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Translational and Rotational Motion


Translational Motion


Translational motion is the motion by which a body shifts from one point in space to another. One example of translational motion is the the motion of a bullet fired from a gun.


An object has a rectilinear motion when it moves along a straight line. At any time, t, the object occupies a position along the line as shown in the following figure. The distance x, with appropriate sign, define the position of the object. When the position of the object at particular time is known, the motion of the particle will be known, and generally is expressed in a form of an equation which relates distance x, to time t, for example x = 6t - 4, or a graph.

Motion in two or three dimensions is more complicated. In two dimensions, we need to specify two coordinates in order to fix the position of any object. The following figure shows a simple example of projectile motion: a ball rolling off a table. Let us define the horizontal direction as the x-axis and the vertical direction as the y-axis. Consider a ball initially rolling on off a flat table with an initial velocity of 10 m/s.


While the ball is on the table we observe that the initial x-component of velocity (v0x) is 10 m/s (constant), the initial y-component of velocity is 0 m/s, the x-component of acceleration is 0 m/s2 and the y-component of acceleration is 0 m/s2. The components of acceleration and velocity are those parts of the velocity or acceleration that points in the x or y direction.Let us observe what happens the instant the ball leaves the table.

The initial velocity in the y-direction is still zero and the initial velocity in the x-direction remains 10 m/s. However, the ball is no longer in contact with the table and it falls freely. The gravitational acceleration of the ball is down. In this case, the motions in the horizontal and vertical directions should be analyzed independently. Horizontally, there is no acceleration in the horizontal direction, therefore, the x-component of velocity is constant 




 In the vertical direction there is an acceleration equal to the acceleration of gravity.  Therefore, the velocity in the vertical direction changes as below 




Rotational Motion


Rotational motion deals only with rigid bodies. A rigid body is an object that retains its overall shape, meaning that the particles that make up the rigid body remain in the same position relative to one another. A wheel and rotor of a motor are common examples of rigid bodies that commonly appear in questions involving rotational motion.


Circular Motion


Circular motion is a common type of rotational motion. Like projectile motion we can analyze the kinematics and learn something about the relationships between position, velocity and acceleration. Newton’s first law states that an object in motion remains in motion at constant velocity unless acted upon by an outside force. If the force is applied perpendicular to the direction of motion, only the direction of velocity will change. If a force constantly acts perpendicular to a moving object, the object will move in a circular path at constant speed. This is called uniform circular motion.


The circular motion of a rigid body occurs when every point in the body moves in a circular path around a line called the axis of rotation, which cuts through the center of mass as shown in the following figure.



Uniform Circular Motion

An online simulation to measure the position, velocity, and acceleration (both components and magnitude) of an object undergoing circular motion.



Translational Motion Versus Rotational Motion

There is a strong analogy between rotational motion and standard translational motion. Indeed, each physical concept used to analyze rotational motion has its translational concomitant.


Translational Motion


Rotational Motion



Angular displacement




Angular velocity




Angular acceleration



Moment of inertia


F = Ma







Kinetic energy


Kinetic energy


Moment of Inertia

Discover the relationships between angular velocity, mass, radius and moment of inertia for collections of point-masses, rings, disks, and more complex shapes.

Torque and Moment of Inertia

Calculate net torque and moment of inertia based on the positions of the objects and the mass of a bar.




Focus on Math



Coordinate Systems

To precisely describe motion, we must be able to say where an object is located within a given reference frame. For example we can locate a chair in a room by saying it is 2 m away from the door, 3 m away from the window, and 0.5 m away from a table. When we say space is three dimensional, we mean we need three numbers to completely locate the position of an object or point. A system for assigning these three numbers, or coordinates, to the location of a point in a reference frame is called a coordinate system. Most frequently, we will use a Cartesian (rectangular) system that describes the position in terms of x, y, z coordinates. However, you are free to choose the coordinate system you wish to use, orient it the way you want, and place its origin wherever you prefer.