Motion in a Plane: Page-by-Page Notes

Page 1

• Scalars vs. Vectors
  • Scalar quantities: specified completely by a number and unit; have magnitude only. Examples: speed, mass, time, density, volume, temperature, etc.
  • Vector quantities: have both magnitude and direction and obey vector addition rules. Examples: displacement, velocity, force, acceleration, electric field, magnetic field, etc.
  • Caution: If a quantity has magnitude and direction but does not obey the triangle/vector addition rule, it is not a vector. Example: electric current in a wire has magnitude and direction but does not obey the triangle rule, so it is not a vector.
• Representation of a Vector
  • A vector quantity is represented by an arrow: length proportional to magnitude under a chosen scale; arrowhead indicates direction.
  • Example: velocity 30 km/h due east; if 1 cm represents 10 km/h, a line OA of 3 cm to the east represents the velocity (Fig. 3.1).
  • Head vs tail: head is the arrow tip; tail is the other end.
• Symbol for a vector
  • In print, a bold letter or a letter with an arrow denotes a vector (magnitude + direction). Example: force may be written as F or F⃗.
  • Magnitude of a vector is often written as |F⃗|; sometimes denoted as F (light face) or F (magnitude of F⃗).

Key idea: Distinguish between scalars and vectors, and know how vectors are graphically represented.

Page 2

• Position Vector
  • The location of a point in space is described by a position vector r. Its magnitude is the distance from the origin to the point, and its direction is from the origin to the point.
  • If a point P has coordinates (x, y, z) and origin O is the reference, then the position vector is
    $\,\boldsymbol{r} = x\hat{i} + y\hat{j} + z\hat{k} \ r = |\boldsymbol{r}| = \sqrt{x^2 + y^2 + z^2}$
• Displacement Vector
  • For a body moving from point A to B, the displacement vector is AB⃗.
  • If r⃗1 and r⃗2 are position vectors of A and B respectively, then
    $\boldsymbol{r}<em>1 = x</em>1\hat{i} + y<em>1\hat{j} + z</em>1\hat{k}, \,\, \boldsymbol{r}<em>2 = x</em>2\hat{i} + y<em>2\hat{j} + z</em>2\hat{k}$
  • Displacement follows the triangle law:
    $\boldsymbol{r}<em>1 + \boldsymbol{AB} = \boldsymbol{r}</em>2 \,\Rightarrow \, \boldsymbol{AB} = \boldsymbol{r}<em>2 - \boldsymbol{r}</em>1$
  • Components of AB: if A=(x1,y1,z1) and B=(x2,y2,z2), then
    $\boldsymbol{AB} = (x<em>2-x</em>1)\hat{i} + (y<em>2-y</em>1)\hat{j} + (z<em>2-z</em>1)\hat{k}$
  • Magnitude of AB:
    $|\boldsymbol{AB}| = \sqrt{(x<em>2-x</em>1)^2 + (y<em>2-y</em>1)^2 + (z<em>2-z</em>1)^2}$

Comment: Position vectors provide a coordinate-free way to describe location; displacement is the change in position.

Page 3

• Type of Vectors
  • (i) Parallel and antiparallel vectors
  • Parallel (like) vectors have the same direction; magnitudes may differ.
  • Antiparallel (unlike) vectors have opposite directions; magnitudes may differ.
  • (ii) Equality of Vectors
  • Two vectors are equal if they have the same magnitude and direction. Equality is independent of position in space (a vector is free to slide parallel to itself).
  • (iii) Negative Vector
  • The negative vector has the same magnitude but opposite direction.
  • (iv) Unit Vector
  • A unit vector has magnitude 1 and points in the direction of a given vector. Written as \hat{A} and read as "A hat".
  • For a vector \boldsymbol{A}, the unit vector is $\hat{\boldsymbol{A}} = \boldsymbol{A}/|\boldsymbol{A}|$. In unit-vector form, a vector can be written as $\boldsymbol{A} = |\boldsymbol{A}| \hat{\boldsymbol{A}}.$

Note: Unit vectors are dimensionless.

Page 4

• Rectangular (Cartesian) Unit Vectors
  • In a right-handed coordinate system, the unit vectors along the X, Y, Z axes are denoted as $\hat{i}, \hat{j}, \hat{k}$.
  • A vector along an axis can be written as a scalar multiple of the magnitude and the corresponding unit vector, e.g., a vector along X is written as $A \hat{i}$.
• Collinear Vectors
  • Collinear vectors act along the same line; if in the same direction, they are like vectors; if opposite, unlike vectors.
• Co-initial and Co-planar Vectors
  • Co-initial: vectors share a common initial point.
  • Co-planar: vectors lie in the same plane.
• Zero Vector
  • A vector of zero magnitude, denoted 0⃗, with arbitrary direction.
• Properties of Zero Vector
  • a) A + 0⃗ = A⃗
  • b) A⃗ − 0⃗ = A⃗
  • c) n·0⃗ = 0⃗ for any non-zero scalar n
• Examples
  • If a particle is at rest, its displacement over an interval is the zero vector.
  • Velocity of a stationary particle is the zero vector.
  • If a body moves with constant velocity, the acceleration is the zero vector.

Remember: If a vector is displaced such that neither magnitude nor direction changes, the vector remains the same.

Page 5

• Multiplication of Vectors by Real Numbers
  • If a vector \boldsymbol{A} is multiplied by a positive real number \lambda, the magnitude scales by \lambda but direction stays the same.
  • If multiplied by a negative real number (−\lambda), magnitude scales by \lambda and direction is reversed.
• Addition and Subtraction of Vectors (Graphical)
  • Addition aims to find the resultant vector R, such that R is the single vector equal in effect to the sum of the vectors.
  • Triangle Law: To add \boldsymbol{A} and \boldsymbol{B}, draw \boldsymbol{A}; then draw \boldsymbol{B} with its tail on the head of \boldsymbol{A}; the closing side (from tail of \boldsymbol{A} to head of \boldsymbol{B}) represents the resultant R = \boldsymbol{A} + \boldsymbol{B}.
  • The resultant is denoted as \boldsymbol{R}.
  • Addition of more than two vectors uses the Polygon Law: place vectors sequentially end to end; the resultant is represented by the closing side of the polygon taken in the opposite order.
• Commutative and Associative Laws
  • Commutative: \boldsymbol{P} + \boldsymbol{Q} = \boldsymbol{Q} + \boldsymbol{P}.
  • Associative: (\boldsymbol{A} + \boldsymbol{B}) + \boldsymbol{C} = \boldsymbol{A} + (\boldsymbol{B} + \boldsymbol{C}).
• Parallelogram Law
  • If two vectors are represented by adjacent sides of a parallelogram, the resultant is the diagonal through the common point.
  • For vectors $\boldsymbol{A}, \boldsymbol{B}$, the resultant is obtained by completing the parallelogram and taking the diagonal OC as the resultant: $\boldsymbol{R} = \boldsymbol{A} + \boldsymbol{B}$.
  • Extends to three or more vectors by placing them head-to-tail and taking the closing side.
• Components of Vectors
  • In Cartesian coordinates, for A = (Ax, Ay, Az) and B = (Bx, By, Bz), the resultant R = A + B has components
    $R<em>x = A</em>x + B<em>x, \; R</em>y = A<em>y + B</em>y, \; R<em>z = A</em>z + B_z.$

Note: The vector addition is a geometric operation; the algebraic sum of components yields the same resultant.

Page 6

• Continued: Vector Addition for Multiple Vectors

  • For vectors A, B, C, D with components along x, y, z, the resultant is
    $\boldsymbol{R} = \boldsymbol{A} + \boldsymbol{B} + \boldsymbol{C} + \boldsymbol{D},$
    with components
    $R<em>x = A</em>x + B<em>x + C</em>x + D<em>x, \; R</em>y = A<em>y + B</em>y + C<em>y + D</em>y, \; R<em>z = A</em>z + B<em>z + C</em>z + D_z.$

• Equality of Vectors and Position Independence

  • Equality of vectors is independent of their position in space; moving a vector parallel to itself does not change its magnitude or direction.

• Note on Vector Subtraction

  • Subtraction is defined via addition with the negative: \boldsymbol{A} - \boldsymbol{B} = \boldsymbol{A} + (−\boldsymbol{B}).
  • Subtraction is not commutative: \boldsymbol{A} - \boldsymbol{B} ≠ \boldsymbol{B} - \boldsymbol{A} (indeed, \boldsymbol{A} - \boldsymbol{B} = −(\boldsymbol{B} - \boldsymbol{A})).
  • Subtraction is not associative either.

Tip: The diagonal of the parallelogram formed by two vectors gives their sum; the other diagonal gives their difference.

Page 7

• Parallelogram Law (continued)
  • If two vectors P⃗ and Q⃗ are represented by the adjacent sides of a parallelogram, the resultant is the diagonal OC = P⃗ + Q⃗; OA⃗ and OB⃗ represent the vectors placed on the parallelogram with common origin O.
  • For multiple vectors A⃗, B⃗, C⃗, the resultant is the vector sum R⃗ = A⃗ + B⃗ + C⃗; the components add accordingly: $R<em>x = A</em>x + B<em>x + C</em>x, \ R<em>y = A</em>y + B<em>y + C</em>y, \ R<em>z = A</em>z + B<em>z + C</em>z.$
• Vector Algebra in 3D
  • For vectors A⃗ = Ax î + Ay ĵ + Az k̂, B⃗ = Bx î + By ĵ + Bz k̂, C⃗ = Cx î + Cy ĵ + Cz k̂, the resultant is $\boldsymbol{R} = \boldsymbol{A} + \boldsymbol{B} + \boldsymbol{C} = (A</em>x + B<em>x + C</em>x) \hat{i} + (A<em>y + B</em>y + C<em>y) \hat{j} + (A</em>z + B<em>z + C</em>z) \hat{k}.$

Page 8

• Subtraction (Graphical) Revisited
  • The negative of a vector is drawn by reversing its arrow. The difference of two vectors can be found using the parallelogram method or by adding the negative of the second vector.
• Self-test 3.1 (example)
  • Given |P⃗| = 4 N, |Q⃗| = 3 N and angle between them = 60°, find |P⃗ - Q⃗|.
  • Concept: The diagonal of the parallelogram formed by P⃗ and Q⃗ gives the addition; the other diagonal gives the subtraction.
• Important reminders
  • Vector addition is commutative, but subtraction is not. Subtraction is not associative either.

Page 9

• Resolution of a Vector
  • When a vector is resolved along two coplanar non-parallel vectors P⃗ and Q⃗, we can write
    $\boldsymbol{A} = \lambda \boldsymbol{P} + \mu \boldsymbol{Q},$
    where λ and μ are real numbers.
• Rectangular components (two axes)
  • Consider A⃗ represented by OP⃗. Draw a tail at O and axes OX and OY at right angles. Drop a perpendicular to OX from P to obtain components along X and Y.
  • If angle θ is between A⃗ and Ox, then
    $A<em>x = A \cos\theta, \quad A</em>y = A \sin\theta.$
  • Vector form: $\boldsymbol{A} = A<em>x \hat{i} + A</em>y \hat{j} = A\cos\theta\,\hat{i} + A\sin\theta\,\hat{j}.$
• Examples
  • If F⃗ has angle 60° with the horizontal and magnitude F, then
    $F<em>x = F \cos 60°, \quad F</em>y = F \sin 60°.$
• Rectangular components along three axes
  • For A⃗ with direction cosines α, β, γ with axes OX, OY, OZ respectively,
    $A<em>x = A\cos\alpha, \; A</em>y = A\cos\beta, \ A<em>z = A\cos\gamma, \ A^2 = A</em>x^2 + A<em>y^2 + A</em>z^2.$

Caution: Vectors are resolved at the tail, not at the head.

• Self-test: If \boldsymbol{A} = 3\hat{i} - 4\hat{j} + 2\hat{k}, find its magnitude. Hint: $A = \sqrt{3^2 + (-4)^2 + 2^2}.$ Result: approximately 5.38.

Page 10

• Rectangular components continued
  • If a vector A⃗ has components (Ax, Ay) in 2D, then A⃗ = Ax \hat{i} + Ay \hat{j} and A^2 = Ax^2 + Ay^2.
• Three-dimensional rectangular components
  • A⃗ = Ax \hat{i} + Ay \hat{j} + Az \hat{k}; and A^2 = Ax^2 + Ay^2 + Az^2.
• Angles with axes (direction cosines) α, β, γ
  • Ax = A cos α, Ay = A cos β, A_z = A cos γ.

Note: In 3D, a vector’s magnitude is obtained via the Pythagorean theorem in three dimensions.

Page 11

• Self-test (3D magnitude)
  • For vector \boldsymbol{A} = 3î − 4ĵ + 2k̂, magnitude is $|\boldsymbol{A}| = \sqrt{3^2 + (-4)^2 + 2^2} = \sqrt{29} \approx 5.38.$
• Summary of 2D/3D components
  • Vector components along axes are additive: the resultant components are the sums of corresponding components from each vector.

Page 12

• Vector Addition: Analytical Method (Magnitude and Direction of the Resultant)
  • Magnitude: If R = P⃗ + Q⃗ and the angle between P⃗ and Q⃗ is θ, then
    $R^2 = P^2 + Q^2 + 2PQ\cos\theta.$
  • Therefore, the magnitude is
    $R = \sqrt{P^2 + Q^2 + 2 P Q \cos\theta}.$
• Direction of the Resultant
  • If β is the angle that R makes with P, then
    $\tan \beta = \frac{Q \sin\theta}{P + Q \cos\theta}.$
• Special Cases (Law of Cosines results)
  • If θ = 0° (same direction): $R = P + Q, \quad \beta = 0.$
  • If θ = 90°: $R = \sqrt{P^2 + Q^2}, \quad \beta = \tan^{-1} \left( \frac{Q}{P} \right).$
  • If θ = 180° (antiparallel): $R = |P - Q|, \quad \beta = 0 \text{ or } \pi \text{ depending on magnitudes}.$
• Law of Sines (brief)
  • In a triangle formed by vectors, e.g., with sides P, Q, R opposite angles α, β, γ respectively:
    $R \sin\beta = Q \sin\gamma, \quad P \sin\beta = Q \sin\alpha, \quad P \sin\alpha = R \sin\gamma.$
  • Consequently, R sin γ = P sin α = Q sin β.

Page 13

• Equilibrium Self-test

  • If two forces F1 and F2 act on a particle in equilibrium and |F1| = 3 N, find |F2|. (Using the given relation F1 sin(90°+30°) = F2 sin(90°+60°) in the hint.)

• Position Vector and Displacement (Intro to non-uniform motion)

  • For motion along a curved plane, the displacement in a small interval Δt is Δr = r(t+Δt) − r(t).
  • Velocity (average): $\boldsymbol{v}_{\text{av}} = \frac{\Delta \boldsymbol{r}}{\Delta t}.$
  • In the limit Δt → 0, the instantaneous velocity is
    $\boldsymbol{v} = \lim_{\Delta t \to 0} \frac{\Delta \boldsymbol{r}}{\Delta t} = \frac{d\boldsymbol{r}}{dt}.$
  • If components are vx and vy along X and Y,
    $\boldsymbol{v} = v<em>x \hat{i} + v</em>y \hat{j}, \quad v<em>x = \frac{dx}{dt}, \quad v</em>y = \frac{dy}{dt}.$

• Acceleration

  • For motion with constant acceleration, the position is
    $\boldsymbol{r} - \boldsymbol{r}_0 = \boldsymbol{u} t + \tfrac{1}{2} \boldsymbol{a} t^2,$
  • If \boldsymbol{r}0 = 0, then the components follow $x = u</em>x t + \tfrac{1}{2} a<em>x t^2, \quad y = u</em>y t + \tfrac{1}{2} a_y t^2.$
  • Differentiating gives acceleration components: $\boldsymbol{a} = \frac{d\boldsymbol{v}}{dt} = a<em>x \hat{i} + a</em>y \hat{j}, \quad a<em>x = \frac{dv</em>x}{dt}, \quad a<em>y = \frac{dv</em>y}{dt}.$

Page 17-18

• Projectile Motion (motion in a plane with gravity, negligible air resistance)

  • Initial velocity u at angle θ; components: $u<em>x = u \cos\theta, \quad u</em>y = u \sin\theta.$
  • Horizontal motion: velocity vx = u cos θ (constant), horizontal position x(t) = u cos θ · t.
  • Vertical motion: vy(t) = u sin θ − g t; vertical position y(t) = u sin θ · t − ½ g t^2.
  • Trajectory equation (eliminate t):
    $y = x \tan\theta − \frac{g}{2u^2 \cos^2\theta} x^2.$

• Maximum height, time of flight, and horizontal range

  • Maximum height H: when vy = 0 at the peak, with uy = u sin θ:
    $H = \frac{u^2 \sin^2\theta}{2g}.$
  • Time of flight T: ascent time equals descent time, so
    $T = \frac{2u \sin\theta}{g}.$
  • Horizontal range R: total horizontal distance when it returns to the initial vertical level:
    $R = u^2 \frac{\sin(2\theta)}{g} = \frac{u^2 \sin 2\theta}{g}.$
  • Note: Mass m cancels out; results are mass-independent.
  • Maximum height, time of flight, and range are often quoted with the understanding that θ is the angle with the horizontal.

• Additional trajectory details

  • For a given speed u, R is maximum when sin 2θ is maximum, i.e., θ = 45°.
  • If angle is given with respect to vertical, replace θ by (90° − θ) in the formulas.

Page 21-22

• Uniform Circular Motion (UCM)
  • Definition: An object moving in a circle at constant speed; the magnitude of acceleration is constant, but direction continuously changes.
• Angular Displacement and Velocity
  • Angular displacement: for a particle P rotating anticlockwise about an axis perpendicular to the plane, if it moves from P1 to P2 in time t sweeping an angle θ, then θ is the angular displacement. 360° corresponds to one revolution; in radians, one revolution is 2π radians.
  • 1 radian is the angle corresponding to an arc length equal to the radius: 1 rad = 360°/(2π) ≈ 57.3°.
• Angular Velocity ω
  • Defined as the time rate of change of angular displacement: $\boldsymbol{\omega} = \frac{d\theta}{dt}.$
  • SI unit: rad s⁻¹. If a body completes N rotations in time t, then $\omega = \frac{2\pi N}{t} = 2\pi \nu,$ where ν is the frequency (rotations per second).
  • If θ increases by Δθ in time Δt, the average angular velocity is
    $\boldsymbol{\omega}_{\text{avg}} = \frac{Δθ}{Δt}.$
  • Instantaneous angular velocity is the limit as Δt → 0: $\omega = \frac{d\theta}{dt}.$
• Right-Hand Rule
  • Angular velocity is an axial vector; its direction aligns with the axis given by curling the fingers in the direction of rotation and pointing the thumb along the angular velocity vector.
• Centripetal (Radial) Acceleration
  • In uniform circular motion, a centripetal force toward the center provides centripetal acceleration toward the center.
  • Magnitude: for speed v and radius r, $a_c = \frac{v^2}{r} = r \omega^2.$
  • Derivation sketch: as the particle moves from A to B, the change in velocity vector Δv is toward the center; |Δv| ≈ v Δθ for small Δt; hence a ≈ v Δθ/Δt = v ω.
• Relationship between Linear and Angular Quantities
  • Linear speed v relates to angular speed ω by $v = r \omega.$
• Non-uniform Circular Motion
  • When speed is not constant, the motion has two components of acceleration:
  • Centripetal (radial) acceleration a_c toward the center (changes direction of velocity).
  • Tangential acceleration a_t along the tangent (changes magnitude of velocity).
  • Net acceleration magnitude: $a<em>R = \sqrt{a</em>c^2 + a_t^2}.$
  • Direction of acceleration is not fixed; ac points toward the center while at is tangent to the circle.
• Angular Acceleration
  • If angular velocity is not constant, angular acceleration is defined as the time rate of change of ω:
    $\alpha = \frac{d\omega}{dt}.$
  • Instantaneous angular acceleration: $\alpha = \frac{d\omega}{dt}.$
• Tangential Acceleration (definition in vector form)
  • Tangential acceleration can also be connected to the rate of change of the tangential velocity: $a_t = \frac{dv}{dt} = r \frac{d\omega}{dt} = r \alpha.$

Page 23-25

• Three-Dimensional Space (brief recap)
  • In 3D, kinematic quantities (displacement, velocity, acceleration) can be resolved into three rectangular components.
  • Position vector: $\boldsymbol{r} = x\hat{i} + y\hat{j} + z\hat{k}.$
  • Velocity vector: $\boldsymbol{v} = v<em>x \hat{i} + v</em>y \hat{j} + v<em>z \hat{k}, \ v</em>x = \frac{dx}{dt}, \; v<em>y = \frac{dy}{dt}, \; v</em>z = \frac{dz}{dt}.$
  • Acceleration vector: $\boldsymbol{a} = a<em>x \hat{i} + a</em>y \hat{j} + a<em>z \hat{k}, \ a</em>x = \frac{dv<em>x}{dt}, \; a</em>y = \frac{dv<em>y}{dt}, \; a</em>z = \frac{dv_z}{dt}.$

Takeaway: Any motion in space can be decomposed into three perpendicular components along the coordinate axes.

Page 26-27

• Recap of analytical methods and quick derivations
  • The Law of Cosines for resultant magnitude remains central:
    $R^2 = P^2 + Q^2 + 2PQ \cos\theta.$
  • The direction of the resultant is given by
    $\tan \beta = \frac{Q \sin\theta}{P + Q \cos\theta}.$
• Instantaneous angular acceleration and tangential acceleration relations reiterated:
  • $\alpha = \frac{d\omega}{dt}, \ a_t = \frac{dv}{dt} = r \alpha.$
• Self-test style problems appeared in the material, e.g., combining two forces, angular acceleration values, etc.

Page 27

• Summary of three-dimensional kinematics
  • In 3D space, all kinematic quantities are expressed as vectors and can be broken into components along the three axes:
    $\boldsymbol{r} = x\hat{i} + y\hat{j} + z\hat{k}, \ \boldsymbol{v} = v<em>x \hat{i} + v</em>y \hat{j} + v<em>z \hat{k}, \ \boldsymbol{a} = a</em>x \hat{i} + a<em>y \hat{j} + a</em>z \hat{k}.$
  • Component relations: $v<em>x = \frac{dx}{dt}, \; a</em>x = \frac{dv_x}{dt} = \frac{d^2x}{dt^2},$ and similarly for y, z.

Overall understanding: Motion in a plane or space can be analyzed via vectors, their magnitudes, directions, and resolutions into components along chosen axes. Core tools include vector addition/subtraction (triangle, polygon, and parallelogram laws), resolution into components, and kinematic equations for linear and projectile motion as well as circular motion (both uniform and non-uniform).

Quick Reference: Key Formulas

• Position and displacement
  • $\boldsymbol{r} = x\hat{i} + y\hat{j} + z\hat{k}, \quad r = |\boldsymbol{r}| = \sqrt{x^2 + y^2 + z^2}$
  • $\boldsymbol{AB} = \boldsymbol{r}<em>2 - \boldsymbol{r}</em>1 = (x<em>2-x</em>1)\hat{i} + (y<em>2-y</em>1)\hat{j} + (z<em>2-z</em>1)\hat{k},$
  • $|\boldsymbol{AB}| = \sqrt{(x<em>2-x</em>1)^2 + (y<em>2-y</em>1)^2 + (z<em>2-z</em>1)^2}.$
• Vector operations
  • $\boldsymbol{R} = \boldsymbol{A} + \boldsymbol{B} \quad \text{with} \quad R^2 = A^2 + B^2 + 2AB\cos\theta$
  • Direction: $\tan\beta = \frac{B\sin\theta}{A + B\cos\theta}$
• Resolution and components
  • 2D: $\boldsymbol{A} = A<em>x \hat{i} + A</em>y \hat{j}, \quad A<em>x = A\cos\theta, \; A</em>y = A\sin\theta$
  • 3D: $A<em>x = A\cos\alpha, \; A</em>y = A\cos\beta, \; A<em>z = A\cos\gamma, \; A^2 = A</em>x^2 + A<em>y^2 + A</em>z^2$
• Projectile motion (no air resistance)
  • Components: $u<em>x = u\cos\theta, \; u</em>y = u\sin\theta$
  • Positions: $x = u\cos\theta \, t, \quad y = u\sin\theta \, t - \tfrac{1}{2} g t^2$
  • Trajectory: $y = x\tan\theta - \frac{g}{2u^2 \cos^2\theta} x^2$
  • Time of flight: $T = \frac{2u\sin\theta}{g}$
  • Maximum height: $H = \frac{u^2 \sin^2\theta}{2g}$
  • Range: $R = \frac{u^2 \sin(2\theta)}{g}$
• Circular motion
  • Uniform: $a_c = \frac{v^2}{r} = r\omega^2, \quad v = r\omega$
  • Instantaneous angular velocity: $\omega = \frac{d\theta}{dt}$
  • Centripetal direction toward center; axis as per right-hand rule
  • Non-uniform: $a<em>R = \sqrt{a</em>c^2 + a<em>t^2}, \quad \tan\beta = \frac{a</em>t}{a_c}$
  • Angular acceleration: $\alpha = \frac{d\omega}{dt}$, $a<em>t = r\alpha, \; a = \boldsymbol{a}</em>c + \boldsymbol{a}_t$
• Three-dimensional motion: components
  • $\boldsymbol{r} = x\hat{i} + y\hat{j} + z\hat{k}, \quad \boldsymbol{v} = v<em>x \hat{i} + v</em>y \hat{j} + v<em>z \hat{k}, \quad \boldsymbol{a} = a</em>x \hat{i} + a<em>y \hat{j} + a</em>z \hat{k}.$