Ground Rules. PC1221 Fundamentals of Physics I. Uniform Circular Motion, cont. Uniform Circular Motion (on Horizon Plane) Lectures 11 and 12

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1 PC11 Fudametals of Physics I Lectues 11 ad 1 Cicula Motio ad Othe Applicatios of Newto s Laws D Tay Seg Chua 1 Goud Rules Switch off you hadphoe ad page Switch off you laptop compute ad keep it No talkig while lectue is goig o No gossipig while the lectue is goig o Raise you had if you hae questio to ask Be o time fo lectue Be o time to come back fom the ecess beak to cotiue the lectue Big you lectueotes to lectue Uifom Cicula Motio (o Hoizo Plae) Uifom Cicula Motio, cot A foce, F, is diected towad the cete of the cicle This foce is associated with a acceleatio, a c Applyig Newto s Secod Law alog the adial diectio gies F = mac = m Hoizotal Plae 3 A foce (fom my had) causes a cetipetal acceleatio to act towad the cete of the cicle It causes a chage i the diectio of the elocity ecto If the foce aishes, the object would moe i a staight-lie path taget to the cicle (d5-14) 4

2 Cetipetal Foce The foce causig the cetipetal acceleatio is called the cetipetal foce This is ot a ew foce, but it is a ew ole fo a foce Cetipetal foce causes cicula motio 5 Coical Pedulum The object is i equilibium i the etical diectio ad udegoes uifom cicula motio i the hoizotal diectio (ball ad stig) = Lgsi ta is idepedet of m This fomula ca be deied 6 Coical Pedulum m Cetipetal foce = ma = = T si -- (1) Sub () ito (1): T cos = m = cos si T = cos = g ta -- (3) But si = L = Lsi -- (4) -- () Sub (4) ito (3): = g ta Lsi = Lg si ta = T L Lg si ta Motio i a Hoizotal Cicle The speed at which the object moes depeds o the mass of the object ad the tesio i the cod The cetipetal foce is supplied by the tesio = T m 8

Motio i a Hoizotal Cicle Tesio (T( ) o the stig: T m T = T = m T = m = T m Hoizotal (Flat) Cue The foce of static fictio supplies the cetipetal foce The maximum speed at

So = m μg f Example. A cate of eggs is located i the middle of the flat bed of a pickup tuck as the tuck egotiates a ubaked cue i the oad.

3 Motio i a Hoizotal Cicle Tesio (T( ) o the stig: T m T = T = m T = m = T m Hoizotal (Flat) Cue The foce of static fictio supplies the cetipetal foce The maximum speed at which the ca ca egotiate the cue is = μg Note, the maximum elocity does ot deped o the mass of the ca m μ = 10 Maximum Velocity o a Flat Race Tack f : fictio f = μ = μg = So = m μg f Example. A cate of eggs is located i the middle of the flat bed of a pickup tuck as the tuck egotiates a ubaked cue i the oad. The cue may be egaded as a ac of a cicle of adius 35.0 m. If the coefficiet of static fictio betwee cate ad tuck is 0.6, how fast ca the tuck be moig without the cate slidig? Aswe: What if this is a shap tu? 1

Baked Cue Baked cues ae desiged to deal with

of the omal foce that supplies the cetipetal

Cue = cos Vetically: cos = m -(1) Hoizotally:

Loop-the-Loop i Vetical Pla This is a example

(b), the upwad foce expeieced by the object is

4 Baked Cue Baked cues ae desiged to deal with ey small o o fictio situatio Thee is a compoet of the omal foce that supplies the cetipetal Baked Cue 13 foce ta = g 14 Speed o a baked Cue = cos Vetically: cos = m -(1) Hoizotally: si is the cetipetal foce si = -() Loop-the-Loop i Vetical Pla This is a example of a etical cicle At the bottom of the loop (b), the upwad foce expeieced by the object is geate tha its weight bot = 1+ g Pilot is at the bottom of the loop 16

poduces the cetipetal acceleatio F t poduces the tagetial acceleatio ΣF = ΣF + ΣF t The gaitatioal foce exets a tagetial foce o the object Look at

5 Loop-the the-loop i etical pla Loop-the-Loop, Pat At the top of the cicle, the foce exeted o the object is less tha its weight Pilot is at the top of the loop (1) Bottom () Top top = 1 g 17 No-Uifom Cicula Motio Vetical Cicle with No- Uifom Speed The acceleatio ad foce hae tagetial compoets F poduces the cetipetal acceleatio F t poduces the tagetial acceleatio ΣF = ΣF + ΣF t The gaitatioal foce exets a tagetial foce o the object Look at the compoets of F g The tesio at ay poit ca be foud Wate ad pail 19 T = m + gcos R This fomula coespods to the peious special cases whe = 0 ad

cos si Vetical Cicle with No-uifom Speed T Top ad Bottom of Cicle The tesio at the

if the elocity is less tha this alue? Examples.

6 cos si Vetical Cicle with No-uifom Speed T Top ad Bottom of Cicle The tesio at the bottom is a maximum The tesio at the top is a miimum If T top = 0, the top = gr What if the elocity is less tha this alue? Examples. A olle coaste ca has a mass of 500 kg whe fully loaded with passeges. (a) If the ehicle has a speed of 0.0 m/s at poit A, what is the foce exeted by the tack o the ca at this poit? (b) What is the maximum speed the ehicle ca hae at B ad still emai o the tack? Aswe C B Mg 10 m 15 m A Mg At B, Mg = 3 4

Motio i Acceleated Fames Cetifugal Foce A fictitious foce esults fom a acceleated fame of efeece A fictitious foce appeas to act o a (fist) object i the same way as a eal foce, but you caot idetify a

What if the obsee is Fom the fame of the passege (b), a foce appeas to push he towad the doo Fom the fame of the Eath, the ca applies a leftwad foce o the passege.

5 6 Which ease will fly off fist? Thee eases ae placed o a otatig disc at diffeet distaces fom cete ad we spi the disc at iceasig speed. Which ease will fly off fist (ie, middle, oute)? Why?

7 Motio i Acceleated Fames Cetifugal Foce A fictitious foce esults fom a acceleated fame of efeece A fictitious foce appeas to act o a (fist) object i the same way as a eal foce, but you caot idetify a secod object that applies the fictitious foce to the fist object. Theefoe a fictitious foce is ot a eal foce that exhibits eal effects. What if the obsee is Fom the fame of the passege (b), a foce appeas to push he towad the doo Fom the fame of the Eath, the ca applies a leftwad foce o the passege. This is the cetipetal foce The outwad foce is ofte called a cetifugal foce It is a fictitious foce due to the acceleatio associated with the ca s chage i diectio i the tai? 5 6 Which ease will fly off fist? Thee eases ae placed o a otatig disc at diffeet distaces fom cete ad we spi the disc at iceasig speed. Which ease will fly off fist (ie, middle, oute)? Why? Aswe: All thee eases expeiece the same fictio o the disc suface. The cetipetal foce o each ease is diffeet: m / = m(ω) / = mω, whee ω is the agula elocity. As the oute ease (with the lagest ) expeieces the lagest cetipetal foce, its fictitious outwad foce (cetifugal foce) is the lagest so it will fly off fist. (d5-15) 7 Coiolis Foce This is a appaet foce caused by chagig the adial positio of a object i a otatig coodiate system The esult of the otatio is the cued path of the ball (d6-14) 8

Fictitious Foces, examples Although fictitious foces ae ot eal foces, they ca hae eal effects Examples: Objects i the ca do slide You ca feel pushed to the outside of a otatig platfom The Coiolis

8 Fictitious Foces, examples Although fictitious foces ae ot eal foces, they ca hae eal effects Examples: Objects i the ca do slide You ca feel pushed to the outside of a otatig platfom The Coiolis foce is esposible fo the otatio of weathe systems ad ocea cuets. How? Ocea Cuets (has othig to do with tides which ae caused by the pull of gaity fom the Moo ad the Su): The Eath spis o its axis fom West to East (coute-clockwise) whe iewed lookig dow o the Noth Pole). Due to the otatio of the eath, cuets ae deflected to the ight i the othe hemisphee ad to the left i the southe hemisphee by coiolis effect. The deflectio leads to highs ad lows of sea leel that causes ocea cuets Fictitious Foces i a Rotatig System Example. A object of mass 5.00 kg, attached to a spig scale, ests o a fictioless, hoizotal suface as i Figue. The spig scale, attached to the fot ed of a boxca, has a costat eadig of 18.0 N whe the ca is i motio. (a) If the spig scale eads zeo whe the ca is at est, detemie the acceleatio of the ca. (b) What costat eadig will the spig scale show if the ca moes with costat elocity? (c) Descibe the foces o the object as obseed by someoe i the ca ad by someoe at est outside the ca. Aswe: 5.00 kg Accodig to the ietial obsee (a), the tesio is the cetipetal foce m T = The oietial obsee (b) sees m T Ffictitious = T =

The study of the motion of a body along a general curve. the unit vector normal to the curve. Clearly, these unit vectors change with time, u ˆ

The study of the motion of a body along a general curve. the unit vector normal to the curve. Clearly, these unit vectors change with time, u ˆ Section. Cuilinea Motion he study of the motion of a body along a geneal cue. We define u ˆ û the unit ecto at the body, tangential to the cue the unit ecto nomal to the cue Clealy, these unit ectos change