Rotational Kinematics

The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged

prev | up | next

Discussion

let's rap

discussion

Translational and Rotational Quantities Compared
  translational connections rotational
base quantities
s, r

  
s =
θ × r
θ

  
coordinate systems
 r =
x ˆi + y ˆj

 x = r cos θ
y = r sin θ
r = √(x2 + y2)
θ = tan−1 (y / x)
 r =
rˆr + θˆθ
velocity
v =
dr / dt
v =
ω × r
ω =
dθ / dt
acceleration
 a =
dv / dt = d2r / dt2
 a =
α × r − ω2r
 α =
dω / dt = d2θ / dt2
equations of motion v = v0 + at
x = x0 + v0t + ½ at2
v2 = v02 + 2a(x − x0)		     ω = ω0 + αt
θ = θ0 + ω0t + ½ αt2
ω2 = ω02 + 2α(θ − θ0)

the museum of obsolete technology

  • constant linear velocity
    • movie film
      • 24 fps
    • audio tape
      • cassette: 15/8 ips
      • 8-track: ??
      • open reel (reel-to-reel): ??
    • data tape
      • 9-track: ??
    • videotape
      • VHS: ??
      • Betamax: ??
      • open reel (reel-to-reel): ??
    • optical disk
      • CD: 1.2 - 1.4 m/s
      • DVD:
        • 3.49 m/s (single layer)
        • 3.84 m/s (double layer)
  • constant angular velocity
    • phonograph
      • Edison cylinder
      • 78 rpm
      • 45 rpm
      • 33 1/3 rpm
      • 16 2/3 rpm
    • video
      • laser disc: 11 m/s
    • magnetic data storage
      • floppy disk: ??
      • hard drive: ??
      • Zip disk: ??
      • Jaz disk: ??

Summary

Problems

practice

  1. opening image from James BondLooking out from the inside of a rifled barrel. The opening of every James Bond film begins with this image.
    A rifle is a long gun whose barrel has been grooved or "rifled" on the inside with spiral channels. (For comparison, a long gun with a smooth bore is called a musket.) Bullets fired from a rifled barrel spin. This gives them greater stability in flight and thus greater accuracy when fired. Since 1964, the standard infantry weapon in the US Army has been the .22 caliber M16 rifle. Due to rifling, a bullet fired from an M16 rotates two and a half times on its journey from the breech to the muzzle. Given a barrel length of 510 mm and a muzzle velocity of 950 m/s, determine …
    1. the average translational acceleration,
    2. the average rotational acceleration (in radians per second squared), and
    3. the final angular velocity (in rotations per second)
    Solutions …
    1. Review basic problem solving techniques. List the relevant known quantities and the identify the goal of the problem. Select an appropriate equation. Substitute values and solve for the unknown quantity. (Watch the units.)
         
      s =  510 mm = 0.510 m   v2 =  v02 + 2as
      v0 =  0 m/s   a =  v2  
      v =  950 m/s 2s
      a =  ??   a =  (950 m/s)2  
          2(0.510 m)  
             a =  8.8 x105 m/s2
         
    2. This part of the problem is similar in style to the previous part, but slightly more difficult since less is known. We don't have enough information "as is" to solve it. Something else is needed and that something else is time. There are several ways to find it, but my personal choice is to use the two average velocity formulas.
                           
      v  =  s  =  v + v0 ⇒ t =  2s  =  2 (0.510 m)  = 0.001074 s
      t 2 v + v0 950 m/s + 0 m/s
                           
      All that remains now is to select an appropriate equation, substitute, and solve. (Once again, watch the units.)
         
      θ =  2.5 rotations = 5π rad   θ =  ω0t + 1  αt2
      ω0 =  0 rad/s 2
      t =  0.001074 s   α =  2θ  
      α =  ?? t2
            α =  2(5π rad)  
       (0.001074 s)2  
             α =  2.7 × 107 rad/s2
               
    3. List. Select. Substitute. Solve. (Watch the units.)
               
      t =  0.001074 s   ω =  ω0 + αt
      α =  2.7 × 107 rad/s2   ω =  0 rad/s + (2.7 × 107 rad/s2)(0.001074 s)
      ω0 =  0 rad/s   ω =  29,000 rad/s
      ω =  ??   ω =  4700 rotations per second
               
  2. Write something else.
    • Answer it.
  3. Write something different.
    • Answer it.
  4. Write something completely different.
    • Answer it.

numerical

  1. Earth
    1. What's the angular speed of the earth's rotation about its axis?
    2. What's the angular speed of the earth in its orbit around the sun?
  2. Clock I
    1. What's the angular speed of the second hand on an analog clock?
    2. What's the angular speed of the minute hand on an analog clock?
    3. What's the angular speed of the hour hand on an analog clock?
  3. Clock II
    1. If the hour and minute hands of a clock are aligned at 00:00 (midnight), at what time will they realign again?
    2. If the hour, minute, and second hands of a clock are aligned at 00:00:00 (midnight), at what time will they realign again?
  4. Clock III
    1. To the nearest order of magnitude, what's the speed of the tip of the minute hand on a wall clock in a classroom?
    2. To the nearest order of magnitude, what's the speed of the tip of the minute hand on a man's wristwatch?

Resources

prev | up | next

Another quality webpage by

Glenn Elert		 eglobe logo home | contact

bent | chaos | eworld | facts | physics