Transcript The Coriolis Effectm

Elliott Chick
Current University of Exeter MPhys Undergraduate
What is it?
 The Coriolis effect is the apparent deflection of an
object in a rotating reference frame.
 Gaspard-Gustave Coriolis was first to consider this
supplementary force.
 When Newton's laws of motion are applied in a
rotation frame of reference, other forces appear.
 These forces are “Fictitious forces” and are used as
correction factors for the simple application of
Newton’s laws in a rotating system.
Newton's Laws in a rotating reference frame.
As we know, in an inertial reference frame:
𝐹𝑖 = 𝑚𝑎𝑖
However, when we look at newton's 2nd law in a rotating
reference frame, several fictitious forces appear:
𝐹𝑟 = 𝐹𝑖 + 𝐹𝑐𝑒𝑛𝑡𝑟𝑖𝑓𝑢𝑔𝑎𝑙 + 𝐹𝐶𝑜𝑟𝑖𝑜𝑙𝑖𝑠 + 𝐹𝐸𝑢𝑙𝑒𝑟 = 𝑚𝑎𝑟
Newton's Laws in a rotating reference frame.
Looking at the equation fully:
𝐹𝑟 = 𝐹𝑖 + 𝐹𝑐𝑒𝑛𝑡𝑟𝑖𝑓𝑢𝑔𝑎𝑙 + 𝐹𝐶𝑜𝑟𝑖𝑜𝑙𝑖𝑠 + 𝐹𝐸𝑢𝑙𝑒𝑟 = 𝑚𝑎𝑟
𝑑𝜔
𝐹𝑟 = 𝑚𝑎𝑖 − 𝑚𝜔 × 𝜔 × 𝑟 − 2𝑚𝜔 × 𝑣𝑟 − 𝑚
× 𝑟 = 𝑚𝑎𝑟
𝑑𝑡
The Coriolis force
The Coriolis acceleration is defined as:
𝑎𝑐𝑜𝑟𝑖𝑜𝑙𝑖𝑠 = −2𝜔 × 𝑣
With the Coriolis force being:
𝐹𝑐𝑜𝑟𝑖𝑜𝑙𝑖𝑠 = −2𝑚𝜔 × 𝑣
Where ω = angular velocity, m = mass of object and v =
velocity of the particle in a rotating system
An example…
• One of the Coriolis effect’s most common appearance
is in ballistics.
• Target: Ciudad Real Madrid
40.479167 N, 3.611667 W
(http://toolserver.org/~geohack/geohack.php?pagename=Ciudad_Real_Madrid&params=40_28_45_N_03_36_42_W_type:landmark)
• Distance from Physics Building:
1141 km
• Tomahawk missile
• Average speed = 244.4 m/s (sub sonic)
• Effective range = 2500 km
• ω of earth = 7.27 x 10-5 rads/s
(http://www.nhc.noaa.gov/gccalc.shtml)
(http://en.wikipedia.org/wiki/Tomahawk_(missile))
(http://hypertextbook.com/facts/2002/JasonAtkins.shtml)
http://www.europemapofeurope.net/europemap-of-europe-large-2008-muck-hole.jpg
An example…
 Time to target =
1141000𝑚
244.4𝑚/𝑠
= 4668.6𝑠
 Acceleration due to Coriolis effect:
 𝑎𝑐 = 2𝜔 × 𝑣 = 2 x (7.27 x 10-5 ) x 244.4m/s = 0.035 ms-2
 Displacement:
 𝑠 = 𝑣0 𝑡 +
𝑎𝑡 2
2
= 381.4km west of target (40.47N, 8.12W)
Actual point of impact…
The Experiment
 The aim of this experiment was to demonstrate the
Coriolis acceleration in a rotating reference frame, and
showing that the Coriolis acceleration is proportional
to the angular velocity of the rotating reference frame.
 Apparatus used: (From experiment ME06)
 Glass turntable (Connected to DC power supply)

Covered with paper
 Metal ramp
 Ball bearing
Experimental setup
Experimental method
 Find the velocity of the ball bearing
 Time taken to travel between 2 points
 Measure the angular frequency of the turntable
 Measure amount of rotations and time taken, use
rotations/time to work out revolutions per second
 Multiply by 2π to find the angular frequency of the
turntable
Experimental method continued
 Coat the ball bearing in a layer of ink
 Place on ramp, with quick release in place
 Start up the turntable
 Release the ball
 The ball will leave behind a trail of dots, allowing easy
observation of the path of the ball
Analysis
 The paper is taken off of the turntable, and this is the
result
 Values for displacement taken every 2 cm
 Every 2 cm along linear path is equal to:

0.956𝑚
0.02𝑚
= 47.8 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑠 𝑒𝑣𝑒𝑟𝑦 𝑠𝑒𝑐𝑜𝑛𝑑
 Therefore, the time interval every 2cm:

1
47.8
= 0.0209𝑠 𝑒𝑣𝑒𝑟𝑦 2𝑐𝑚
Results 1
0.2
Demonstration of repeatable results
for ω = 1.2 rads/s
0.18
0.16
Displacement (m)
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.05
0.1
0.15
Time (s)
0.2
0.25
Results 2
0.5
ω = 1.2 rads/s
0.4
Velocity (m/s)
0.3
0.2
y = 2.4128x
0.1
0
-0.05
0
-0.1
0.05
0.1
Time (s)
Theoretical value: 2 x 0.956 x 1.2 = 2.29 𝑚𝑠 2 ± 0.12
0.15
0.2
Results 3
0.9
ω = 1.67 rads/s
0.8
0.7
y = 2.9287x
Velocity (m/s)
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.05
0
-0.1
0.05
0.1
0.15
Time (m/s)
Theoretical value: 2 x 0.956 x 1.67 = 3.19 𝑚𝑠 2 ± 0.12
0.2
0.25
0.3
Results 4
0.35
ω = 0.55 rads/s
0.3
Velocity (m/s)
0.25
0.2
0.15
y = 1.0013x
0.1
0.05
0
-0.05
0
-0.05
0.05
0.1
0.15
0.2
Time (s)
Theoretical value: 2 x 0.956 x 0.55 = 1.05 𝑚𝑠 2 ± 0.12
0.25
0.3
0.35
Conclusions
 The Coriolis effect can be easily proven using this
experimental method
 From my results I can see that the Coriolis acceleration
is proportional to the angular velocity of the rotating
reference frame.
 The velocity was kept constant, thus the only factor
effecting the Coriolis acceleration was ω showing its
proportionality.