RECTILINEAR MOTION Linear motion (also called rectilinear motion) is motion along a straight line, and which can be described mathematically using only one spatial dimension i.e Is a linear motion in which the direction of the velocity remains constant and the path is a straight line. Linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; non uniform linear motion with variable velocity or non-zero acceleration. The motion of a particle (a point-like object) along a line can be described by its position , which varies with (time).
Displacement
The motion in which all the particles of a body move through the same distance in the same time is called translatory motion. There are two types of translatory motions: rectilinear motion; curvilinear motion. Since linear motion is a motion in a single dimension, the distance traveled by an object in particular direction is the same as displacement. If is the initial position of an object and is the final position, then mathematically the displacement is given by:
The equivalent of displacement in rotational motion is the angular displacement measured in radian. The displacement of an object cannot be greater than the distance. Consider a person travelling to work daily. Overall displacement when he returns home is zero, since the person ends up back where he started, but the distance travelled is clearly not zero.
Velocity
Velocity is defined as the rate of change of displacement with respect to time. The SI unit of velocity is metre per second.
Average velocity
The average velocity is the ratio of total displacement taken over time interval . Mathematically, it is given by:
where: is the time at which the object was at position is the time at which the object was at position
Instantaneous velocity
The instantaneous velocity can be found by differentiating the displacement with respect to time.
Speed
Speed is the absolute value of velocity i.e. speed is always positive. The unit of speed is metre per second. If is the speed then,
The magnitude of the instantaneous velocity is the instantaneous speed.
Acceleration
Acceleration is defined as the rate of change of velocity with respect to time. Acceleration is the second derivative of displacement i.e. acceleration can be found by differentiating position with respect to time twice or differentiating velocity with respect to time once. The SI unit of acceleration is metre per second squared.
If is the average acceleration and is the average velocity over the time interval then mathematically,
The instantaneous acceleration is the limit of the ratio and as approaches zero i.e.,
Equations of kinematics
In case of constant acceleration, the four physical quantities acceleration, velocity, time and displacement can be related by using the Equations of motion
here, is the initial velocity is the final velocity is the acceleration is the displacement is the time These relationships can be demonstrated graphically. The gradient of a line on a displacement time graph represents the velocity. The gradient of the velocity time graph gives the acceleration while the area under the velocity time graph gives the displacement. The area under an acceleration time graph gives the change in velocity.
Analogy between linear and rotational motion
Linear motion Rotational motion Displacement = Angular displacement = Velocity = Angular velocity = Acceleration = Angular acceleration = Mass = Moment of Inertia = Force = Torque = ANGULAR MOTION
Angular displacement
A particle moves in a circle of radius . Having moved an arc length , its angular position is relative to its original position, where . In mathematics and physics it is usual to use the natural unit radians rather than degrees or revolutions. Units are converted as follows:
An angular displacement is a change in angular position:
where is the angular displacement, is the initial angular position and is the final angular position.
Angular speed and angular velocity
Angular velocity is the change in angular displacement per unit time. Symbol for angular velocity is and the units are rad/s. Angular speed is the magnitude of angular velocity.
The instantaneous angular velocity is given by
Using the formula for angular position and letting , we have also
where is the translational speed of the particle. Angular velocity and frequency are related by .
Angular acceleration
A changing angular velocity indicates the presence of an angular acceleration in rigid body, measured in rad/s2. The average angular acceleration over a time interval Δt is given by
The instantaneous acceleration α(t) is given by
Thus, the angular acceleration is the rate of change of the angular velocity, just as acceleration is the rate of change of velocity. The translational acceleration of a point on the object rotating is given by where r is the radius or distance from the axis of rotation. This is also the tangential component of acceleration: it is tangential to the direction of motion of the point. If this component is 0, the motion is uniform circular motion, and the velocity changes in direction only. The radial acceleration (perpendicular to direction of motion) is given by
. It is directed towards the center of the rotational motion, and is often called the centripetal acceleration. The angular acceleration is caused by the torque, which can have a positive or negative value in accordance with the convention of positive and negative angular frequency. The ratio of torque and angular acceleration (how difficult it is to start, stop, or otherwise change rotation) is given by the moment of inertia: .
Equations of kinematics
When the angular acceleration is constant, the five quantities angular displacement , initial angular velocity , final angular velocity , angular acceleration , and time can be related by four equations of kinematics: