) 0.5 m/s D) 0.0 m/s E) 0.72 m/s (50kg)(1m /s)ˆ i + (70kg)(1m /s) ˆ j = (120kg) v f v f = 5 12 i ˆ ˆ j v = (5/12) 2 + (7 /12) 2.72m /s

6 Clicker question 2 Set frequency to BA Four identical square tiles are glued together like a J, as shown. Each tile has edge length a and mass m. The origin is at the bottom right, as shown. What is the x component of the center of mass of the system of tiles? A) -(1/2) a B) -(3/4) a C) - a D) -(3/2) a E) -3 a

7 Clicker question 2 Set frequency to BA Four identical square tiles are glued together like a J, as shown. Each tile has edge length a and mass m. The origin is at the bottom right, as shown. What is the x component of the center of mass of the system of tiles? A) -(1/2) a B) -(3/4) a C) - a D) -(3/2) a E) -3 a x(cm) = (m1 x1 + m2 x + m3 x3 + m4 x4 ) / (mtotal) Numbering the squares 1, 2, and 3 walking down, and #4 is the one on the left, gives x(cm) = m*(-a/2)+m*(-a/2)+m*(-a/2)+m*(-3/2a)/ (4m) = -3a/4.

8 Using concept of negative mass A square of side 2R has a circular hole of radius R/2 removed. Relative to the center of the square the center of the hole is located at (R/2,R/2). Locate the center of mass with respect to the center of the square. CoM = m i x i m i CoM = m sq x sq m cir x cir m sq m cir Treat the hole as negative mass: x(cm) = [4R 2 (0) π (R/2) 2 (R/2)]/[4R 2 -π(r/2) 2 ] πr 8 /[4 π ] =.393R /3.21 =.122R 4 2R R/2 y(cm)=-.122r (same as x(cm)

9 Angular kinematics In chapters 2 and 3 we dealt with kinematics which involved displacement, velocity, and acceleration. Angular kinematics is the same thing but for objects which are rotating (rather than translating). For something to rotate, it must have an axis about which it rotates like the axle for a wheel. Only sensible place for the origin is along the axis. Use polar coordinates (r,θ) instead of Cartesian coordinates (x,y). Axis is in the z-direction. r θ

10 Angular velocity Angular velocity tells us how fast (and in what direction) something is spinning. Δθ r θ The z subscript indicates the axis is in the z- direction (and the rotation is therefore in the xy plane) Counterclockwise is positive Also have angular acceleration which describes how the spinning rate changes

11 Using Angular Variables The angular position of a line on a disk of radius 6 cm is given by θ =10 5t + 4t 2 rad. Find the average angular speed between 1 and 3 s: θ(1) = = 9rad. θ(3) = = 31rad. ω = θ(3) θ(1) 3 1 = =11rad /s

12 Using Angular Variables The angular position of a line on a disk of radius 6 cm is given by θ =10 5t + 4t 2 rad. Find the linear speed of a point on the rim at 2 s: ω = dθ ;ω(2) = 5 + 8t =11rad /s dt v = rω =.(06m)(11rad /s) v(2) =.66m /s

13 Using Angular Variables The angular position of a line on a disk of radius 6 cm is given by θ =10 5t + 4t 2 rad. Find the radial and tangential acceleration of a point on the rim at 2 s: dθ dt = 5 + 8t α = dω dt = d 2 θ ;α(2) = 8rad /s2 dt 2 a tan = rα =.(06m)(8rad /s 2 ) a a tan (2) =.48m /s 2 rad = v 2 r a total = 2 a rad 2 + a tan = (.66m /s)2.06m a rad (2) = 7.26m /s 2

14 Clicker question 3 Set frequency to BA Big Ben and a little alarm clock (synchronized to an atomic clock) both keep perfect time. Which minute hand has the largest angular velocity? A. Big Ben B. little alarm clock C. same D. Impossible to tell

15 Clicker question 3 Set frequency to BA Big Ben and a little alarm clock (synchronized to an atomic clock) both keep perfect time. Which minute hand has the largest angular velocity? A. Big Ben B. little alarm clock C. same D. Impossible to tell Angular velocity is only a measure of how quickly the angle changes. Both minute hands complete 1 revolution every hour. 1 rph = 2π rad/hr = 2π/3600 rad/s x(cm) = (m1 x1 + m2 x + m3 x3 + m4 x4 ) / (mtotal) Numbering the squares 1, 2, and 3 walking down, and #4 is the one on the left, gives x(cm) = m*(-a/2)+m*(-a/2)+m*(-a/2)+m*(-3/2a)/ (4m) = -3a/4.

16 Angular velocity & acceleration vectors Note the subscript z on the angular velocity and acceleration which indicates the axis of rotation Also, note that ω and α can be positive or negative This is like 1D motion. Is there an equivalent 3D motion? Yes. The axis of rotation can change orientation; it is not always along the z-axis (and therefore neither is ω). Angular velocity and acceleration are vectors: points perpendicular to the plane of rotation in the direction given by the right hand rule (direction of your thumb when fingers curl in direction of rotation).

17 Summary of angular kinematics Angular displacement Δθ measured in radians Δθ Angular velocity: r θ Angular acceleration: Angular velocity and acceleration are vectors: points perpendicular to the plane of rotation in the direction given by the right hand rule (direction of your thumb when fingers curl in direction of rotation).

18 Angular kinematics The same equations which were derived for constant linear (or translational) acceleration apply for constant angular (or rotational) acceleration Constant linear acceleration only! Constant angular acceleration only!

19 Clicker question 4 Set frequency to BA A flywheel with a mass of 120 kg, and a radius of 0.6 m, starting at rest, has an angular acceleration of 0.1 rad/s 2. How many revolutions has the wheel undergone after 10 s? What equation should the student use to obtain the answer? A. B. C. D. E. None of them will work

20 Clicker question 4 Set frequency to BA A flywheel with a mass of 120 kg, and a radius of 0.6 m, starting at rest, has an angular acceleration of 0.1 rad/s 2. How many revolutions has the wheel undergone after 10 s? What equation should the student use to obtain the answer? A. θ 0 =0, ω 0z =0, α z =0.1 rad/s 2 and we want θ B. C. D. E. None of them will work

21 Relationships to linear velocity If we want the linear displacement or velocity of a point on a rotating object, we need r and either θ or ω. What is the speed of the tire rim if the radius is 0.35 m and the tire rotates at 20 rad/s? Radians measure distance around a unit circle r θ s Therefore (only for θ in radians!) Time derivative of both sides (r is constant):. Linear speed is radius times angular speed Rim speed is

22 Relationships to linear acceleration What about acceleration? If speed is then We know that centripetal (or radial) acceleration is. Using v = rω, this can be rewritten as. Total linear acceleration is composed of both tangential and radial acceleration which are always perpendicular to each other.

23 Clicker question 5 Set frequency to BA CD slows from an angular velocity of 4 rad/s to a stop in 4 seconds with constant angular acceleration. What is the magnitude of the angular acceleration (α) of the CD? A. 1 rad/s 2 B. 4 rad/s 2 C. 16 rad/s 2 D. 1 cm/s 2 E. None of the above 5 cm

24 Clicker question 5 Set frequency to BA CD slows from an angular velocity of 4 rad/s to a stop in 4 seconds with constant angular acceleration. What is the magnitude of the angular acceleration (α) of the CD? A. 1 rad/s 2 B. 4 rad/s 2 C. 16 rad/s 2 D. 1 cm/s 2 Remember E. None of the above constant α we have and for 5 cm but want magnitude: 1 rad/s 2.

25 Clicker question 6 Set frequency to BA CD has an angular acceleration of -1 rad/s 2 and an angular velocity of 2 rad/s. What is the magnitude of the acceleration of the dust particle 5cm out? A. 20 cm/s 2 B. 5 cm/s 2 C. 21 cm/s 2 D. 0.8 cm/s 2 E. None of the above 5 cm Total acceleration is composed of radial and tangential acceleration. Can now get net force by multiplying acceleration by mass