r CM = ir im i i m i m i v i (2) P = i

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1 Physics 121 Test 3 study guide Thisisintendedtobeastudyguideforyourthirdtest, whichcoverschapters 9, 10, 12, and 13. Note that chapter 10 was also covered in test 2 without section 10.7 (elastic collisions), so please see the test 2 study guide as well. Chapter 9: Conservation of momentum New concepts: Center of mass; momentum of a system of particles and momentum conservation; impulse The center of mass of a system of things (including ordinary objects, since they re made up of lots of littler particles) is defined to be r CM = ir im i. (1) The expression in the denominator is just the total mass M = i m i, and r i is the position vector of the particle labeled by i (and m i its mass). Like all vector formulas, this is really three equations in one; it holds for each component of the vector equation separately. For instance, taking the dot product of both sides with ŷ gives (since r ŷ = y) the y component of the equation: y CM = ( i y im i )/M. The center of mass is important for many reasons. To name just a few: it is the point at which gravity acts like it is pulling on any extended object (that is, everything which is made up of lots of littler pieces), it is the point around which an object will rotate in free space, and the center of mass will have a simple overall motion even when individual particles are moving in a complicated way. The total momentum of a system of particles is defined to be i m i P = i m i v i (2) but because each particle s velocity v i is just d(r i )/dt, we can pull d/dt outside the sum and use the definition of the center of mass to write P = Mv CM (3) where v CM = dr CM /dt, and M is the total mass, as above. The total momentum changes with time according to F netexternal = dp dt. (4) The reason only external forces appear is that every internal force (that is, one caused by some member of the system, acting on another member of the system) is always cancelled by an equal and opposite internal force (by Newton s Third Law). A special case of this is when the external forces vanish: then we have conservation of momentum: total momentum P of a system of particles is constant with time when there are no net external forces. Together with 1

2 conservation of energy, conservation of momentum is one of the cornerstone rules which seem to be true in physics: no experiment has ever found any violation of the principles of conservation of energy and momentum. In other words, it seems to be really, really, REALLY true, ALL the time! In applying the rule of conservation of momentum to collisions between objects, add up the momenta ( i m iv i ) of all objects before the collision, and make it equal to the sum of the momenta of all objects after the collision. Exception: if one of the objects is VERY much more massive than the other (such as a something small colliding with a brick wall or the whole Earth), then treat the massive object as having constant momentum (usually zero), and reverse the momentum of the incoming object(s) if the collision is perfectly elastic (see below). (This is really only an exception to how momentum conservation is used in practice in these situations; it s just the limiting behavior of the rule of conservation of momentum when one of the masses approaches infinity, and NOT an exception to conservation of momentum!) Momentum conservation (in each direction) usually forms one (or more) of the equations needed to analyze a general collision, but we usually need more information to be able to completely specify what happens. Often, that information can be what happens with the kinetic energy. A perfectly elastic collision has conservation of kinetic energy, while a totally inelastic collision isoneinwhichtheobjectssticktogether. Forexample, ifadroppedballbounces back to its original height, its kinetic energy must have been conserved when it collided with the floor, so the collision was perfectly elastic. If a ball of clay is dropped on the floor, it often sticks without bouncing at all; that s a totally inelastic collision. Most collisions of large objects are only partly elastic; kinetic energy is lost to heat (and sound, and possibly other forms) in any collision that s not perfectly elastic. In one dimension, conservation of momentum together with information about the kinetic energy lost (such as whether the collision was elastic or not) is enough to solve for two unknowns (such as the final velocity of two objects, or one velocity and one unknown mass) in a collision. But if a collision uses two or three dimensions, additional information is needed to analyze the system completely. This extra information is often given as the direction of one of the final velocites (such as in a billiards collision, where the forces act parallel to the line between the centers of the two balls at impact), or as one known component of a final velocity in some direction. In collisions (such as a cue stick with the cue ball in billiards), the details of the forces and accelerations are usually not as important as the total amount of momentum transferred from one object to the other. This difference of momentum is called the net impulse. The definition of impulse comes from undoing the equation F net = dp (5) dt and undoing a derivative always involves doing an integral, so impulse (total 2

3 momentum transfer, or p is equal to p = F net dt. (6) One can also define the impulse transferred by each force when several forces are involved; just replace the net force above by the individual force. Total impulse is the sum of all individual impulses. When a collision is 2-D (such as a billiard ball collision that s not head-on), we end up with extra unknowns coming from needing to know the individual x and y components of the final velocities. So, we need more information! This extra information is usually supplied in terms of the angle (of the line between the ball centers) at the moment of impact. The net impulse is along this direction, so that s usually the extra bit of information needed. For example, if the target ball was initially at rest, it will travel along the direction given by that line (and that s how to aim in billiards!). In collisions in particle physics, the extra information similarly comes from the direction the final particles are moving compared with the directions of the initial ones. Chapter 10: Conservation of Energy New Concepts: elastic collisions (section 10.7) Recall the definition of kinetic energy: K = 1 2 mv2 (7) Please see the chapter 10 notes in the test 2 study guide. The new material for test 3 is elastic collisions: those whose overall kinetic energy is conserved during the collision. Most actual collisions of macroscopic bodies are only partly elastic, but on the microscopic scale collisions are nearly always totally elastic. The difference is that in large bodies, kinetic energy can be transferred into microscopic kinetic energy of the individual molecules of the objects, which we call heat. The simplest case is where two objects have known initial velocities, and then collide with one another but do not stick together. Then in one dimension, there are two unknowns to solve for (the final speeds of each particle). Conservation of momentum (which always applies in a rapid collision) supplies one necessary equation, and for elastic collisions, conservation of kinetic energy gives a second independent equation which allows us to solve for the final velocities. Because kinetic energy depends upon speed squared, a quadratic equation typically will arise. Quadratic equations usually have two independent solutions; typically, one of these will give the post-collision velocities while the other will correspond to no collision having happened (which also conserves both momentum and kinetic energy). In more than one dimension, there are more unknowns (due to the final velocities having several components), so we need more equations to solve for 3

4 what happens. Typically this can be given in terms of some angle: for instance, the angle of the vector impulse that was transferred (along the line connecting the centers of two billiard balls, for collisions of those), or the final directions of the particles can be used. Chapter 11: Rotational Vectors and Angular Momentum(Angular Momentum in Three Dimensions) New concepts: Moment of inertia, parallel-axis theorem; rotational kinetic energy; angular velocity and torque as vectors; mathematical cross product (vector product); angular momentum; conservation of angular momentum when net external torque vanishes Recall the definition of angle in radians: θ = s/r, where s is the arc length along the portion of a circle spanned by angle θ, and r is the radius of the circle. So, radians are dimensionless (and can be counted as no units at all in equations). We can apply the same calculus definitions to anglular things that we apply to distance things, so we define the angular velocity ω = dθ dt (8) (units of ω = rad/s = 1/s), and the angular acceleration α = dω dt = d2 θ dt 2 (9) (units of α = rad/s 2 = 1/s 2 ). For rotations in a single plane, these are conventionally measured as counterclockwise from the direction that the plane is being viewed. In more than two dimensions, these will become vector quantities (see below). If the angular acceleration α is constant, there are constant angular acceleration equations to match each of the constant linear acceleration equations: α = constant : (10) ω = ω 0 +αt (11) θ = θ 0 +ω 0 t+ 1 2 αt2 (12) ω 2 = ω α(θ θ 0 ) (13) The angular equivalent of mass (inertia) is called the moment of inertia I (also called angular inertia or rotational inertia in some books), defined for a bunch of point masses m i to be the sum I = i m i r 2 i (14) 4

5 (units of I = kgm 2 ), where r i is the distance of mass m i from the axis of rotation. For this reason, I depends upon the axis of rotation! For an example, if all mass is at a common distance R from the axis of rotation (such as for a hoop), then we have simply I = MR 2, where M is the total mass. The values of I (already summed or integrated) for some common extended (non-pointlike) objects are listed in Table 12.2 on page 318 of your book. If the moment of inertia I CM is known for an axis passing through the center of mass (CM), then the moment of inertia about some axis parallel to that which does not pass through the CM is given by the parallel-axis theorem: I = I CM +Md 2 (15) where d is the distance of the new axis from the center of mass, and M is the total mass of the object. For the angular equivalent of Newton s Second Law, we need the angular equivalent of force, which is torque (τ). The magnitude of the torque produced by a force F at a distance r away from the axis of rotation is given by τ = F r sinθ (16) where θ is the angle between F and r, the vector pointing from the axis of rotation to where the torque is. Torque in a plane is simply considered to be clockwise or counterclockwise in the plane of rotation. Below, we shall consider torque in more than two dimensions where it will become a vector, and we shall see that the above equation gives the magnitude of the cross product of the vectors r and bf. Torque is measured in units of Newton meters (N m), one of which is one Newton times one meter. This is equivalent to a Joule (the unit of energy), but Newton meters is preferred for torque to keep the concepts separate (torques are almost never added to nor subtracted from energies, despite having the same units). If the moment of inertia I is constant, we have the equivalent of Newton s Second Law when m is constant: τ net = Iα (17) (exactly equivalent to F = ma for constant m). We will see how to generalize this formula to accomodate changing I in terms of angular momentum, defined below. We can construct the kinetic energy due to rotation by simple analogy: K rot. = 1 2 Iω2. (18) It is simply regular kinetic energy (measured in Joules), taking into account the fact for constant angular velocity ω, the actual linear speed v of each piece of the object depends upon the distance r that the object is away from the center of rotation. The relation between the two is v = ωr. (19) 5

6 A very similar formula is that for rolling motion, which says that an object of radius R rolling without slipping at velocity v has an angular velocity ω according to v = ωr. (20) When using conservation of energy for objects that are both rolling and moving, the correct way to add the kinetic parts is K tot. = 1 2 Mv2 CM I CMω 2 CM (21) and to use the above relation for rolling motion between v and ω. The general result of this is that objects with large rotational inertia I compared to MR 2 (that is, large k where k is defined by I = kmr 2 ) store a larger fraction of their kinetic energy in rotational form, and so they have smaller changes in their velocity (compared to low-k objects) for the same change of potential energy(thisisgivenbythechangeinheighttimesmg forgravitationalpotential energy). Generalizing rotations to more than one plane requires making torque, angular velocity, and angular acceleration into vectors. The rule for assigning a direction to a vector angular velocity is one of several things you will learn which all go by the name of the right-hand rule: If your right hand s fingers curl in the direction of rotation along the plane of rotation, then your right hand s thumb (pointing at right angles to the fingers) points in the direction of vector angular velocity ω, by convention. Put another way, it is the direction a normal screw would move if turned in the direction of rotation (into the page for a clockwise rotation, for example). Practice the right-hand rule until you are sure you know it very clearly! Similarly, angular acceleration has a direction which can be compared with the direction of angular velocity. Ifω and α are parallel, the rotation is speeding up; if the vectors are opposite, the rotation is slowing down. Like regular acceleration and velocity, the angular versions can also be at right angles. In thatcase, theeffectcanbesimilartocircularmotionbutfortheangularvelocity vector (see the discussion on tops below). Next, we introduced vectors for some of the other rotational things. But first, we needed the mathematical definition of the cross product, a function which takes two vectors and spits out a third vector at right angles to the first two. (Since it gives a vector for an answer unlike the scalar dot product which gives a scalar, it is also known as the vector product.) Since it is a vector, we need both its magnitude and direction. The magnitude is given by v w = v w sinθ (22) where θ is the angle between v and w. Notice that the only difference from the dot product here is that sin instead of cos of the angle shows up. For this reason, cross products are the largest when the two vectors are perpendicular (not parallel, as the dot product had). The direction of the cross product is given by yet another right-hand rule: 6

7 Curl your right hand fingers from the direction of v toward the direction of w along the smallest angle possible, and your right hand thumb will point in the direction of v w. Again, it is a good idea to practice this (especially with torques and angular momenta, as defined below) until you are sure that you know how to use this rule, for it is often used for angular things! Like the dot product, the cross product also has a component version. If the individual components of the vectors to be crossed are known, the dot product is also given by the component formula (v w) x = v y w z v z w y (23) (v w) y = v z w x v x w z (24) (v w) z = v x w y v y w x (25) This is completely equivalent to the magnitude and direction version given before. Since this gives the components of v w explicitly, the magnitude and direction information is implicitly included. Now for the physics using the cross product: the vector torque (τ) caused by a force F at a location r away from the rotation axis is τ = r F. (26) (Notice that the rule for the magnitude of the cross product shows that this agrees with the definition of torque in a plane given earlier.) The right hand rule for cross products shows that a torque which tends to produce clockwise angular acceleration points into the page by this convention, in agreement with the sense of the right hand rule for angular velocity vectors (the other right hand rule in this study guide!). So, the two right hand rules are related. Another important concept is the vector angular momentum (L) of an object with momentum p at a location r away from the axis of rotation: L = r p (27) showing that the units of L are kgm 2 /s. Notice that this definition allows that objects can have nonzero angular momentum even if they are not rotating, depending on the axis of rotation chosen! If there are lots of objects with different momenta p i at different distances r i, then just add the pieces up: L total = i r i p i. (28) For an object that is actually rotating with angular velocity ω, this recipe gives where I is the angular inertia about the axis of rotation. L = Iω (29) 7

8 Now, for the dynamics of angular momentum. Adding all the torques on an object (or collection of objects) gives τ netexternal = dl dt (30) Only external net torque shows up for the same reason that only external force did for change of ordinary momentum: internal torques come from internal forces, which cancel by Newton s Third Law. Notice that this equation is almost in the form of being essentially r the familiar equation F = dp/dt; the only difference is that the time derivative also acts on the r bit in the angular version. A nifty application of this is the top: if a top is spinning rapidly and is tilted away from being straight upright, the gravitational torque (and so also α) can be at right angles to the angular momentum L that is already there from the spin. So the top s angular momentum can have a component which moves in a circle, and the top precesses instead of falling over. The rotational inertia of a top is important for this effect; objects with a small I will fall over rather than precess (as can be seen theoretically when energy conservation is also considered). If all external torques vanish, we have conservation of angular momentum: L does not change with time, just as regular total momentum is constant when there is no external force. The constant direction of the spin of a gimbalmounted gyroscope and the fact that Earth s spin vector (along the north pole) point the same direction in space are static consequences of the conservation of angular momentum. For a more dramatic consequence, when the rotational inertia I changes, ω must change to keep L constant; this is how a spinning figure skater speeds up when moving their arms in (to decrease I). Chapter 13: Newton s Law of Universal Gravitation New concepts: Gravitational force between any two masses; potential energy of gravity; orbital speed and escape velocity Concentrate on: force and potential energy between two objects with mass (M and m) F = GMm r 2 attractive (31) U = GMm r Use of potential energy is conservation of energy, as always: (32) K i +U i = K f +U f = E total (aconstant) (33) Types of gravitational orbits: open (E > 0 hyperbola shape, or E = 0 parabola shape), closed (E < 0 ellipse or circle shape) Speed of a circular orbit at radius r: v circ = GM/r 8

9 Escape speed from radius r: v esc. = 2GM/r Gravitationalforceisradial(alongr), soitprovideszerotorque(r F G = 0). Therefore, it conserves angular momentum, as shown in the chapter 12 notes. L = r p = constant. This is often a more convenient way than conservation of energy to find the speed of an orbiting object, especially at points where p is perpendicular to r (such as the near and far points of an elliptical orbit). 9

= o + t = ot + ½ t 2 = o + 2

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