Angular Acceleration

      When we switch on an electric fan, we notice that its angular velocity goes on increasing. We say that it has an angular acceleration ( change in velocity called acceleration ). Angular acceleration can be defined as "the rate of change of angular velocity". If ω_{i and} ω_{f are the values of instantaneous velocity of a rotating body at instants} t_{i and} t_{f, the average angular acceleration during the interval} t_f-t_{i is given by}

       The instantaneous angular acceleration is the limit of the ratio as approaches zero. Therefore, instantaneous angular acceleration is given by

      The angular acceleration is also a vector quantity whose magnitude is given by the above equation and whose direction is along the axis of rotation, given by right hand rule. Angular acceleration is expressed in units of /.

Angular Velocity

            Very often we are interested in knowing how fast or how slow a body is rotating. It is determined by its angular velocity which is defined as "the rate at which the angular displacement is changing with time". Referring to figure given below, if ∆Ѳ is the angular displacement during the time interval ∆t, the average velocity ω_av during this interval is given by,

        The instantaneous angular velocity ω is the limit of the ratio ∆Ѳ/∆t as ∆t, following instant t, approaches to zero.

        In limit when ∆t approaches zero, the angular displacement would be infinitesimally small, so it would be a vector quantity and the angular velocity as defined by the above equation would also be a vector. Its direction is along the axis of rotation and is given by right hand rule.

Angular velocity (ω) is measured in radians per second which is SI unit. Sometimes it is also given in terms of revolution per minute.

Units of Angular Displacement

          Three units are generally used to express angular displacement, namely degrees, revolution and radian. We are familiar with the first two. As regards radian which is SI unit, consider an arc of length S of a circle of radius r which subtends an angle Ѳ at that center of the circle given in radians (rad) as shown in the figure given below:

Or

       If OP is rotating, the point P covers a distance in one revolution of P. In radian it would be,

1 revolution =

Circular Motion

Introduction:

                  Many objects move in circular paths and their direction is continually changing. Since acceleration is a vector quantity, this change of direction means that their acceleration are not constant. A stone whirled around by a string, a car turning around a corner and satellites in orbits around the Earth are all examples of Circular Motion.

Angular Displacement:

                Consider the motion of a single particle P of mass m in a circular path of radius r . Suppose this motion is taking place by attaching the particle P at the end of a mass less rigid rod of length r whose other end is pivoted at the center O of the circular path, as shown in fig (a). As the particle is moving on the circular path, the rod OP rotates in the plane of circle. The axis of rotation passes through the pivoted O and is normal to the plane of rotation. Consider a system of axes as shown in fig (b). The z-axis is taken along the axis of rotation with the pivot O as origin of coordinates. Axes x and y are taken in the plane of rotation. While OP is rotating, suppose at any instant t, its position is , making angle with x-axis. At later time ,let its position be making angle with x-axis fig (c).

Angle defines the angular displacement of OP during the time interval . 

    The angular displacement ∆Ѳ is assigned a positive sign when the sense of rotation of OP is counter clockwise.

For very small values of ∆Ѳ, the angular displacement is a vector quantity.

    The direction associated with ∆Ѳ is along the axis of rotation and is given by right hand rule which states that

     Grasp the axis of rotation in right hand with fingers curling in the direction of rotation; the thumb points in the direction of angular displacement, as shown in figure-(d) 

Conditions of Equilibrium

There are two conditions of equilibrium.

First Condition of Equilibrium

"An object is said to be in equilibrium if the sum of all the forces acting on it in one direction balances the sum of all the forces acting in the opposite direction”. 

        It means that if the net force of all acting forces or the resultant force is zero then the object will be in static equilibrium.

        If the forces are acting in two dimensions i.e. x-component and y-component, then the first condition of equilibrium can be given as:

                                                  
                                                  

Examples: 

            If two forces and are equal in magnitude and they are acting on a ring in opposite direction then the ring will remain at rest.

Second Condition of Equilibrium

“For an object to be in equilibrium the net turning effect of forces acting on the object must be zero i.e. the sum of the clockwise torques must be equal to the sum of the anticlockwise torques”

Mathematically it can be expressed as,

                                                            
Example:

              A meter rod with a hole at its centre is suspended by means of a nail on a wall. We then apply two equal but opposite forces at point X and Y. As the upward force at Y is equal to the downward force at X, the body executes rotational motion.

Centre of Gravity:            

                 The centre of gravity of a body is defined as a point at which the whole weight of the body appears to act. This point may be inside or outside the body.

Three States of Equilibrium:

                   There are three states of equilibrium.

1-Stable Equilibrium:

            A body on being displaced if it has tendency to come back to its initial position then it is said to be in Stable Equilibrium.

OR

            It can also be defined as if the point of support is above the centre of gravity then it is said to be in Stable Equilibrium.

Example:

           If a book is lifted from its edges and then allowed to fall it will come back to its original position. Chair, table and other bodies lying on the floor are the examples of stable equilibrium.

2-Unstable Equilibrium:

           When a body on being displaced topples over and takes a new position then it is said to be in state of Unstable Equilibrium.

OR

               It can also be defined as if the point of support is below the centre of gravity then the body is said to be in Unstable Equilibrium.

Example:

          If a pencil standing vertically is slightly disturbed from its position, it will not come back to its original position.

3-Neutral Equilibrium:

           When a body on being displaced neither it topples over nor it takes its original position then the body is said to be in Neutral Equilibrium.

OR

        It can also be defined as if the point of support is on the centre of gravity then the body is said to be in Neutral Equilibrium.

 Example:

         If a ball is pushed slightly to roll, it will neither come back to its original position nor it will roll forward rather, it will remain at rest.

Equilibrium

When two or more than two forces act on a body in such a way that there is no change its translational or rotational motion then the body is said to be in equilibrium. OR An object is said to be in equilibrium if it has zero acceleration.

Kinds of Equilibrium 
1-Static Equilibrium

Everybody is in static equilibrium if all the forces acting on it cancel the effect of each other i.e. when there is no unbalanced force acting on the body then body is said to be in “Static Equilibrium”
 

Examples

            A book lying on a table is in static equilibrium because its weight is cancelled by the normal reaction of table.

                If the two forces and are equal in magnitude and they are acting on a ring in opposite direction. The ring will remain at rest.

2-Dynamic Equilibrium

If the two forces act on a body maintains its state of uniform motion under the influence of the acting forces then this type of equilibrium is called “Dynamic Equilibrium”.

Examples

           A car moving with uniform velocity is the example of Dynamic Equilibrium. The force of engine acts in forward direction while the force of friction between road and tires acts backward. These two forces, being equal and opposite cancel the effect of each other and the car moves with uniform velocity.

        Jumping out of a paratrooper from an aeroplane is another example of dynamic equilibrium. At a particular velocity the reaction of air on parachute becomes equal to the weight of the weight of the paratrooper. At this stage, both the forces cancel effect of each other and the parachute falls down with a uniform velocity.

Torque (Moment of Force)

The object which can rotate about an axis will start rotating under the influence of a suitable force; the turning effect of a force is called “Torque or Moment of force”. Torque may rotate an object in clockwise or anticlockwise direction.

                                               OR

          The moment of a force about a point is the product of force and perpendicular distance of its line of action from the fixed point.

          Torque is represented by "" (Tau) and it is the product of force (F) and moment arm (d).

                 Torque ( )= Force x Perpendicular Distance

                                                      OR  

                                                           

Factors on which Torque ( ) depends:

The torque depends upon the following two factors:

         1-The magnitude of the applied force (F).

         2-The moment arm (d). The perpendicular distance between the axis of rotation of the body and line of action of the force is called the “moment arm”.

Resultant of two forces acting at an angle (Parallelogram of forces)

Consider two forces acting on an object make a certain angle. The forces and can be represented in magnitude and direction by two adjacent sides of a parallelogram.

 By adding and also and by Head-to-Tail rule, the resultant force is a vector represented by the diagonal from the points of intersection. This is called “Parallelogram of forces”.