AH Mechanics Checklist (Unit 2) AH Mechanics Checklist (Unit 2) Circular Motion

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1 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) Cicula Motion No. kill Done 1 Know that cicula motion efes to motion in a cicle of constant adius Know that cicula motion is conveniently descibed using the pola coodinate system with pola coodinates (, θ ), whee > 0 is the adial coodinate (aka adius) and θ ( π, π] is the angula coodinate (aka pola angle) 3 Know that the displacement of a paticle in pola coodinates, whee e is the adial unit vecto, is: = 4 Know that, fo the case of cicula motion, is constant 5 how that the adial unit vecto can be witten in tems of Catesian unit vectos as: e e = cos θ i + sin θ j 6 how that the displacement of a paticle in Catesian coodinates is: = ( cos θ ) i + ( sin θ ) j 7 Know that angula displacement in pola coodinates, whee e is the θ tangential unit vecto (aka angula unit vecto), is: θ = θ e θ 8 Know that the I unit of angula displacement is the dimensionless deived unit the adian (ad) 9 Know that, fo cicula motion, θ depends on time 10 how that the tangential unit vecto can be witten in tems of Catesian unit vectos as: e = sin θ i + cos θ j θ 11 how that the angula displacement of a paticle in Catesian coodinates is: θ = ( θ sin θ ) i + (θ cos θ ) j 1 Know that (the) angula velocity (vecto) is defined as the deivative of angula displacement wt time: ω θ & def = d θ dt M Patel (eptembe 01) 1 t. Macha Academy

2 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) 13 Know that the I unit of angula velocity is the deived unit ad s 14 Know that, in this couse, ω is taken to be constant 15 Know that, if ω is constant, the elation ω = dθ dt θ = ω t 16 olve poblems using the pevious equation 17 Know that angula acceleation is defined as the deivative of angula velocity wt time: α & ω = d θ dt becomes: 18 Know that the I unit of angula acceleation is the ad s 19 Know that, in this couse, α is usually taken to be 0 0 how that, fo motion in a cicle of given adius and constant angula velocity, the velocity of a paticle is: 1 v = ( θ & sin θ ) i + ( θ & cos θ ) j 1 how that the pevious equation can be witten as: v = θ & e θ how that, fo motion in a cicle of given adius and constant angula velocity, the magnitude of the velocity is: v = ω 3 olve poblems using the pevious equation 4 how that, fo motion in a cicle of given adius, the acceleation of a paticle is: a = ( θ && sin θ θ & cos θ ) i + ( θ && cos θ θ & sin θ ) j 5 how that the above can be witten as: a = θ & e + θ && e θ 6 how that, fo motion in a cicle of given adius and constant angula velocity, the magnitude of the acceleation is: v a = ω = 7 olve poblems using the pevious equations 8 Know that the wod centipetal means cente-seeking 9 Know that a centipetal foce is a foce that tends to make a body follow a cuved path M Patel (eptembe 01) t. Macha Academy

3 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) 30 how that, fo an object of mass m moving with constant angula velocity ω in a cicle of adius, the centipetal foce is: mv F = m ω e = 31 how that the magnitude of the centipetal foce is: mv F = m ω = 3 olve poblems using the pevious equations 33 Know that the angle of banking (aka banking angle) is the angle a banked cicula cuve makes with the hoizontal in ode to pevent a moving vehicle fom slipping up o down the cuve 34 Know that, fo banking poblems, it is simple to esolve foces hoizontally and vetically 35 Daw a fee-body diagam fo an object moving on a ough unbanked cuve of adius in a gavitational field of stength g, whee µ is the coefficient of static fiction between the object and the cuve 36 how that, fo an object moving on a ough unbanked cuve of adius in a gavitational field of stength g, whee µ is the coefficient of static fiction between the object and the cuve, the speed v equied to move in a hoizontal cicle (i.e. no slipping out) satisfies: e v µ g 37 olve poblems using the pevious inequation 38 olve poblems using the pevious inequation and the angula velocity 39 Given µ, g and, deduce that the maximum speed v is: max. v = µ g max. 40 olve poblems using the pevious equation 41 Daw a fee-body diagam fo an object to move in a hoizontal cicle on a smooth banked cuve (with given banking angle θ ) of adius in a gavitational field of stength g without slipping down o skidding up the cuve 4 how that, fo an object moving on a smooth banked cuve of adius in a gavitational field of stength g, the angle of banking θ equied to move in a hoizontal cicle (i.e. no slipping down the cuve o skidding up the cuve) satisfies: M Patel (eptembe 01) 3 t. Macha Academy

4 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) tan θ = v g 43 olve poblems using the pevious equation 44 olve poblems using the pevious equation and the angula velocity 45 Daw a fee-body diagam fo an object to move in a hoizontal cicle on a ough banked cuve (with given banking angle θ ) of adius in a gavitational field of stength g without slipping up the cuve 46 how that, fo an object moving on a ough banked cuve (with given banking angle θ ) of adius in a gavitational field of stength g, whee µ is the coefficient of static fiction between the object and the cuve, the speed v equied to move in a hoizontal cicle without slipping up the cuve satisfies: v g (sin θ + µ cos θ ) (cos θ µ sin θ ) 47 olve poblems using the pevious inequation 48 olve poblems using the pevious inequation and the angula velocity 49 Hence deduce fom the pevious inequality that the maximum such speed v is: max. v = max. g (sin θ + µ cos θ ) (cos θ µ sin θ ) 50 olve poblems using the pevious equation 51 olve poblems using the pevious equation and the angula velocity 5 Daw a fee-body diagam fo an object to move in a hoizontal cicle on a ough banked cuve (with given banking angle θ ) of adius in a gavitational field of stength g without slipping down the cuve 53 how that, fo an object moving on a ough banked cuve (with given banking angle θ ) of adius in a gavitational field of stength g, whee µ is the coefficient of static fiction between the object and the cuve, the speed v equied to move in a hoizontal cicle without slipping down the cuve satisfies: v g (sin θ µ cos θ ) M Patel (eptembe 01) 4 t. Macha Academy (cos θ + µ sin θ ) 54 olve poblems using the pevious inequation 55 olve poblems using the pevious inequation and the angula velocity 56 Hence deduce fom the pevious inequality that the minimum such speed v is: min.

5 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) v = min. g (sin θ µ cos θ ) (cos θ + µ sin θ ) 57 olve poblems using the pevious equation 58 olve poblems using the pevious equality and the angula velocity 59 Know that a light, inextensible sting is one that has negligible mass and cannot be extended 60 Know that a conical pendulum consists of a mass suspended by a light, inextensible sting fom a point O whee the mass moves in a hoizontal cicle with cente vetically below O 61 how that, fo a conical pendulum of sting length L with tension Q, mass m, in a gavitational field of stength g, moving with constant angula speed ω in a cicle of adius, whee the sting makes an angle θ with the vetical: tan θ = ω g = v g 6 olve poblems using the pevious equations 63 how that, fo the above conical pendulum, the sting tension is: Q = mω L 64 olve poblems using the pevious equation 65 how that the pevious equation can be witten in any of the following altenative foms: Q = mω sin θ = mv sin θ = mv Lsin θ = mv L 66 olve poblems using the pevious equations 67 how that, fo the above conical pendulum, the otation peiod T is: T = π g tan θ M Patel (eptembe 01) 5 t. Macha Academy = π Lcos θ g 68 olve poblems using the pevious equations 69 Know that Newton s Law of Univesal Gavitation states that any point masses attact each othe with a foce that is diectly popotional to the poduct of thei masses and invesely popotional to the squae of thei distance 70 Know that Newton s Law of Univesal Gavitation fo two point masses m and m with espective position vectos and elative to O, 1 1 whee G kg 1 m 3 s is Newton s constant of univesal gavitation (aka gavitational constant, univesal

6 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) gavitational constant o Big G), def =, 1 ˆ def 1 = 1 a unit vecto fom m to m, 1 and F the foce of m on m, is: 1 1 G m m 1 F = ˆ 1 71 Know that Newton s Law of Gavitation applies to spheical bodies, whee is now the distance between the bodies centes 7 Know that Newton s Law of Gavitation applies to a spheical body and a point mass 73 Know that F = F how that the magnitude of F (o F ), denoted by F, is: 1 1 G m m 1 F = 75 olve poblems using the pevious equation 76 Know that a body is in obit aound an object if the body follows a closed path aound the object 77 Know that a pimay is an object that is being obited 78 Know that a satellite is any obiting body; a natual satellite is one that occus natually, any othe satellite being an atificial satellite 79 Know that, unde Newton s Law of Gavitation, obits ae ellipses 80 Know that, in this couse, obits will be assumed cicula, any pimay will be assumed spheical and any satellite assumed to be a point mass 81 how that, fo Newton s Law of Gavitation, the acceleation due to gavity at distance fom the suface of a gavitating mass M, whee ˆ is a unit vecto pointing outwads fom the mass is given by: GM g = ˆ 8 Use Newton s Law of Gavitation togethe with the equations fo cicula motion to solve poblems involving a satellite in obit about a pimay such as, the time fo one obit, the speed and angula velocity of the satellite, the height of the satellite above the suface of the pimay and the foce acting on the satellite M Patel (eptembe 01) 6 t. Macha Academy

7 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) imple Hamonic Motion No. kill Done 1 Know that a estoing foce is a non-constant foce that tends to bing a system back to an equilibium state Know that an oscillation is a epetitive vaiation (nomally peiodic) of some quantity (e.g. displacement, cuent, intensity) ove time about some cental position (aka cente of oscillation o equilibium point o equilibium position) 3 Know that a system that undegoes oscillation is called an oscillato 4 Know that simple hamonic motion (HM) is 1D peiodic motion with a estoing foce that is diectly popotional to the displacement of the paticle fom the equilibium point 5 Know that a simple hamonic oscillato (HO) is a system that undegoes HM 6 Know that when esistive foces (popotional to velocity) occu in HM, the system is a damped simple hamonic oscillato 7 Know that when extenal foces (time-dependent) occu in HM, the system is a diven (foced) simple hamonic oscillato 8 Know that, in this couse, damped and diven systems will not be consideed 9 Know that a cycle of oscillation (aka oscillation cycle) of a HO is any pat of the motion that stats fom a given position and etuns to that position fo the fist time 10 Know that the peiod of a HO is the time taken to undego one complete cycle of oscillation 11 Know that peiod is measued in seconds 1 Know that the fequency of a HO is the numbe of complete oscillation cycles pe unit time 13 Know that the I unit of fequency is the deived unit the Hetz (Hz), equivalently, s 1 14 Know that the amplitude of a HO is the maximum distance fom the equilibium position 15 Know that amplitude is measued in metes 16 how that the equation descibing HM fo a HO whose displacement fom the equilibium point O is x, the estoing foce is k x and whee the angula fequency ω def = k, is given by: m x&& = ω x 17 Know that angula fequency has units s 1 18 Know that the scala vesion of the pevious equation is: M Patel (eptembe 01) 7 t. Macha Academy

8 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) x&& = ω x 19 Veify that a geneal solution of the pevious diffeential equation, whee A is the amplitude and θ the phase angle (which fixes the stating point of motion), is: x = A sin ( ωt + θ ) 0 Know that θ is a dimensionless quantity 1 how fom the geneal solution that the peiod of a HO is: T = π ω olve poblems using the pevious equation 3 how fom the geneal solution, o fom the definition of fequency, that the fequency of a HO is: ω f = π 4 olve poblems using the pevious equation 5 how fom the geneal solution that the amplitude of a HO is A 6 Know that the motion of a HO stats when: x = A sin θ 7 Know the special case of the geneal solution when the HO stats at O, i.e. when x = 0 o θ = 0: x = A sin ( ωt ) 8 Know the special case of the geneal solution when the HO stats at O, i.e. when x = A o θ = π : x = A cos ( ωt ) 9 how that the speed of a HO is: x& = ωa cos ( ωt + θ ) 30 how that the magnitude of the maximum speed of a HO occus at the equilibium position and is given by ωa 31 Veify, whee v is the speed of the HO, the elation: v = ω ( A x ) 3 olve poblems using the pevious equation 33 how fom the pevious elation that the maximum M Patel (eptembe 01) 8 t. Macha Academy

9 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) speed of a HO is given by ωa 34 how that the acceleation of a HO is: x&& = ω A sin ( ωt + θ ) 35 how that the magnitude of the maximum acceleation of a HO occus at x = A and x = A and is given by ω A 36 how fom the defining equation of a HO that the magnitude of the maximum acceleation of a HO is ω A 37 Know that a mateial is elastic if it etuns to its oiginal state afte a foce is applied then eleased 38 Know that when an elastic sping o elastic sting is extended o compessed, the foce poduced in the sping o sting is called tension 39 Know that Hooke s Law (of Elasticity) is an appoximate empiical law which states that the tension T in an elastic sping o elastic sting is popotional to the extension o compession x (fom the equilibium position) of the end of the sping o sting, whee k is the stiffness constant of the sping o sting: T = k x 40 Know that mateials which obey Hooke s Law ae called Hookean 41 Know that the tension in Hooke s Law is a estoing foce 4 Know that the unextended o uncompessed length of the sping o sting is called the natual length l of the sping o sting 43 Know that the modulus of elasticity (aka elastic modulus) λ of a sping o sting is a popety of the sping o sting that indicates its tendency to be defomed non-pemanently 44 Know that the elastic modulus is defined by: λ def = k l 45 how that Hooke s Law can be witten as: T = l λ x 46 olve poblems petaining to unknown tensions in Hookean spings and stings using the pevious equation 47 how that, fo a Hookean sping o sting suspended vetically in a gavitational field of stength g (so that its natual length is now l ), when a mass m is attached to the end (so that the equilibium position is x ), whee x is the 0 displacement of the mass fom x, the 0 M Patel (eptembe 01) 9 t. Macha Academy

10 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) tension in the sping o sting T is: T = λ l ( x + x 0 ) 48 how that, fo the pevious situation, the system consisting of the sping o sting is a HO with equation of motion: λ x&& = ml x 49 olve poblems using the pevious equation 50 olve poblems using the equations of HM with the pevious equation 51 Know that a sting is said to be inextensible if it cannot be extended 5 Know that a simple pendulum is a system consisting of a mass suspended by a light, inextensible sting fom a fixed point O whee the mass moves in a small ac in a vetical plane 53 how that, fo a simple pendulum of length l and mass m in a gavitational field of stength g, whee s is the ac length and θ the (small) angle made by the sting with the vetical, the equation of motion is that of a HO: s&& = g l s 54 olve poblems using the pevious equation 55 olve poblems using the equations of HM with the pevious equation M Patel (eptembe 01) 10 t. Macha Academy

11 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) Momentum and Impulse No. kill Done 1 Recall that momentum is the poduct of mass and velocity Recall that foce is the ate of change of momentum 3 Know that the impulse I of a foce F acting ove a time inteval Δt def = t t is defined as: 1 I def = t F dt t1 4 Know that impulse is a vecto quantity 5 Know that impulse has units kg m s 1 6 Calculate the impulse of a non-constant foce acting on an object ove a time inteval Δt 7 Know that the impulse fo a constant foce acting ove time Δt is : I = F Δt 8 olve poblems using the pevious equation 9 Pove the Impulse-Momentum Theoem, namely, that the impulse of a foce acting on an object, whee u is the initial velocity of the object at time t 1 (when the foce stats to act) and v is the final velocity of the object at time t (when the foce stops acting), equals the change in momentum of that object: I = mv mu 10 olve poblems using the pevious equation 11 Know that, fo constant foce: F Δt = mv mu 1 olve poblems using the pevious equation 13 Know that a closed system is a system in which no extenal foces ae pesent and in which thee is no inteaction with matte outside the system 14 Know that the Pinciple of Consevation of (Linea) Momentum (aka Consevation of Momentum o Momentum Consevation) states that the total momentum of a closed system is constant ove time: (Total momentum) t1 = (Total momentum) t ( t 1, t R ) 15 Know that, fo collision poblems, consevation of momentum can be M Patel (eptembe 01) 11 t. Macha Academy

12 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) fomulated as the statement that the total momentum befoe the collision equals the total momentum afte the collision 16 Know the special fom of momentum consevation in 1D o D involving a collision between objects of masses m and m which have velocities 1 u and u (espectively) befoe the collision and velocities v and v 1 1 (espectively) afte the collision: m u + m u = m v + m v olve poblems using the pevious equation 18 Know the special fom of momentum consevation in 1D o D involving a collision between objects of masses m and m which have velocities 1 u and u (espectively) befoe the collision and which stick togethe 1 (coalesce) and move with the same velocity v afte the collision: m u + m u = ( m + m ) v olve poblems using the pevious equation 0 Use momentum consevation to solve othe poblems M Patel (eptembe 01) 1 t. Macha Academy

13 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) Wok, Powe and Enegy No. kill Done 1 Know that the wok done by a foce F to move an object ove a displacement s def = Δ x def = x x 1, equivalently, ove a time inteval Δt def = t t, whee v is the velocity 1 of the object, is defined as: W def = x F. dx = x 1 t F. v t1 Know that wok can be done against a foce 3 Know that wok is a scala quantity 4 Know that the I unit of wok is the Joule (J), equivalently, kg m s 5 Calculate the wok done by a non-constant foce acting on an object ove distance s 6 Know that the wok done by a constant foce, whee θ is the angle between F and s, is: W = F. s = F s cos θ 7 olve poblems using the pevious equations 8 Calculate the wok done by a constant foce acting on an object ove a displacement s 9 how that the wok done by a constant foce in moving an object a displacement s, whee the foce and the displacement ae in the same diection, is: W = F s 10 olve poblems using the pevious equation 11 Know that powe is the ate at which a foce does wok: dt P def = dw dt 1 Know that powe is a scala quantity 13 Know that the I unit of powe is the deived unit 14 the Watt (W), equivalently, kg m s 3 how that the powe fo a constant foce, whee v def = d s, is: dt P = F.v = F v cos θ 15 olve poblems using the pevious equation M Patel (eptembe 01) 13 t. Macha Academy

14 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) 16 how that the powe exeted by a constant foce in moving an object with velocity v, whee the foce and the velocity ae in the same diection, is: P = F v 17 olve poblems using the pevious equation 18 Know that enegy is the ability o capacity fo a system to do wok on anothe system 19 Know that enegy is a scala quantity 0 Know that the I unit of enegy is the Joule 1 Know that the kinetic enegy E K of an object is the enegy it has by vitue of its motion, moe pecisely, the kinetic enegy is the wok done in moving an object fom est (at position x 1 ) to a cetain velocity v (at position x ): E def = K x F. dx = m x 1 v v %. dv% 0 Know that the above integal can be evaluated to give: 1 E = mv K 3 olve poblems using the pevious equation 4 Know that the Wok-Enegy Theoem (aka Wok-Enegy Pinciple) states that the wok done by all foces in moving an object fom velocity v 1 to v equals the change in its kinetic enegy: 1 1 W = mv mv 1 5 olve poblems using the pevious equation 6 Know that the potential enegy E P of an object is the enegy it has by vitue of its position o by vitue of the aangement of paticles within the object 7 Know that the gavitational potential enegy of an object of mass m placed at height h in a unifom gavitational field of field stength g is: E = mgh P 8 olve poblems using the pevious equation 9 Know that the gavitational potential enegy of an object of mass m placed a distance fom the cente of a gavitating body of mass M in a non-unifom gavitational field is: M Patel (eptembe 01) 14 t. Macha Academy

15 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) E = GMm P 30 olve poblems using the pevious equation 31 Know that the elastic potential enegy stoed in a Hookean object of stiffness constant k when extended o compessed a distance x fom the equilibium position is: 1 E = kx P 3 olve poblems using the pevious equation 33 Know that the Pinciple of Consevation of Enegy (aka Consevation of Enegy o Enegy Consevation) states that, in a closed system, enegy can be neithe ceated no destoyed 34 Know the altenative fomulation of enegy consevation, namely, that in a closed system, the total enegy is constant ove time: Total enegy at time t 1 = Total enegy at time t ( t 1, t R ) 35 Know that mechanical enegy means E o E P K 36 Know that, fo mechanical systems, enegy consevation states that the total mechanical enegy of a closed system is constant ove time: (Total E ) P t1 + (Total E ) K t1 = (Total E ) P t + (Total E ) K t ( t 1, t R ) 37 olve poblems using the pevious equation, such as motion in a vetical cicle, object dopped in a constant o non-constant gavitational field with no ai esistance, object sliding down a smooth o ough plane, object being pojected up a smooth o ough plane 38 Know that a foce is consevative if the wok done by that foce in moving an object between points is independent of the path taken to go between those points 39 Know that a foce is consevative if the total mechanical enegy is conseved (and vice vesa) 40 Know that gavitation is a consevative foce 41 Know that a foce is non-consevative if the wok done by that foce in moving an object between points depends on the path taken to go between those points 4 Know that fiction and ai esistance ae non-consevative foces M Patel (eptembe 01) 15 t. Macha Academy

16 AH Mechanics Checklist (Unit ) AH Mechanics Checklist (Unit ) Futhe Rectilinea Motion No. kill Done 1 Know that acceleation in 1D can be witten as: a = dv = dv ds = v dv dt ds dt ds Use Newton s nd Law, when acceleation depends on displacement, to fom 1 st ode diffeential equations of the fom: v dv dx = a (x) 3 olve the above sepaable DE, given the specific function a (x) 4 Use Newton s nd Law, when acceleation is a function of velocity, to fom 1 st ode diffeential equations of the fom: v dv dx = a (v) 5 olve the above sepaable DE, given the specific function a (x) 6 Deive the equation v = ω ( A x ) by solving the defining equation fo a HO, namely, x&& = v dv dx = ω x, by sepaating vaiables 7 Know that the teminal velocity of an object is the maximum velocity that it eaches unde cetain conditions, nomally involving ai esistance 8 Calculate the teminal velocity of an object given an expession 3 fo its velocity, fo example, v 17 ( 1 e t ) = i 9 Know that the escape velocity of an object fom a gavitating body is the minimum velocity equied to escape fom the gavitational field of that body 10 how that, fo a spheical gavitating body of mass M and adius R, whee g is the gavitational field stength nea the suface of the gavitating body and G the gavitational constant, the escape velocity v of a body of mass m is: escape v = gr = escape GM R 11 olve poblems using the above equalities M Patel (eptembe 01) 16 t. Macha Academy