High speed cameras are used to photograph events which take place too quickly to

Question

High speed cameras are used to photograph events which take place too quickly to be seen by the human eye. One such high speed camera is composed of a rotating film drum, a rotating shutter and some miscellaneous optics. The film is fastened along the outside of the drum (along its entire circumference) and the drum is driven by high pressure gas.

You desire to photograph the dynamics of a bumblebee in flight and estimate that its wings flap somewhere about 10,000 times a second. The radius of the film drum is r = 0.501 m and you've loaded it with 35mm film .

1)How many photographs will fit along the circumference of the drum? (Assume 35mm length is one photo)
Number =

2)You desire to take the photographs at a frequency of 20,000 Hz. What is the angular frequency of the shutter? (Note: shutter has one hole in it, and angular frequency is the same as angular velocity)
=

3)What must the angular frequency of the film drum be?
=

4)When the drum has taken its pictures, the gas is turned off and the drum slows down uniformly, finally setting to a standstill after 2 minutes.
What was the angular acceleration of the film drum after turning off the gas?
a = rad/s2

5)How many times does the film drum rotate after the gas is turned off and before coming to rest? Give your answers in revolutions.

Explanation / Answer

Linear distance Lr the rotating point has travelled during rotation, is described with the equation: L = An*R (1), where An - angle of rotation, and R - radius of rotating point (the distance between rotating point and the center of rotation). As the radius of rotation R is constant, the velocity and acceleration, defined as derivatives (linear operators) will be described with the equations derived from (1). Based on just said, the equation for full angle An of rotating point looks similar to the formula for its linear distance with linear speed and acceleration replaced with angular ones. Ommitting all the intermediate operations, we get equation (2): An = Van * t + Aan * ((t^2) / 2) (2) Now, we are ready to solve problems a) - d). Problem a) is solved by you, and we just do it here for completeness. Solution a) The number of photographs taken is calculated as L/35mm, were L = 2 * Pi * R: Nphotos = L / 35, where L must be in millimeters: Nphotos = (2 * 3.14 *501) /35 = 90 (closely rounded) Solution b): The single hole of the shutter allows to take one photograph each time it makes one full rotation, i., e, has travelled angle of 2Pi radians. For 20000 pictures per second the angular speed of the shutter will be 20000 * 2Pi = 125663.71 rad/s Solution c): When the shutter makes one rotation, the drum travelles jusst one photograph, i.e. 1/90 fraction of its rotation. Therefore, the angular speed of the drum Van is 1/90 of the angular speed of the shutter: Van = 125663.71 / 90 = 13962.63 rad/s Solution d): As the angular speed of the drum is going down, the angular acceleration (although, negative, SEE NOTE BELOW) is constant. This means , that within 2 minutes (120 seconds), the angular acceleration of the drum Aan is calculated as its angular speed divided by 120 seconds: Aan = 13962.6 / 120 = 116.3 rad/(s^2) Solution of problem e): From the solution c) we know the angular speed of the drum, and from the solution d) we know its angular acceleration. Applying formula (2) and NOTE below, we calculate An - full angle of the drum made for 2 minutes (120 sec) The number of rotations of the drum for 120 s is Nrotations = An / 2Pi. We get: An = 1396.26 * 120 - 11.64 * ((120^2) / 2) = 1396.26 * (120 / 2`) = 83775.6 And Nrotations = 83775.6 / (2*Pi) = 13333.3

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