A Level Exam-style Practice Paper

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Prudence Lynch
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1 A Leel Exam-style Pactice Pape a i The peiod is gien by the time lapse between high tide and low tide which is.5 hous. ii The amplitude is gien by half the total displacement and so is 5 m. b The safe mooing height make is m aboe low tide, m below mean sea leel, the cente of the motion (see diagam). a = 5 m, x = m, T =.5 hous, ad hou T a x The wate leel is ising at a speed of.6 m hou (3 s.f.) when it passes the make. c Taking the displacement to be zeo at mean sea leel we want the inteal between the two times afte low tide when x = m. Taking low tide as t =, usingx acost allows us to wok without a phase constant (see diagam) and the equation of motion becomes: 5cos 5 t This is fist tue t hous afte low tide, whee: cos t 5 5 t t Using symmety (see diagam), the wate falls to an unsafe depth t hous afte low tide whee: t Tt t tt So the total length of time fo which boats can moo safely between two consecutie low tides is 7.5 hous (3 s.f.). Peason Education Ltd 8. Copying pemitted fo puchasing institution only. This mateial is not copyight fee.

2 a AD = EC = CB = a Since AB = a AC = DE = a Split the lamina along EC. Both sections hae equal mass (since the tiangle has been folded oe), call this m. The total mass of the lamina is theefoe m. The cente of mass of ACED, G is a fom both AC and AD. Fo the tiangle. taking C as the oigin, the coodinates of the etices ae: C:(,) B:( a,) E:(, a) So, with C as oigin, the coodinates of the cente of mass a a a a of CBE, G, ae,, The lamina can be eplaced by two paticles: a a ACED of mass m, fom A a a CBE of mass m, 3 3 fom A Using m i i mi, the cente of mass of the composite lamina has coodinates elatie to A as the oigin of: a a 3 x m m m a a 3 y x a 6 5a 6 y x a 5a y G, The cente of mass of the lamina lies: a i fom AD. 5a ii fom AB. b When suspended fom A, the lamina hangs as shown, with G diectly below A and AC (and theefoe DE) making an angle of θ with the etical whee: tan 5a a 5 tan.3... The edge DE makes an angle of with the etical (to the neaest whole degee). Peason Education Ltd 8. Copying pemitted fo puchasing institution only. This mateial is not copyight fee.

3 3 a a k ms a d d x d k dx Sepaating aiables and integating: d dx k ln k xb At t = s, = U ms Use these initial conditions to find the constant of integation, B: ln k U B ln B k U So displacement, x, is gien by: lnk x lnk U x lnk U lnk k U x ln k When the paticle eaches A, x = OA and = U ms Substituting these alues gies: k U OA ln m, as equied. k U d b Using the elationship a dt d k dt Sepaating aiables and integating: U U k d U dt t actan t k k U U U actan actan t k k k k U U t actan actan k k k t Peason Education Ltd 8. Copying pemitted fo puchasing institution only. This mateial is not copyight fee. 3

4 = 36 m, 3 3 tan sin and cos 5 5 a Let the mass of the ca be m and the nomal eaction foce be R Resoling etically: Rcos R cos Resoling hoizontally: m Rsin m sin cos gtan The ca taels at a speed of 7. ms (to 3 s.f.). b The fictional foce, F R,.5 If the ca is about to slide up the slope, F acts down the slope. Resoling etically: Rsin Rcos RcosRsin R cos sin R Resoling hoizontally: m RsinRcos m sincos 3 g Peason Education Ltd 8. Copying pemitted fo puchasing institution only. This mateial is not copyight fee.

b continued If the ca is about to slide down the slope, F acts up the slope. Resoling etically: RcosRsin R 3 5 5 Resoling hoizontally: m RsinRcos m sincos g3 36 5 5 36 9.8 5 3.35.

5 b continued If the ca is about to slide down the slope, F acts up the slope. Resoling etically: RcosRsin R Resoling hoizontally: m RsinRcos m sincos g If the ca is not to skid up o down the slope, the speed must emain between 3. ms and 77. ms (to 3 s.f.). 5 a y cosx V y dx V cosxdx V sin x V The olume of the solid is m3, as equied. Peason Education Ltd 8. Copying pemitted fo puchasing institution only. This mateial is not copyight fee. 5

6 5 b Cente of mass: y xdx x y dx x xcosxdx cosxdx Using integation by pats to find xcosxdx d f( x) g( x) d x f( x) g( x)d x f( x) g( x)dx dx dx d xcosxdxxcosxdx x cosxdx dx dx sinx sinx x dx x sinx sinxdx x cosx sinx xsinxcosx Theefoe, cente of mass is at: x x x x x sinx sin cos x The x coodinate of the cente of mass is c When the solid is about to topple, the weight acts though the base (see diagam). The adius of the base is the alue of y when x = y cos y tan tan The solid is on the point of toppling when the plane is inclined at an angle of 7 (to the neaest whole degee) to the hoizontal. Peason Education Ltd 8. Copying pemitted fo puchasing institution only. This mateial is not copyight fee. 6

7 6 The stating position is a distance acos6 = a beneath O (see diagam). Taking this as the zeo leel fo potential enegy, the total enegy of the paticle is the kinetic enegy in this position. K.E. m K.E. m3ga 3a So total enegy If the tension in the sting is T and the angle with the upwad etical is θ, esoling towads the cente of the cicle gies: m m T cos a When sting fist becomes slack, T = so this becomes: m cos a agcos ( ) This happens a distance acosθ aboe O (see diagam), so the potential enegy at this point is: P.E. h( acos acos6 ) Since total enegy emains constant: 3a KE. PE.. 3a m acos a ga ga 3 cosga gagacos Substituting fom () fo gies: gacos gagacos cos cos 3cos cos The sting fist becomes slack when OP makes an angle of 8 with the upwad etical (to neaest whole degee) and P theefoe does not each the top of the cicle. Peason Education Ltd 8. Copying pemitted fo puchasing institution only. This mateial is not copyight fee. 7

Radian Measure CHAPTER 5 MODELLING PERIODIC FUNCTIONS

Radian Measure CHAPTER 5 MODELLING PERIODIC FUNCTIONS 5.4 Radian Measue So fa, ou hae measued angles in degees, with 60 being one eolution aound a cicle. Thee is anothe wa to measue angles called adian measue. With adian measue, the ac length of a cicle is