#### 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¶ms=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.