🌊College Physics II – Mechanics, Sound, Oscillations, and Waves Unit 4 – Motion in 2D and 3D

Key Concepts and Definitions

• Vectors represent physical quantities with both magnitude and direction, while scalars only have magnitude
• Displacement is the change in position of an object, a vector quantity
• Velocity is the rate of change of displacement with respect to time, also a vector quantity
• Acceleration is the rate of change of velocity with respect to time, another vector quantity
• Projectile motion is the motion of an object launched at an angle to the horizontal, subject only to the force of gravity
• Circular motion is the motion of an object along a circular path at a constant speed
  • Involves centripetal acceleration, which is always directed towards the center of the circle
• Rotational motion is the motion of an object around a fixed axis, characterized by angular displacement, velocity, and acceleration

Vector Analysis in Multiple Dimensions

• Vectors in 2D and 3D can be represented using Cartesian coordinates (x, y) and (x, y, z) respectively
• Vector addition and subtraction in multiple dimensions follow the same rules as in one dimension, but are applied component-wise
• Scalar multiplication of a vector involves multiplying each component of the vector by the scalar
• The magnitude of a vector in 2D is given by $\sqrt{x^2 + y^2}$, and in 3D by $\sqrt{x^2 + y^2 + z^2}$
• The dot product of two vectors $\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$ is a scalar quantity
  • It represents the projection of one vector onto another, multiplied by the magnitude of the other vector
• The cross product of two vectors $\vec{a} \times \vec{b}$ is a vector quantity, perpendicular to both $\vec{a}$ and $\vec{b}$
  • Its magnitude is given by $|\vec{a}| |\vec{b}| \sin \theta$, where $\theta$ is the angle between the vectors

Kinematics in 2D and 3D

• Position, velocity, and acceleration are vector quantities in 2D and 3D
• The position vector $\vec{r}(t)$ describes an object's position as a function of time
• Velocity is the first derivative of position with respect to time: $\vec{v}(t) = \frac{d\vec{r}}{dt}$
• Acceleration is the first derivative of velocity or the second derivative of position with respect to time: $\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}$
• Kinematic equations for constant acceleration in 2D and 3D are similar to those in 1D, but are applied component-wise
  • $\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2}\vec{a}t^2$
  • $\vec{v}(t) = \vec{v}_0 + \vec{a}t$
  • $\vec{v}^2 = \vec{v}_0^2 + 2\vec{a} \cdot (\vec{r} - \vec{r}_0)$
• Relative motion involves analyzing the motion of objects from different frames of reference
  • Velocities in different frames are related by the velocity of one frame relative to the other

Forces and Newton's Laws in Higher Dimensions

• Newton's laws of motion apply to forces and motion in 2D and 3D
• The net force on an object is the vector sum of all forces acting on it: $\vec{F}_{net} = \sum \vec{F}_i$
• Newton's second law states that the net force on an object equals its mass times its acceleration: $\vec{F}_{net} = m\vec{a}$
• Forces can be resolved into components along the coordinate axes (x, y, z)
  • The net force in each direction determines the acceleration in that direction
• Friction, tension, and other forces can act in multiple dimensions
• The normal force is always perpendicular to the surface of contact
• Static and kinetic friction forces oppose the motion or attempted motion between surfaces

Projectile Motion

• Projectile motion is a combination of horizontal and vertical motion, analyzed independently
• The horizontal velocity remains constant (neglecting air resistance), while the vertical velocity changes due to gravity
• The time of flight, range, and maximum height of a projectile can be calculated using kinematic equations
  • Time of flight: $t = \frac{2v_0 \sin \theta}{g}$
  • Range: $R = \frac{v_0^2 \sin 2\theta}{g}$
  • Maximum height: $h = \frac{v_0^2 \sin^2 \theta}{2g}$
• The trajectory of a projectile is a parabola in the absence of air resistance
• Projectile motion problems often involve finding the initial velocity, launch angle, or range

Circular and Rotational Motion

• Uniform circular motion is characterized by a constant speed and a constant radius
• Centripetal acceleration is directed towards the center of the circle and is given by $a_c = \frac{v^2}{r}$
• Centripetal force is the net force causing centripetal acceleration, directed towards the center of the circle
  • It could be provided by tension, gravity, friction, or other forces
• Angular displacement ($\theta$), angular velocity ($\omega$), and angular acceleration ($\alpha$) describe rotational motion
  • $\omega = \frac{d\theta}{dt}$ and $\alpha = \frac{d\omega}{dt}$
• Tangential velocity ($v$) and acceleration ($a_t$) are related to angular quantities by $v = r\omega$ and $a_t = r\alpha$
• Torque ($\tau$) is the rotational equivalent of force, causing angular acceleration
  • $\tau = I\alpha$, where $I$ is the moment of inertia, a measure of an object's resistance to rotational acceleration

Applications and Real-World Examples

• Projectile motion examples include sports (basketball, football), ballistics, and fireworks
• Circular motion examples include planets orbiting the sun, satellites orbiting Earth, and a mass on a string
  • Banked curves on roads and roller coasters also involve circular motion
• Rotational motion examples include wheels, gears, and flywheels in engines
• Vector analysis is used in navigation (airplanes, ships), engineering (bridges, buildings), and computer graphics
• Forces in multiple dimensions are essential in understanding structures, machines, and biomechanics
  • Examples include cranes lifting objects, forces on joints in the human body, and wind loading on buildings

Problem-Solving Strategies

• Identify the given information, unknowns, and the quantity to be calculated
• Draw diagrams to visualize the problem, including coordinate axes, vectors, and forces
• Break the problem into smaller, manageable parts (e.g., horizontal and vertical components)
• Apply relevant concepts, equations, and principles to solve for the unknowns
  • Use kinematic equations, Newton's laws, and vector operations as needed
• Check the units and reasonableness of the answer
• Practice solving a variety of problems to develop proficiency and understanding
  • Work through examples in textbooks, online resources, and past exams
• Collaborate with classmates and seek help from the instructor or tutors when needed