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Opus in profectus

Rotational Kinematics

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Practice

practice problem 1

Looking out from the inside of a rifled barrel. From the opening of every James Bond film.
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 angular acceleration (in radians per second squared)
  3. the final angular velocity (in rotations per second)

solution

  1. Review basic problem solving techniques. List the relevant known quantities and the identify the goal of the problem.

    ∆s =  510 mm = 0.510 m
    v0 =  0 m/s
    v =  950 m/s
    a =  ?

    Select an appropriate equation. Substitute values and solve for the unknown quantity. (Watch the units.)

    v2 =  v02 + 2a∆s  
     
    a = 
    v2
    2∆s
    		 
     
    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 2

    Rearrange it to make time the subject.

    ∆t = 
    2∆s
    v + v0
    		 
     
    ∆t =  2 (0.510 m)  
    950 m/s + 0 m/s  
    ∆t = 0.001074 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
    ω0 =  0 rad/s
    ∆t =  0.001074 s
    α =  ?
       
    θ = 
    ω0t + 1  α∆t2
    2
    		 
     
    α = 
    2θ
    ∆t2
    		 
     
    α =  2(5π rad)  
     (0.001074 s)2  
    α = 2.7 × 107 rad/s2  
     

  3. Givens, unknown, equation, substitute, solve. (Watch the units.)

    ∆t =  0.001074 s
    α =  2.7 × 107 rad/s2
    ω0 =  0 rad/s
    ω =  ?
    ω =  ω0 + α∆t
    ω =  0 rad/s + (2.7 × 107 rad/s2)(0.001074 s)
    ω =  29,000 rad/s
    ω = 4700 rotations per second

practice problem 2

Write something else.

solution

Answer it.

practice problem 3

Write something different.

solution

Answer it.

practice problem 4

Write something completely different.

solution

Answer it.