Physics 121 Test 1 study guide This is intended to be a study guide for your first test. The concepts are organized into the relevant chapters of the book. Chapter 1: Concepts of motion; units and dimensions Know how to analyze units of any formula. If any two quantities are added, subtracted, or set equal, they must have the same physical units! If other mathematical operations (multiply, divide, raising to a power, etc.) are performed on physical quantities, perform the same operations on the units that you perform on the numbers. Know how to convert units. A handy rule is that any conversion factor (equivalence of units) can be made into the number 1 by dividing both sides of the equation by what appears on one side. For example, 1 inch = 2.54 cm, so 1 = 1 inch/(2.54 cm) and 1 = 2.54 cm /(1 inch). Use multiplication or division by the number 1 (which never changes the quantity there). If a unit has been raised to a power, convert it by raising the appropriate version of 1 to the same power. For example, 1 cm = 10 mm, so we can write 1 = 10 mm/(1 cm). So 3

3

1 cm = 1 cm



10 mm 1 cm

3

= 1000 mm3 .

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Know how to use scientific notation. For example, we write 32140 as 3.214 × 104 and 0.023 as 2.3 × 10−2 in scientific notation. The first part is called the “base” and the power of 10 the “exponent.” To add or subtract numbers, convert them so that they have the same exponent first. To multiply, multiply the bases and add the exponents. To divide, divide the bases and subtract the exponents. Know what significant figures are, and how to apply the correct number of significant figures to a result. The rules for mulitiplying and dividing are to use the fewest number of significant figures that appear in the things being multiplied or divided. For addition and subtraction, use the least significant digits place when the numbers are lined up so that their decimal points are directly above one another. Know the defintion of the radian: Angle in radians = (arc length of a portion of the circumference of a circle spanned by that angle) divided by (the radius of the circle). Know what motion diagrams are: drawings showing the location of the object at different times, with the times labeled. From a motion diagram, you can construct average velocities and accelerations in the vicinity of each point listed (see the next chapter’s notes for details). Chapter 2: Kinematics in one dimension Know the definitions of average and instantaneous velocity, and average and instantaneous acceleration: vave. =

∆x ; ∆t

v= 1

dx dt

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dv ∆v ; a= (3) ∆t dt Average velocity between two points is their distance apart divided by the time it took to get between them, and is the average slope of the graph of position versus time between the points. Instantaneous velocity is the slope of the position versus time graph. SI units of velocity are m/s. Similar remarks apply to acceleration, if “position” is replaced by “velocity” above. Thus, acceleration is the slope of the velocity versus time graph (and the curvature of the position versus time graph). SI units of acceleration are m/s2 . Know the formulas for constant acceleration: aave. =

1 x = x0 + v0 t + at2 2

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v = v0 + at

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These equations can be combined; for instance one may solve for t in the second equation and plug the result in for t in the first to get an equation without time in it: v 2 = v02 + 2a(x − x0 ) (6) Do not apply these formulae when acceleration is not a constant! Instead, use the calculus definitions of velocity and acceleration as derivatives (given above). Special case of constant acceleration: acceleration due to gravity = g = constant near the Earth’s surface = 9.8 m/s2 directed downward. For motion on a plane with angle θ made from the horizontal, know that gravity (which points straight down) will make the angle θ as well but with respect to the direction that is perpendicular to the surface on the plane (see fig. 2.31 on p. 54 for this). Chapter 3: Vectors and coordinate systems Know what a vector is: a quantity with direction in addition to its magnitude (length of the vector). Vector notation may be given as magnitude and direction, separate components for each direction in space, or a combined notation where ˆ the components are multiplied by the unit vectors in each direction ˆi, ˆj, and k which are vectors of length 1 (and no physical units!) pointing in the positive x, y, and z directions (respectively). For example, a velocity vector v might be written in any of the following equivalent ways: |v| = 10 m/s; θ = 0 rad CCW from + x axis v = 10 m/s ˆi + 0 ˆj vx = 10 m/s; vy = 0 . Know how to add and subtract vectors, and how to multiply vectors by scalars (ordinary numbers). To add: total x component is sum of individual x components, and the same applies to the y and z directions. To add graphically, put tail of second vector onto tip of first vector and draw vector from tail of first vector to tip of second. To subtract, add the negative of the vector. To multiply by a scalar, multiply the magnitude by the scalar (including units!), and keep 2

the same direction (exception: direction will flip if scalar is a negative number; this is because vector magnitudes are always considered to be positive). Know what a vector formula stands for: the equation is true separately for each x, y, and z component of the equation. Know how to reconstruct the magnitude and direction of a vector given its components. Magnitude of a vector (for example, force F) is given from its x and y components by the Pythagoras theorem: q (7) |F| = Fx2 + Fy2 . Direction angle measured counterclockwise from the +x axis is given by θ = arctan

Fy Fx

(possibly, + π rad or 180 deg.)

(8) (9)

where the rule is to add 180 degrees (π radians) to the result if your calculator returns the angle in the wrong quadrant (always check with a sketch!). Know also how to break magnitude and direction form of a vector into its components. If the direction angle θ is measured counterclockwise from the +x axis, then the rules (for a force F) are Fx = |F| cos θ

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Fy = |F| sin θ .

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If the direction is not measured counterclockwise from the +x axis, use the general trigonometry formulas to find components. Draw a right triangle with the hypotenuse (side opposite to the right angle) as the total vector, with the sides joined by the right angle aligned with the coordinate directions representing the components of the vector in those directions. Then, the trigonometry definitions are: sine = opposite/hypotenuse cosine = adjacent/hypotenuse tangent = opposite/adjacent = sine/cosine and can be used to find the components in the coordinate system used. Chapter 4: Kinematics in two dimensions For constant acceleration in several dimensions, we just need to make acceleration, velocity, and position all into vectors (already done in the ch. 1 notes). Understand that this applies to each direction separately: a vector equation is shorthand notation for what’s happening in each independent coordinate direction. Know how to apply the constant-acceleration formulae in the special case of acceleration due to gravity (projectile motion): if a = −g ˆj, then the separate component equations are x = x0 + vx0 t (12) 3

vx = vx0 1 y = y0 + vy0 t − gt2 2 vy = vy0 − gt

(13) (14) (15)

Know the properties of purely circular motion: acceleration is directed toward the center of the circle (of radius r) and has the magnitude acirc. = v 2 /r

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when the object has velocity v around the edge of the circle. Circular acceleration changes the direction of the velocity, but not its magnitude! Purely circular acceleration is always perpendicular to the velocity vector. Note that circular (also called centripetal) acceleration is not an example of constant acceleration! Although the magnitude of centripetal acceleration is a constant, its direction is constantly changing (staying pointed toward the center of the circle) – so the equations for constant acceleration as given above don’t apply to circular motion. However, if we work with angles as the variables (instead of linear positions), there are some analogous equations. First, 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 angular things that we apply to distance things, so we define the angular velocity ω=

dθ dt

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(units of ω = rad/s = 1/s), and the angular acceleration α=

d2 θ dω = 2 dt dt

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(units of α = rad/s2 = 1/s2 ). For motion in a circle, velocity is motion along the arc length of the circle, so using the definition of θ in radians above we find for circular motion v = ωr

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Also, circular motion is periodic (repeating), so the time for one full cycle carries a special name, the period T . Time is distance over speed, so for a circle of radius r we have 2πr 2π T = = . (20) v ω

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If the angular acceleration α is constant, there are constant angular acceleration equations to match each of the constant linear acceleration equations: α = constant :

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ω = ω0 + αt

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1 θ = θ0 + ω0 t + αt2 2

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ω 2 = ω02 + 2α(θ − θ0 )

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Note that any point on an object experiencing angular acceleration α will have two components to its ordinary acceleration vector: one component either parallel or opposite to its velocity (showing that the ordinary velocity is speeding up or slowing down) and the “usual” component of centripetal acceleration, of strength v 2 /Rcircle , pointing toward the center of the best-fit circle to the object’s path. Neither one of these is a constant vector – they will both be changing direction in general – so that’s why the linear constant acceleration equations for x given in chapter 2 do not apply. (The above constant ANGULAR acceleration equation for θ will apply though, if α is a constant.) Chapter 5: Force and motion Know Newton’s laws of motion and what they mean: FIRST LAW: A body moving at constant velocity will continue to do so unless acted upon by a nonzero net outside force. In particular, a body at rest will remain at rest unless a net force acts on it. SECOND LAW: Force acts to change momentum p (see definition below) according to the equation F = dp/dt. In the special case that the object is not losing (or gaining) mass, this reduces to F = ma. (Not until chapter 7 will our book cover the third:) THIRD LAW: For every force of action, there is an equal and opposite reaction. If object A exerts a force on object B, then object B exerts an equal and opposite force on object A. Know the defintion of momentum: p = mv (valid for velocities small compared with the speed of light). This will be introduced formally in chapter 9, but a description of Newton’s Second Law is not complete without it, so I will give it to you earlier. Definition of a body’s weight = force due to gravity = mg. Know what the normal force N is (called n in your book): a contact force exerted by surfaces which are touching. The normal force exerted by a surface is always pointed perpendicular to and outward from the surface. It can adjust itself as necessary to keep two solid objects from passing through one another. Know how to draw a free-body diagram for any object or set of objects: a diagram showing all the separate forces acting on an object (neglecting internal forces which cancel by Newton’s Third Law). Drawing the free-body diagram 5

is very often the first place to begin when analyzing a system’s state of motion and acceleration. Chapter 6: Dynamics in 1-D and using Newton’s laws (Section 6.5 on drag may be skimmed in this chapter.) Except for the introduction of frictional forces (see below), most of this chapter is devoted to analyzing forces and motions in one dimension, and makes heavy use of the following concepts from previous chapters (all of the following up to frictional forces is old material). The basic Newton’s Law at the heart of this chapter is the SECOND LAW: P The net force on an object changes its momentum p according to F = dp/dt, where p = mv for objects moving much P less than the speed of light. If the object is not losing mass, this reduces to F = ma. One of the most important (and easiest) applications of this is statics: if an object is at rest or moving at constant velocity, its acceleration vanishes, so the sum of all forces on it must add up to zero. To use the second law, the first step is usually to draw the free-body diagram: a diagram showing all the separate forces acting on an object or collection of objects (and internal forces, which cancel by Newton’s Third Law, can be neglected if a collection of objects is being analyzed). Step 2 is to draw a coordinate system for each free-body diagram, with x and P y axes perpendicular to each other; then step 3 is to apply the components of F = ma separately to the x and y directions. If the direction of acceleration is known, it’s often easiest to align one of the axes with the acceleration direction. If the acceleration is zero or not yet known, it’s usually easiest to align the axes with the most forces possible (so the forces don’t have to be broken up with trigonometry). For breaking a force (or other vector, such as velocity or acceleration) into components, the rules are Fx = |F| cos θ (25) Fy = |F| sin θ

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if the direction angle θ is measured counterclockwise from the +x axis. If the direction angle is measured differently, use trigonometry directly: make the vector be the hypotenuse of a right triangle whose two sides are aligned with the x and y directions, and then sin = opp./hyp. ; cos = adj./hyp. ; tan = opp./adj. = sin/cos. To reconstruct the magnitude and direction from components: magnitude of a vector (in 2-D) comes from Pythagoras’ Theorem: q (27) |F| = Fx2 + Fy2 . (In three dimensions, Fz2 would also be added to the stuff inside the square root above.) Direction angle measured CCW from the +x axis is θ = arctan

6

Fy Fx

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(possibly, + π rad or 180 deg.)

(29)

where the rule is to add 180 degrees (π radians) to the result if your calculator returns the angle in the wrong quadrant (always check with a sketch!). (Note that if you use arccos or arcsin instead of arctan to find angle, the quadrantfixing rules are different than this; only arctan behaves precisely this way.) For motion at constant speed in a circle of radius R, the acceleration points directly toward the center of the circle and has magnitude acirc. = v 2 /R. New Material: Frictional Forces Frictional forces from surface contact always point parallel to the surface (that is, perpendicular to the normal force, which points perp. to the surface). If the two surfaces are sliding against one another, kinetic friction is appropriate; if not, static friction is relevant. (For instance, an object that rolls without slipping is undergoing static frictional forces at the point of contact, even though the object is moving.) For static friction: the frictional force always points oppositely to any unbalanced net force (parallel to the surface). Static frictional force adjusts itself to be as strong as necessary (to oppose the unbalanced force), up to a maximum strength of µs |N|, so that fs ≤ µs |N|. (30) N is the normal force to the surface (but remember that friction forces point perpendicular to N), and µs is a dimensionless constant called the “coefficient of static friction.” The value of µs depends upon the two surfaces which are touching. For kinetic friction: the surfaces must be sliding against each other. The kinetic friction force on each surface always points opposite to the velocity of that surface. In constrast to static friction, the kinetic friction force is approximately constant, with value fk = µk |N|, (31) where N is again the normal force and µk is a dimensionless constant which depends on the surfaces called the “coefficient of kinetic friction.” It’s usually true that µk < µs , so that if an object begins to slide, it will usually continue to do so until the force causing it to slide is somehow reduced. For rolling friction: the surfaces are not actually sliding where they touch, but there is still friction that opposes the direction of relative motion: the rolling friction force on an object always points opposite to the velocity of that object with respect to the surface it is rolling on, much like kinetic friction. Also like kinetic friction, the rolling friction force is approximately constant, with value fr = µr |N|,

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where N is again the normal force and µr is again a dimensionless constant. It’s usually true that µr < µk < µs for two given materials. A few representative coefficients of friction are given in table 6.1 on page 149 of your book.

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