LINEAR MOTION_TA

MECHANICS

NEWTON'S FIRST LAW OF MOTION

  • Newton's First Law states that in the absence of external forces and when observed from an inertial reference frame:

    • An object at rest remains at rest.

    • An object in motion continues in motion with constant velocity (constant speed in a straight line).

TYPES OF MOTION

  • Rectilinear Motion: Movement along a straight line.

  • Circular Motion: Movement along the circumference of a circle.

  • Spin or Rotational Motion: A body rotates about its central axis.

  • Simple Harmonic Motion: Oscillatory movement where the restoring force is proportional to the displacement.

  • Random Motion: Irregular motion examples include Brownian motion.

LINEAR MOTION

RECTILINEAR MOTION

Scalar and Vector Quantities

  • Scalar Quantity: Has only magnitude.

    • Examples: Distance, Speed, Mass, Work.

  • Vector Quantity: Has both magnitude and direction.

    • Examples: Displacement, Velocity, Acceleration, Force.

Important Parameters

  1. Distance: The linear space between two points.

  2. Displacement: The linear space in a specified direction; change in position.

  3. Speed: The rate of change of distance over time.

  4. Velocity: Speed in a specified direction.

  5. Acceleration: The rate of change of velocity.

Distance vs Displacement

  • Distance: Path traced by a particle, often a broken line.

  • Displacement: Change in position from an initial point A to an end point B in a specified direction.

EXAMPLE DISCUSSIONS

Average Velocity

  • Formula:[ v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1} ]

    • Example: If a taxi travels 258 m in 3.0 s, then average velocity is 86 m/s.

Example Problem: Water Taxi

  • A taxi travels from the west to the east bank of a channel, returning after a set time:

    • Displacement: Zero (returns to starting point).

    • Distance: 240 m each way, totaling 480 m.

Constant Velocity Motion

  • Definition: Movement at a constant speed in a straight line.

  • For a distance moved at time ( t = t_f - t_0 ):

    • Distance covered: ( s = x_f - x_0 ) where ( x_0 ) is initial and ( x_f ) is final position.

MOTION UNDER CONSTANT ACCELERATION

  • Definition: Changes in speed occur with time under constant acceleration.

  • Motion can be defined as:

    • ( v_x = u + at ) where ( u ) is initial velocity, ( a ) is acceleration, and ( t ) is time.

  • The distance covered can also be calculated using: [ x = x_0 + ut + \frac{1}{2}at^2 ]

    • Where ( x_0 ) is the initial position.

EXAMPLE APPLICATIONS OF ACCELERATION

  • A metro train decelerating from 23 m/s with a braking force can be described using: [ F = ma \Rightarrow a = \frac{F}{m} ]

    • Calculating the stopping distance using:[ v^2 = u^2 + 2ax ]

  • A signal passing through a communication system with varying speeds can be calculated graphically or algebraically.

VECTOR ADDITION

  • Vectors can be added using graphical methods:

    • Triangle law for vector addition:

      1. Draw vector A.

      2. From the tip of A, draw vector B.

      3. The resulting vector R = B + A.

SIMPLE PENDULUM

  • A simple pendulum consists of a mass suspended from a string that swings back and forth.

  • The governing equations for a simple pendulum include the relationship between its period ( T ) and the length of the string ( L ):[ T = 2\pi \sqrt{\frac{L}{g}} ]

    • Where ( g ) is the acceleration due to gravity.

PRACTICAL PROBLEMS

  • Laboratory experiments can measure the acceleration due to gravity by analyzing pendulum motion.

  • Various projectile problems (e.g., heights, speeds) can also be solved using the principles of kinematics and the equations of motion.

INSTANTANEOUS VELOCITY AND ACCELERATION

  • Instantaneous Velocity: The speed of an object at a certain point in time, found using derivatives of position functions.

  • Instantaneous Acceleration: The rate of change of velocity over time, also determined using derivatives.