Motion in a Plane Study Notes

Introduction to Motion in a Plane

• Dimensional Constraints:

  • In one-dimensional motion, only two directions are possible, represented by positive ($+$) and negative ($-$) signs.

  • To describe motion in two dimensions (a plane) or three dimensions (space), vectors are required to describe physical quantities such as position, displacement, velocity, and acceleration.

• Scalars vs. Vectors:

  • Scalar Quantity: Has only magnitude and no direction. It is specified by a single number and a proper unit. Examples include distance, mass, temperature, and time.

  • Vector Quantity: Has both magnitude and direction. It must obey the triangle law of addition or the parallelogram law of addition. A vector is specified by its magnitude (a number) and its direction. Examples include displacement, velocity, acceleration, and force.

Representation and Equality of Vectors

• Notation:

  • A vector is represented by a bold letter ($\mathbf{A}$) or a letter with an arrow placed over it ($\vec{A}$).

  • The magnitude of a vector is called its absolute value, indicated by $| \text{A} | = A$.

• Graphical Representation:

  • Represented by a line segment with an arrowhead.

  • The tail of the vector is denoted as node $P$.

  • The head of the vector is denoted as node $Q$.

  • The length of the line segment represents the magnitude, and the arrow mark indicates the direction.

• Position and Displacement Vectors:

  • Let $P$ and $P'$ be positions of an object at times $t$ and $t'$ respectively.

  • Position Vector ($\vec{r}$): The vector $OP$ from the origin to the object's position at time $t$. At time $t'$, the position vector is $OP' = \vec{r}'$.

  • Displacement Vector: If an object moves from $P$ to $P'$, the vector $\vec{PP'}$ is the displacement vector.

  • Displacement is the straight line joining the initial and final positions; it does not depend on the actual path taken between the two positions.

• Equality of Vectors:

  • Vectors $\vec{A}$ and $\vec{B}$ are equal if and only if they have the same magnitude and the same direction.

  • Two vectors can have the same length but be unequal if their directions differ.

Multiplication of Vectors by Real Numbers

• Positive Scaling: Multiplying a vector $\vec{A}$ by a positive number $\lambda$ results in a vector whose magnitude is changed by the factor $\lambda$, but the direction remains the same as $\vec{A}$.

  • Formula: $\lambda \times \vec{A} = \lambda\vec{A}$ if \lambda > 0.

  • Example: If $\vec{A}$ is multiplied by $2$, the resultant $2\vec{A}$ is in the same direction with twice the magnitude $| \vec{A} |$.

• Negative Scaling: Multiplying a vector $\vec{A}$ by a negative number $\lambda$ results in a vector $\lambda\vec{A}$ whose direction is opposite to $\vec{A}$ and whose magnitude is $-\lambda$ times $| \vec{A} |$.

  • Example: Multiplying vector $\vec{A}$ by $-1$ or $-1.5$ reverses the direction and scales the magnitude accordingly.

Vector Addition and Subtraction: Graphical Methods

• Triangle Law of Vector Addition:

  • If two vectors are represented in magnitude and direction by two sides of a triangle taken in order, their resultant is the third side of the triangle taken in the reverse order.

  • This is known as the "head-to-tail" method.

  • Commutative Law: Vector addition is commutative: \vec{A} + \n\vec{B} = \vec{B} + \vec{A}.

  • Associative Law: Vector addition is associative: $(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$.

• Subtraction of Vectors:

  • Subtraction is defined in terms of addition.

  • The difference of two vectors $\vec{A}$ and $\vec{B}$ is the sum of $\vec{A}$ and the negative of $\vec{B}$.

  • Formula: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$.

• Parallelogram Law of Vector Addition:

  • If two vectors are represented in magnitude and direction by the adjacent sides of a parallelogram, their resultant is given by the diagonal of the parallelogram passing through their common point.

• Example: Rain and Wind:

  • Scenario: Rain falls vertically at $35\,m/s$. Wind starts blowing at $12\,m/s$ in an East-to-West direction.

  • Problem: Find the direction a boy should hold his umbrella.

Unit Vectors and Resolution

• Definition of Unit Vector:

  • A vector of unit magnitude pointing in a particular direction. It has no dimensions or units; it only specifies direction.

  • If $\hat{n}$ is a unit vector and is multiplied by a scalar, the result is a vector.

  • General form: $\vec{A} = |\vec{A}|\hat{A}$.

  • Unit vector calculation: $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$.

• Orthogonal Unit Vectors:

  • Unit vectors along the $x$, $y$, and $z$ axes of a rectangular coordinate system are denoted by $\hat{i}$, $\hat{j}$, and $\hat{k}$.

  • $|\hat{i}| = |\hat{j}| = |\hat{k}| = 1$.

• Resolution of a Vector:

  • A vector $\vec{A}$ can be resolved into component vectors lying along unit vectors $\hat{i}$ and $\hat{j}$.

Analytical Method: Vector Addition

• Magnitude of Resultant:

  • Consider vectors $\vec{A}$ and $\vec{B}$ in the $x$-$y$ plane with an angle $\theta$ between them.

  • Using geometry from a figure where $OS$ is the resultant, $OP = A$, and $PS = B$:

  • $OS^2 = ON^2 + SN^2$

  • $ON = OP + PN = A + B \cos(\theta)$

  • $SN = B \sin(\theta)$

  • $R^2 = (A + B \cos(\theta))^2 + (B \sin(\theta))^2$

  • $R^2 = A^2 + 2AB \cos(\theta) + B^2 \cos^2(\theta) + B^2 \sin^2(\theta)$

  • $R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)}$

• Direction of Resultant:

  • The direction $\alpha$ of the resultant relative to vector $\vec{A}$ is given by:

  • $\tan(\alpha) = \frac{SN}{ON} = \frac{B \sin(\theta)}{A + B \cos(\theta)}$

• Example: Motorboat Velocity:

  • Motorboat speed: $25\,km/h$ North.

  • Water current: $10\,km/h$ at $60^{\circ}$ East of South.

  • Goal: Find resultant velocity.

Motion in a Plane: Kinematics

• Position Vector:

  • At time $t$: $\vec{r} = x\hat{i} + y\hat{j}$.

  • At time $t'$: $\vec{r}' = x'\hat{i} + y'\hat{j}$.

• Displacement Vector:

  • $\Delta \vec{r} = \vec{r}' - \vec{r} = (x' - x)\hat{i} + (y' - y)\hat{j}$.

  • $\Delta \vec{r} = \Delta x\hat{i} + \Delta y\hat{j}$.

• Velocity Vector:

  • Average velocity: $\vec{v} = \frac{\Delta \vec{r}}{\Delta t} = \frac{\Delta x}{\Delta t}\hat{i} + \frac{\Delta y}{\Delta t}\hat{j} = v_x\hat{i} + v_y\hat{j}$.

  • Instantaneous velocity: $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j}$.

• Acceleration Vector:

  • $\vec{a} = a_x\hat{i} + a_y\hat{j}$.

  • Instantaneous acceleration: $\vec{a} = \frac{d\vec{v}}{dt}$.

Projectile Motion

• Definitions:

  • Projectile: An object in flight after being thrown or projected.

  • Trajectory: The path of a projectile, which is a parabola.

• Velocity Components:

  • Initial velocity $u$ at angle $\theta$:

  • Horizontal: $u_x = u \cos(\theta)$.

  • Vertical: $u_y = u \sin(\theta)$.

• Acceleration Components:

  • Horizontal: $a_x = 0$ (velocity $u \cos(\theta)$ remains constant).

  • Vertical: $a_y = -g$. At maximum height, the vertical component of velocity is zero.

• Equation of Path:

  • $x = u \cos(\theta)t \implies t = \frac{x}{u \cos(\theta)}$.

  • $y = u \sin(\theta)t - \frac{1}{2}gt^2$.

  • Substituting $t$: $y = \tan(\theta)x - \frac{g}{2u^2 \cos^2(\theta)}x^2$.

  • This follows the form $y = ax + bx^2$, defining a parabola.

• Time of Flight ($T$):

  • Total time the projectile is in flight.

  • Using $s = ut + \frac{1}{2}at^2$ for vertical motion where displacement $s=0$:

  • $0 = u \sin(\theta)T - \frac{1}{2}gT^2$

  • $T = \frac{2u \sin(\theta)}{g}$.

• Horizontal Range ($R$):

  • Horizontal distance traveled during Time of Flight.

  • $R = \text{Horizontal component of velocity} \times \text{Time of Flight}$.

  • $R = u \cos(\theta) \times \frac{2u \sin(\theta)}{g} = \frac{u^2 (2 \sin(\theta) \cos(\theta))}{g}$.

  • $R = \frac{u^2 \sin(2\theta)}{g}$.

  • Maximum Range ($R_{max}$): Occurs when $\sin(2\theta)$ is max ($1$), which happens at $\theta = 45^{\circ}$. $R_{max} = \frac{u^2}{g}$.

  • Range Symmetry: For a given velocity, the range is the same for angles $\theta$ and $90^{\circ} - \theta$.

    • Proof: Range for $(90 - \theta)$ is $\frac{u^2 \sin(2(90 - \theta))}{g} = \frac{u^2 \sin(180 - 2\theta)}{g} = \frac{u^2 \sin(2\theta)}{g}$.

• Maximum Height ($H$):

  • The highest point reached by the projectile.

  • Using $v^2 - u^2 = 2as$ for vertical motion ( $v=0$ ranges to max height):

  • $0 - u^2 \sin^2(\theta) = -2gH$

  • $H = \frac{u^2 \sin^2(\theta)}{2g}$.

• Example: Cricket Ball:

  • $u = 28\,m/s$ at $30^{\circ}$ above horizontal.

  • (a) Max height: $H = \frac{28^2 \sin^2(30)}{2 \times 9.8} = 10\,m$.

  • (b) Time to return: $T = \frac{2 \times 28 \sin(30)}{9.8} = 2.9\,s$.

  • (c) Distance (Range): $R = \frac{28^2 \sin(60)}{9.8} = 69\,m$.

Uniform Circular Motion

• Definition: Motion of an object following a circular path at a constant speed.

  • "Uniform" refers specifically to the constant speed.

• Period ($T$): The time taken for one full revolution.

• Frequency ($\nu$): Number of revolutions per second.

  • $\nu = \frac{1}{T}$.

  • Unit: hertz ($Hz$).

• Angular Velocity ($\omega$):

  • Rate of change of angular displacement.

  • $\omega = \frac{\Delta \theta}{\Delta t}$. In the limit: $\omega = \frac{d\theta}{dt}$.

  • $\omega = \frac{2\pi}{T} = 2\pi\nu$.

  • Unit: $rad/s$.

• Relation between v and $\omega$:

  • $\text{Angle} = \frac{\text{arc}}{\text{radius}} \implies \Delta \theta = \frac{\Delta r}{r}$.

  • $\Delta r = r \Delta \theta \implies \frac{\Delta r}{\Delta t} = r \frac{\Delta \theta}{\Delta t}$.

  • $v = r\omega$.

• Angular Acceleration ($\alpha$):

  • Rate of change of angular velocity.

  • $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$.

• Centripetal Acceleration ($a_c$):

  • Acceleration directed toward the center of the circle along the radius.

  • Comparison of triangles shows: $\frac{\Delta v}{v} = \frac{\Delta r}{r} \implies \Delta v = \frac{v \Delta r}{r}$.

  • Dividing by $\Delta t$: $a = \frac{v^2}{r}$.

  • Formulae for acceleration:

    • $a_c = \frac{v^2}{R}$

    • $a_c = \omega^2 R$ (using $v = R\omega$)

    • $a_c = v\omega$

• Example: Insect in a Groove:

  • Radius $r = 12\,cm = 0.12\,m$. 7 revolutions in $100\,s$.

  • Period $T = \frac{100}{7}\,s$.

  • (a) Angular speed ($\omega$): $\omega = \frac{2\pi \times 7}{100} = 0.44\,rad/s$.

  • Linear speed ($v$): $v = \omega R = 0.44 \times 0.12 = 5.3 \times 10^{-2}\,m/s$.

  • (b) Acceleration: $a = \omega^2 R = (0.44)^2 \times 0.12 = 2.3 \times 10^{-2}\,m/s^2$.

  • Is acceleration constant? No, because while the magnitude remains constant, the direction changes continuously (always pointing toward the center).