Chapte 7-8 Rotational Motion What is a Rigid Body? Rotational Kinematics Angula Velocity ω and Acceleation α Unifom Rotational Motion: Kinematics Unifom Cicula Motion: Kinematics and Dynamics The Toque, τ Elements of Statics Conditions of Mechanical Equilibium Cente of Mass Rotational Dynamics. Moment of Inetia, I Rotational Kinetic Enegy
Rigid Body and its Motions Until now we consideed only the motion of point-like objects. Objects with extended size can be consideed as a collection of many point-like paticles. When these paticles do not move with espect to each othe, the system is called a igid body: it cannot be defomed. The geneal motion of a igid body can be split into two types: Tanslational (Linea) Motion Rotational (Angula) Motion we need an angula fomalism Object and teminology:,, v, a, F, p...,,,,, L... Linea velocity Rigid body A paticle of the body Tanslation velocity Cente o axis of otation
Angula Kinematics Angula Position θ In puely otational motion, all points on the object move in cicles aound the axis of otation ( O ), with each point descibed by a vecto position Def: The angle θ made by the position vecto with espect to an abitay axis (say x) is called angula position θ O y θ x Ex: see the two points on the adjacent bicycle wheel: notice that, as long they ae not on the same adius, the points on the igid body will have diffeent angula positions In ou appoach to otations, angles will be measued in adians: 1 adian is the angle at the cente of a cicle subtending an ac equal to the adius of the cicle When the angle at the cente is expessed in adians, the length of the ac subtended is given by: l Ex: The cicumfeence π of a complete otation subtends an angle of π Convention: angles measued counteclockwise ae positive angle measued clockwise ae negative + l = θ θ = 1 ad l = θ θ
Angula Kinematics Angula Displacement and Velocity How can we use angula positions to descibe otations? Notice that even though the angula positions of diffeent points of a wheel ae in geneal diffeent, when the wheel otates, all points otate though the same angle Def: The change in angula position of all the points on a otating igid body is called angula displacement: 1 Δθ θ θ1 Abitay adius x Then, if we want to efe to how fast the angula position changes we have to define fist the aveage angula velocity as the angula displacement divided by time: t 1 t t t 1 t vey shot Ex: If the bicycle wheel makes two complete otations evey second, we say that it has a constant angula speed of (π/1 s) = 4π ad/s SI ad s Theefoe, like in the linea case, the instantaneous angula velocity is given by:
Angula Kinematics Angula Acceleation. Angula vecto diections Hence we can define the aveage angula acceleation as the ate at which the angula velocity changes with time: 1 t t t 1 Hence, the instantaneous angula acceleation is: SI ad s t t vey shot Although it is not as intuitive as in the tanslational case, the angula velocity and acceleation ae vectos, pependicula on the cicle of otation: The diection of angula velocity is given by a ight hand ule 0 0 The diection of angula acceleation is paallel o antipaallel with the angula velocity depending on whethe ω inceases o deceases
Relating Linea and Angula Kinematics So, the angula displacement, velocity and acceleation chaacteize the entie igid body: all points have the same Δθ, ω and α Howeve, the linea distance, velocity and acceleation of points at vaious adii fom the axis of otation ae diffeent: each point has a diffeent Δl, v and a The linea kinematics of each point on a igid body can be elated to the oveall angula chaacteistics based on the elationship l = θ Fo instance, conside a wheel otating with constant angula speed ω. A point at distance fom the cente of otation will otate with constant linea speed v taveling an ac Δl in a time Δt: theefoe, we obtain l v t t So, as long as they ae not at the same distance fom the cente of otation, the diffeent points on a igid body have diffeent linea speeds, inceasing fom zeo in the cente of otation to a maximum value on the oute im of the otating igid body v 3 = ω 3 v v = ω Δl = Δθ v 1 = ω 1 Δθ 3 1 v ω ω x
How about Acceleation? Tangent component The acceleation is a bit moe complex since in geneal the vecto linea acceleation of a paticle in cicula motion is not tangent to the tajectoy. Howeve, it can be consideed as having two components one tangent to the tajectoy (paallel o anti-paallel with the velocity) and one pependicula on the velocity: a t : tangent, descibes how the magnitude of the linea velocity vaies a : adial (o centipetal), descibes how the diection of the velocity vaies The component a t of a paticle at distance fom the axis of otation can be easily elated to the angula acceleation α of the igid body: a a t a t v t t a x The component a (also called centipetal since it always point towad the cente of otation) makes necessay a sepaate discussion a few slides futhe ω
Execise 1 : Components of acceleation A paticle moves as shown in the figue. Between points B and D, the path is a staight line. Let s figue out the net acceleation vectos in points A, C and E along the path epesented below, fo each of the following cases: a) the paticle moves with steadily constant speed b) the paticle moves with steadily inceasing speed c) the paticle moves with steadily deceasing speed a 0
Angula Kinematics Unifomly acceleated otation The linea-angula elationship povides an easy way to descibe cicula motion with constant angula acceleation α, by simply noticing that each point on the otating igid body acceleates unifomly along the espective ac of cicle Assume that the motion stats at t 0 = 0 when the otation is chaacteized by θ 0, ω 0 v 0 ω 0 t 0 = 0 At time t ω Δl x v Δθ x Then, if the angula acceleation α is constant, at a late instant t, The linea motion of one paticle of the igid body at distance fom the cente of otation l v t a t 1 0 v v a t 0 v v a l 0 1 0 l v v t t t t if each linea quantity is divided by, we obtain t t t t 0 1 0 0 t 0 1 0 0 t The otational motion of the entie igid body (valid fo any of its pats)
Poblem: 1. Unifomly acceleated otation: An automobile engine slows down fom 4500 pm to 500 pm in.5 s. Calculate a) its angula acceleation (assumed constant) b) the total numbe of evolutions the engine makes in this time. 4500 0 4500pm ad 60s 500 1 500pm ad 60s s 150 ad s 83 ad s s
How about Acceleation? Centipetal Acceleation v So, the adial (o centipetal) acceleation descibes how the diection of the velocity changes: that is, if the object moves in a staight line, a = 0 The centipetal acceleation a is elated to the instantaneous speed v of the paticle moving in a cicle of adius by v 1 a v 1 a v path a v This expession is valid fo any paticle moving along a cuved tajectoy: if the cuvatue of the path can be fitted locally by a cicle of adius and the instantaneous speed is v, the expession above gives the adial component of the acceleation in the espective point a a v 1 1 1 v 1 a v a v path
Unifom Cicula Motion Kinematics The unifom cicula motion is the motion of a paticle in a cicle of constant adius at constant speed, such that α is zeo Being always tangent to the cicula path the vecto instantaneous velocity changes diection, albeit its magnitude stays constant Hence, the tangential acceleation a t is zeo, and the net acceleation is given only by the centipetal acceleation a pointing eveywhee pependicula on the velocity Comments: The magnitude of the centipetal acceleation is lage if the speed is lage The magnitude of the centipetal acceleation is lage if the adius of otation is small path a Constant speed Ex: if a ca takes a tun at high speed, it will have a lage centipetal acceleation than when taking it slowly Ex: if a ca takes a tun shap tun (small adius), it will have a lage centipetal acceleation than when taking a wide tun v a v
By Newton s nd Law, since in the cicula motion the acceleation is necessaily not zeo (since the velocity must change in diection), we see that fo an object to be in unifom cicula motion thee must be a net foce acting on it. We aleady know the acceleation, so can immediately wite the foce: F ma m v This centipetal foce is not a new foce: any net foce pointing adially inwad the cicula tajectoy (pependicula on the velocity) can play the ole of centipetal foce, since it has as a esult a change in diection of velocity Comments: Unifom Cicula Motion Dynamics A common misconception is to assume that an object on a cuved tajectoy is thown out of it by an outwad centifugal foce. We now see that the foce must be actually inwad The objects taking tuns ae appaently pushed outwad by thei inetia, while the centipetal foce keeps it on the tajectoy path If the centipetal foce vanishes, the object flies off tangent to the cicle, not outwad as if a centifugal foce wee pesent F v
Execise : How Angelina Succumbed to Bad Physics In the movie Wanted, bullets ae cuved by skilled tattooed assassins. Fo instance, Angelina kills heself by fiing a bullet in a cicle passing though the skulls of some bald dudes befoe hitting he. Say that the bullet has a mass m = 5 g and a muzzle speed of 500 m/s. Also, say that the dudes aanged themselves conveniently in a cicle with adius = 5.0 m. a) How lage should be the centipetal foce keeping the bullet on the cicula tajectoy? Meditate about the possible oigin of such a foce and how ealistic is such a scenaio b) How fast should Angelina toss the gun to the sweaty guy in the middle fo the scene to make sense?
Poblem:. Nomal as a centipetal foce: A small emote-contol ca with mass m moves with a constant speed v in a vetical cicle inside a hollow metal cylinde of adius. What is the magnitude of the nomal foce exeted on the ca by the walls of the cylinde in tems of these given quantities at a) an abitay point on the loop b) point A (bottom of the vetical cicle) c) point B (top of the vetical cicle)
What causes otations? To otate an object which is initially at est, a foce is needed Beside the magnitude of the foce, the otational effect of the foce depends on the position of the foce with espect to the axis of otation, and on its diection So, in ode to study how one can poduce otations, one must take into consideation all these quantities, not only the simple foce as in the puely tanslational case Let s fist define some geometical chaacteistics: Def 1: A line along the vecto foce is called the line of action Def : The pependicula distance fom the axis of otation to the line of action is called the leve am Ex: Conside a doo acted upon by a foce F. The otational effect depends on a) the stength (magnitude) of the foce b) the position whee the foce is applied with espect to the hinges (axis of otation) smalle effect F c) the diection of the foce (o line of action) zeo effect maximum effect F F lage effect F
F sin Toque Definition Def: The physical quantity that models the otational effect of a foce is called toque, τ, with magnitude defined by taking into account all the influences upon the otation: mn Magnitude of foce F Distance to the axis Angle between F and SI θ F Axis of otation What about the diection of the toque? The vecto toque is pependicula on the plane of the otation that it attempts to poduce We can descibe the diection using a sign convention: τ > 0 τ < 0 if it ties to otate the if it ties to otate the object counteclockwise object clockwise Ex: A longe leve am is vey helpful in otating objects: this is why a wench can be used to loosen a bolt Fomally, this is due to the fact that a longe inceases the toque fo the same F and θ Also, applying the foce pependicula on the am will maximize the toque fo the same F and, since sinθ is maximum when θ = 90 +
Poblem: 3. Net toque on a wheel: Two foces, F 1 = 7.50 N and F = 5.30 N, ae applied to a wheel with adius = 0.330 m, with F half-adius fom the axle, as in the figue. What is the net toque on the wheel due to these foces fo an axis pependicula on the wheel and passing though its cente? F 1 F Axis of otation
Statics Elements Based on ou discussion about diffeent types motion, we see that the motion is contolled by foces in the case of tanslations and toques in the case of otations Theefoe, we can intoduce the conditions fo an object to be completely at est (o moving with constant velocity) A solid body that is at est is said to be in motional equilibium To be in complete equilibium, a solid body must satisfy both tanslational and a otational conditions of equilibium: 1. if the net foce is zeo thee is no tanslation F net 0 F F x y. if the net toque is zeo with espect with any abitay cente of otation (pivot), thee is no otation net 0 Since the pivot is abitay, we can choose its such that the equation won t contain unknown foces 0 0 Ex: Rotation without tanslation: Even when the net foce is zeo, the object can still otate. So, a zeo balance of foce does not gant complete equilibium
Poblem: 4. Rotational equilibium: A unifom, 56 kg beam is suppoted using a cable connected to the ceiling, as shown in the figue. The lowe end of the beam ests on the floo. What is the tension in the cable? 56 kg
Rotational Dynamics Toque and Rotational Inetia Let s fist conside a paticle pefoming cicula motion of adius Now we know that, if its speed vaies, thee must be a tangential foce acceleating it, such that, by Newton s nd law If we conside the paticle as a igid body, what is the coesponding toque? F ma m I t F ma m t t t path v F t This is fo a single point mass; what about a igid body seen as an extended object containing many paticles of mass m i at positions i fom the cente of otation? Then, since the angula acceleation is the same fo the whole object, we can wite: net i m i i Def: moment of inetia of a paticle I m Def: moment of inetia of a system of paticles I I kg m SI m i i i
So, the otational inetia of an object made of many paticles is given by its moment of inetia I m i i Notice that the same object can have diffeent moments of inetia with espect to diffeent axes of otation The otational inetia inceases the futhe the mass is distibuted fom the axis of otation Quiz: The two cylindes shown have the same mass. Which has a lage otational inetia? In this class, the moments of inetia of igid bodies with continuously distibuted mass will be given (see the adjacent table) The net moments of inetia of a system of objects with known individual moments I i is I I I I net Moment of Inetia Concept and fomulas 1 3...
Comments: Newton s nd Law fo Rotations A net toque Στ acting on a igid body with moment of inetia I will detemine an angula acceleation α popotional to the toque as given by: net I This law woks only fo igid bodies (such that α is the same fo all pats) Like foces that can modify the motion of an object only they ae extenal foces, the angula acceleation is detemined only by extenal toques: the intenal toques cannot modify the otation of the system Notice that the angula acceleation has the same vecto diection as the net toque (pependicula on the cicle of otation) Ex: The weight of a igid body will play a ole in the otation in most cases. Since the weight can be consideed as acting in the cente of mass (cm), the weight will have a toque and otate the body only if the if the cente of otation doesn t pass though the cm. Pivot mg cm
Poblems: 5. Net moment of inetia: A helicopte oto blade can be consideed a long thin od, as shown in the figue. a) If each of the thee oto helicopte blades has length L = 3.75 m and has a mass m =160 kg, calculate the moment of inetia of the thee oto blades about the axis of otation. b) How much toque must be applied to bing the blades up to a speed ω in a time t?
The otational kinetic enegy of a igid body fomed of paticles indexed by i (as the wheel on the ight) is the sum of the kinetic enegies of all its pats 1 1 1 KEot m1v 1 mv... mivi By substituting the otational quantities, we find that the otational kinetic enegy can be witten KE m m 1 1 1 ot i i i i I KE mv I net Enegy in Rotational Motion Kinetic enegy Net moment of inetia I of the system of paticles Theefoe, a igid body that has both tanslational motion (motion of its cm) and otational motion (about its cm) has both tanslational and otational kinetic enegies: 1 1 cm cm Tanslation Rotation Physical situation: Conside some igidly connected paticles otating about thei cente of mass with angula speed ω, and in the same time tanslating with speed v cm Then, an abitay paticle of mass m i located at distance i fom the cente of otation, moves with speed v i Rotational velocity, ω By the adjacent agument, the net kinetic enegy is the sum of the kinetic enegies of the two motions v i i Tanslational velocity v cm
Enegy in Rotational Motion Consevation When evaluating the consevation of enegy fo otating igid bodies, the only change fom ou pevious appoach is that, beside the kinetic enegy associated with the tanslation, the total kinetic enegy contains a otational tem So the expession fo the mechanical enegy becomes E KE KE PE mv I kx mgy 1 1 1 tansl ot cm The consevation of enegy can still be witten E W fiction Execise 3: Rolling motion. Assume that all the objects on the figue have the same mass m and ae all eleased down the fictionless incline fom est, fom the same initial height. If the adius of each olling object is the same, which object will move faste at the bottom of the incline?