Acceleration Due to Gravity

When any object is moved then its speed is changed which is called velocity while the change in velocity is acceleration. When it is changed continues then it is in accelerating condition which is called the rate of acceleration. It is equal to the ratio of change in velocity with respect to time between the given paths. The vector quantity acceleration is used to show the increasing or decreasing speed or changing direction of object.

As we know that a free falling object is under the influence of gravity with an acceleration of 9.8 m/s in downward direction. The free falling objects are free from air resistance. This is the gravitational acceleration or acceleration under gravity denoted with the symbol of g with the standard value of 9.8 m/s but it is varied in different gravitational environments. Gravitational Fields widget is used for investigating the effect of location on the value of g. Let’s discuss about the calculation of acceleration due to gravitational force and its properties.

What is Acceleration Due to Gravity?

The most common type of force is the force due to gravity. It is experienced by all of us in every day life. For everyone, falling of an apple from its tree was a common sight and was considered as a natural phenomena. But for Sir Isaac Newton, even though he was a boy at that time, this phenomena made him think inquisitively. Ultimately, he found that the reason of objects falling towards ground is that the Earth has an inherent property of exerting an attraction of force on the objects. This force is called force of gravity and the acceleration generated on the objects due to this force is called Acceleration due gravity .

As per the definition of force, the equation of force due to gravity is given by,
W = mg
where m = mass of the Object
g = acceleration due to gravity.

In this case, the force is better known as the weight of the object. Also the acceleration in this case is the acceleration due to gravity. It is a constant and denoted by the letter g. The approximate value of acceleration due to gravity in feet is 32 ft/s2 or in meters is 9.8 m/s2.These are the values of acceleration of gravity on the Earth.

Acceleration Due To Gravity Formula

The acceleration due to gravity formula or the acceleration due to gravity equation can be derived from the fundamental equations of motion.
They are,
v = u + at,
s = ut +at2 and
v2 – u2 = 2as

Where v = Final Velocity
u = Initial Velocity
a = Acceleration
t = time taken.

In case of motion under gravity, the acceleration a and the distance s in the above equations are replaced by gravity g and the height of the object h
Thus the acceleration due to gravity equations are,
v = u + gt,
h = ut + gt2 and
v2 – u2 = 2gh

Where,
h = Height from ground level and
g = acceleration due to gravity.

When an object is thrown vertically upwards with an initial velocity u, the acceleration due to gravity acts as a Negative acceleration. That is, the velocity of the object gets reduced progressively, becomes 0 at a certain height and then the object starts falling like a free fall. The height at which the final velocity becomes 0 is the maximum height that the object can reach for a given initial velocity.

For a vertical throw, the acceleration due to gravity formulas can be formed by plugging in v = 0 and g = -g in the fundamental equations of motion. In such a case, the acceleration due to gravity equations are,
t =
h = ut -
gt2 and
u2 = 2gh

Acceleration due to Gravity Formula

The acceleration observed in any body due to gravity is called Acceleration due to gravity.

The force due to gravity is given by,
Where m = mass of the body and
g = gravity.

The Universal Law of gravitation gives,
Where G is a constant equal to 6.67  10-11 N-m2/kg2,
= mass of the body 1,
= mass of body 2,
r = radius between the two bodies.

Equating both the force formula we get
Hence Acceleration due to gravity formula is given as
The above formula can also be called Gravitational Acceleration Formula. It is used to find the acceleration due to gravity any where in space. On earth the acceleration due to gravity is 9.8 m/s2.

Acceleration due to gravity Problems

Below are given some problems based on acceleration due to gravity using the above formula.

Solved Examples

Question 1: Calculate the acceleration due to gravity for a planet having mass 4  1023 Kg and is 3  106 m away from sun?
Solution:

Given: Mass of planet M = 4  106 Kg,
Radius r = 3  106 m,
Gravitational Constant G = 6.67  10-11 N-m2/kg2
Acceleration due to gravity g =
=
= 2.964 m/s2.

Question 2: Calculate the acceleration due to gravity for the planet pluto if it is having mass of 1.22  1022 Kg and radius is 1.15 106 m.
Solution:

Given: Mass M = 1.22  1022 Kg and
Radius r = 1.15  106 m
The acceleration due to gravity g =
=
= 0.61m/s2.

Calculating Acceleration Due To Gravi

Experimentally

One of the acceleration due to gravity equations for free fall can be used for calculating acceleration due to gravity. The equation,
h = gt2 can be solved for g, as,
g =

Experiment Conducted to find Acceleration due to gravity:
An object of considerable weight (because the air resistance can be neglected), be dropped from fairly tall building whose height is accurately measured. The observer at the top of the building switches on a digital timer at the same instant he drops the object. The observer at ground level switches off the same timer when the object hits the ground and records the time of fall. Plugging the values of h and t, the acceleration due to gravity can be calculated. The experiment may be conducted for a few times and the mean of the readings may be taken. There is another theoretical method of calculating acceleration due to gravity.

Theoretically

Bodies allowed to fall freely were found to fall at the same rate irrespective of their masses (air resistance being negligible). The velocity of a freely falling body increased at a steady rate, i.e., the body had acceleration. This acceleration is called acceleration due to gravity 'g'.

We know,

From Newtons Second Law
F = mg ............ (1)

Equation of Force is given by Newton's law of gravity, which states that every point mass in the universe is attracts to every other point mass in the universe with a force which is directly proportional to the product of their masses and inversely proportional to square of the distance between them. It can be written as,
F =  .............(2)

Where
F is the force,
m is the mass of the body,
g is the acceleration due to gravity,
M is the mass of the Earth,
R is the radius of the Earth and
G is the gravitational constant

By knowing these constants, we can calculate acceleration due to gravity theoretically. From equations (1) and (2), we can also conclude that 'g' varies with
(a) Altitude
(b) Depth and
(c) Latitude.

An object acquires acceleration due to gravity because of the force of gravity on that object. There is another type of force of attraction that exists between two objects and the same is called as gravitational force and the acceleration created is called Gravitational acceleration

The equation of force is stated as "The force of attraction is directly proportional to the product of the masses of the two objects but inversely proportional to the square of the distance between them."

Mathematically, the equation of force in this case is given by,
F = G where
m1 and m2 are the masses and d is the distance between them.
G is a constant, known as Gravitational acceleration constant. Its approximate value is 6.674×10-11 N m2 kg-2.

Centrifugal Acceleration

The acceleration is the rate of change in the velocity of a moving body with respect to the given time. The acceleration can be classified according to the type of velocity and speed. If the body is in constant velocity then it is in the condition of constant acceleration because there is no change in velocity in the given time period. If a body is moved on circular path then it has a rotational acceleration due to rotational acceleration.

Similarly, some other types of accelerations are centripetal and centrifugal acceleration. Here we are discussing about the centrifugal acceleration. The centrifugal acceleration is the acceleration that is on the curved path. When a car takes a U turn on the curved way then it does not fall towards the centre. This is due to the centrifugal acceleration which makes the motion of the car. Let’s discuss the detail description of the centrifugal acceleration.

What is Centrifugal Acceleration?

Centripetal force is the force acting on the body which makes the body move towards the center of axis of rotation. This force acts along the radius towards the center of the circle.
There is another pseudo force acting opposite to that of centripetal force, which keeps the body move in a circular path with linear speed, away from the center of axis of rotation. This Force is called centrifugal force and if we consider its velocity we can see that the body moving in a circular path will be having linear velocity, so, such a change in the linear velocity taking place in this body with respect to time is called centrifugal acceleration.

Thus, centrifugal acceleration is the acceleration or change in velocity produced by the body moving in a circular path with respect to time, which keeps the body moving in a circular path without falling in to the center.

When an object is in motion along a curved path, it experiences a force and thereby an acceleration because the mass of the object remains constant. We will better explain taking a circular motion, as the concept is same. First, let us understand the concepts involved in a circular motion.

In case of circular motion, the velocity involved is called angular velocity
An angular velocity is measured in terms of the angle covered by the object per unit time. Normally, an angular velocity is denoted by the Greek letter . The direction of angular velocity is limited to clock wise direction or counter clock wise direction.

Hence the definition of of average angular velocity is,

= ,where,  is the angle rotated in the time t

Let us remember, although, the overall motion is circular, at any instant the object has a linear velocity v in a direction that is tangent to the circle of radius, say r, at that point.

Let  be the actual distance moved by the object along the circumference. As per the geometry of circles,

= Therefore,
=

or, = ,
where, V =  is the linear velocity of the object at any point.

Now, an angular acceleration is defined as the rate of change of angular velocity with respect to time. It is denoted by another Greek letter . Now considering an infinitesimal study,
= and hence,
= ,or = But an object is subjected to two types of acceleration in a circular motion.

Look at the following diagram. Here,
ar = Direction of Centripetal Acceleration
at = Direction of Centrifugal Acceleration

Let's say an object is tied at A, one end of a string OA and rotated keeping the other end O as center. When the string is rotated fast, the string gets completely stressed out and the length of the string becomes the radius of rotation. It means a force is exerted on the object from the center to the end and there by an acceleration AO in a radial direction is faced by the object. This is called centrifugal acceleration. To counter this force a tension force developed in the string acting in the opposite direction. This tension force is called the centripetal force and the acceleration generated on the object is called the centripetal acceleration and denoted as ar. The opposite force which is acting tangentially is called centrifugal force which is denoted by at.

As per Newtonâ€™s third law, the magnitudes of centripetal and centrifugal forces and hence their acceleration components are equal.

Centrifugal Acceleration Equation

For finding the centrifugal acceleration equation or a centrifugal acceleration formula, let us study the following diagram and derive the equation for centripetal acceleration and equate it to centrifugal acceleration. In figure (i), A and B are the positions of an object and the positions are infinitesimally close.
Figure (ii) shows the translated vector diagram of the centripetal velocity vectors at A and B. As per the similar triangles property,

= Since A and B are very close we can approximate AB, to the length of the arc AB and hence AB = v  dt
in fig (ii), since A and B are very close, Hence v + dv  dv.

Therefore, = becomes as,
= or,
= .Since  is the centripetal acceleration, we arrive at the formula as,
Centripetal Acceleration ar = , and hence,

Centrifugal Acceleration at =
(since centrifugal and centripetal forces are equal and opposite to each other).

It is expressed in m/s2.

The Effects of Centrifugal Acceleration

The Centrifugal Accelerations due to centrifugal forces are perceived by us in everyday life. When you are traveling in a car and it takes a curve, you are pushed in a direction opposite to the radius of the curve. As long as this force of push is within a control you are safe because the inherent centripetal force takes care.

But as this force is directly proportional to the square of the linear speed of the car and inversely proportional to the radius of the curvature, there can be a danger of the car itself being thrown out radial if the linear velocity is high and the radius of curvature is small.

This is the reason why the highways are banked wherever there is a curve on the high way and also why airplanes fly in an inclined positions when they take a turn. The idea behind such cases is, the horizontal component of the reaction force of earth on the car/airplane traveling/flying at the design velocity is more than the centrifugal force.

Acceleration Due to Gravity

When any object is moved then its speed is changed which is called velocity while the change in velocity is acceleration. When it is changed continues then it is in accelerating condition which is called the rate of acceleration. It is equal to the ratio of change in velocity with respect to time between the given paths. The vector quantity acceleration is used to show the increasing or decreasing speed or changing direction of object.

As we know that a free falling object is under the influence of gravity with an acceleration of 9.8 m/s in downward direction. The free falling objects are free from air resistance. This is the gravitational acceleration or acceleration under gravity denoted with the symbol of g with the standard value of 9.8 m/s but it is varied in different gravitational environments. Gravitational Fields widget is used for investigating the effect of location on the value of g. Let’s discuss about the calculation of acceleration due to gravitational force and its properties.

What is Acceleration Due to Gravity?

The most common type of force is the force due to gravity. It is experienced by all of us in every day life. For everyone, falling of an apple from its tree was a common sight and was considered as a natural phenomena. But for Sir Isaac Newton, even though he was a boy at that time, this phenomena made him think inquisitively. Ultimately, he found that the reason of objects falling towards ground is that the Earth has an inherent property of exerting an attraction of force on the objects. This force is called force of gravity and the acceleration generated on the objects due to this force is called Acceleration due gravity .

As per the definition of force, the equation of force due to gravity is given by,
W = mg
where m = mass of the Object
g = acceleration due to gravity.

In this case, the force is better known as the weight of the object. Also the acceleration in this case is the acceleration due to gravity. It is a constant and denoted by the letter g. The approximate value of acceleration due to gravity in feet is 32 ft/s2 or in meters is 9.8 m/s2.These are the values of acceleration of gravity on the Earth.

Acceleration Due To Gravity Formula

The acceleration due to gravity formula or the acceleration due to gravity equation can be derived from the fundamental equations of motion.
They are,
v = u + at,
s = ut +at2 and
v2 – u2 = 2as

Where v = Final Velocity
u = Initial Velocity
a = Acceleration
t = time taken.

In case of motion under gravity, the acceleration a and the distance s in the above equations are replaced by gravity g and the height of the object h
Thus the acceleration due to gravity equations are,
v = u + gt,
h = ut + gt2 and
v2 – u2 = 2gh

Where,
h = Height from ground level and
g = acceleration due to gravity.

When an object is thrown vertically upwards with an initial velocity u, the acceleration due to gravity acts as a Negative acceleration. That is, the velocity of the object gets reduced progressively, becomes 0 at a certain height and then the object starts falling like a free fall. The height at which the final velocity becomes 0 is the maximum height that the object can reach for a given initial velocity.

For a vertical throw, the acceleration due to gravity formulas can be formed by plugging in v = 0 and g = -g in the fundamental equations of motion. In such a case, the acceleration due to gravity equations are,
t =
h = ut -
gt2 and
u2 = 2gh

Calculating Acceleration Due To Gravity

Experimentally

One of the acceleration due to gravity equations for free fall can be used for calculating acceleration due to gravity. The equation,
h = gt2 can be solved for g, as,
g =

Experiment Conducted to find Acceleration due to gravity:
An object of considerable weight (because the air resistance can be neglected), be dropped from fairly tall building whose height is accurately measured. The observer at the top of the building switches on a digital timer at the same instant he drops the object. The observer at ground level switches off the same timer when the object hits the ground and records the time of fall. Plugging the values of h and t, the acceleration due to gravity can be calculated. The experiment may be conducted for a few times and the mean of the readings may be taken. There is another theoretical method of calculating acceleration due to gravity.

Theoretically

Bodies allowed to fall freely were found to fall at the same rate irrespective of their masses (air resistance being negligible). The velocity of a freely falling body increased at a steady rate, i.e., the body had acceleration. This acceleration is called acceleration due to gravity 'g'.

We know,

From Newtons Second Law
F = mg ............ (1)

Equation of Force is given by Newton's law of gravity, which states that every point mass in the universe is attracts to every other point mass in the universe with a force which is directly proportional to the product of their masses and inversely proportional to square of the distance between them. It can be written as,
F =  .............(2)

Where
F is the force,
m is the mass of the body,
g is the acceleration due to gravity,
M is the mass of the Earth,
R is the radius of the Earth and
G is the gravitational constant

By knowing these constants, we can calculate acceleration due to gravity theoretically. From equations (1) and (2), we can also conclude that 'g' varies with
(a) Altitude
(b) Depth and
(c) Latitude.

Gravitational Acceleration Constant

An object acquires acceleration due to gravity because of the force of gravity on that object. There is another type of force of attraction that exists between two objects and the same is called as gravitational force and the acceleration created is called Gravitational acceleration

The equation of force is stated as "The force of attraction is directly proportional to the product of the masses of the two objects but inversely proportional to the square of the distance between them."

Mathematically, the equation of force in this case is given by,
F = G where
m1 and m2 are the masses and d is the distance between them.
G is a constant, known as Gravitational acceleration constant. Its approximate value is 6.674×10-11 N m2 kg-2.

Instantaneous Acceleration

When an object is moving, then it changes its speed and thus there is a change in velocity. An acceleration is the rate of velocity change in the given time period. It is a vector quantity with magnitude and direction and its changes with changing the direction of the velocity. It is positive with increasing velocity while negative with decreasing velocity. The negative directed acceleration is called retardation and deceleration.

When velocity is constant for a particular period of time then the object is in constant acceleration position. If we talk about one dimension, the rate at which objects get slower speed or speed up is equal to the acceleration. In case of instantaneous acceleration, it is the change at a particular moment. The object with different acceleration at different moments of time then the object is in a variable acceleration state. This is instantaneous acceleration. Here we are discussing more about on instantaneous acceleration with its mathematical formula and problem based on it.
ACCELERATION
We know that Velocity is the rate of change of distance with respect to time. So what can we call if there is rate of change of Velocity? Its nothing but Acceleration.

The Acceleration is the rate of change of velocity with respect to time.

The Acceleration can also be defined mathematically by,
a = ,Where,
a = acceleration of the moving body
v = velocity of the moving body
t = time

Instantaneous Acceleration Formula

To know what is Instantaneous Acceleration, let us consider acceleration of a body at a particular instant of time. So, at that instant of time, let us find its acceleration, that will be its Instantaneous AccelerationThe acceleration of the moving body at any instant of time is defined as its instantaneous acceleration.

The Instantaneous acceleration equation is given by,
ains =
.

The acceleration of the body can be defined in terms of Average Acceleration. The Average Acceleration is defined as the difference between velocities at two points divided by the time taken to travel the distance between them.

The Average acceleration is given by,
aavg = In order to find the instantaneous acceleration, we find the average acceleration for both initial and final point, when we reach that interval of time.

How to Find Instantaneous Acceleration?

If the body is moving with the Constant velocity then its instantaneous acceleration is zero, since acceleration is measure of how fast a body is changing its velocity. For the body moving with varying velocity, the instantaneous acceleration can be obtained by using the above formula of the instantaneous acceleration.
Given below are some numerical problems which will helps us have a better understanding of it.

Solved Examples

Question 1: A diesel locomotive is moving on the rail track. The distance traveled by it is defined by the following equation: x = 4t2 + 4t + 12,
a. Find the instantaneous velocity at time t = 10 seconds?
b. Find the instantaneous acceleration at time t = 18 seconds?
Solution:

a. The instantaneous velocity is defined by,
V =  ,
V =
V =  (8t + 4),
V = 8  10 + 4
V = 84 m/s.

Similarly,
b. The instantaneous acceleration is defined by,
a =  ,
=  ,
=  {8t + 0}
=  (8  1),
= 8 m/s.

Question 2: Find the instantaneous acceleration of a truck, at time t = 6 seconds, moving with the velocity v given by, v = 3t3 + 4t2 + 5t + 6.
Solution:

Let us first find the acceleration by finding the first derivative of the velocity equation,
a =  ---------(1)
a = 9t + 8t + 5 ---------(2)
So the instantaneous acceleration at t = 6 seconds is given by;
a =  (9t + 8t + 5),
a = 9  36+ 8  6 + 5
= 324 + 48 + 5
= 377 m/s.

Question 3: A car is moving on a linear track with the varying acceleration. Its velocity is given by, v = 2t2 + 2t + 4. Find the instantaneous acceleration of the car at t = 8 seconds?
Solution:

The instantaneous acceleration is given by, a =
a =  ,
a =  (4t + 2),
a = 4  8 + 2,
a = 34 m/s.
The acceleration in this is varying, since the acceleration is a function of time (t).

Instantaneous Acceleration Graph

As we have seen from the previous examples, the acceleration could be constant or it could be varying. We can prepare a instantaneous acceleration graph for the previous example where instantaneous acceleration is varying.
For this let us assume that the car is moving for 20 seconds and we will calculate the acceleration for 20 second with the interval of 2 seconds.

 Time 0 2 4 6 8 10 12 14 16 Acceleration 2 10 18 26 34 42 50 58 66 The graph shown above is the instantaneous acceleration graph for the acceleration as shown in above table.

Magnitude of Acceleration

Acceleration means the change of velocity of an object with respect to time. The ratio of change in velocity to the change in time in a given interval is the average acceleration in that time interval. Suppose the change in time is made infinitesimally small and tends to be 0, then the acceleration is the instantaneous acceleration at that point.
We know that magnitude is something where only distance covered is considered but not the displacement.
Generally, the motions of objects are classified as motion in a straight line in a given time interval or the motion may be along a curve. The most common motion along a curve is the circular motion or rotation.
The accelerations are broadly of two types :

• Linear acceleration in case of linear motion
• Angular acceleration in case of circular motion
The magnitude of acceleration is something where we are considering only magnitude but not the direction. Of course, there will be change in direction in velocity in acceleration but we are not going to deal with that since we are considering only the circular motion, where direction is limited to two. In case of linear motions, the direction may be anything but in case of circular motion the directions are limited to two. The object may move in a clockwise direction or in a counter clock wise direction around the center of motion. The latter is considered positive as per convention.

In this article, let us concentrate on circular motion and hence the angular acceleration. We will consider only the counter clock wise rotation and will focus only on magnitude of angular acceleration.

Magnitude of Acceleration Formula

The Magnitude of Acceleration is defined as Increase in Velocity to the corresponding short interval of time.
The Magnitude of Acceleration is given by
a =
.

In terms of dimensions, we can define it as,
a =
LT -2
where,
v =
and
t = .
Thus, Dimensional Formula for Magnitude of Acceleration is [LT -2].

In uniform circular motion, the direction of acceleration is towards the center of the circular path that is called centripetal acceleration and the constant of magnitude of acceleration a is given by,
a = .
Where, v denotes the speed in m/s, and
r is the radius of circular path in m.

If we Consider free fall of body, the magnitude of acceleration is conventionally denoted by g and has approximate value of 9.8m/s2.

Magnitude of Angular Acceleration

Before proceeding to the magnitude of angular acceleration and deriving a formula for the magnitude of angular acceleration, let us refresh the fundamentals of the circular motion.

In case of circular motion the velocity involved is called angular velocity. Angular velocity is measured in terms of the angle covered by the object per unit time. Normally, an Angular velocity is denoted by the Greek letter . The direction of angular velocity is just two, limited to clockwise direction or counter clockwise direction.

Hence the definition of of average angular velocity is,
= ,

where  is the angle rotated in the time t

Now, we will bring in the linear velocity v and the radius of the circle r, in a circular motion.

Let  be the actual distance moved by the object along the circumference. As per the geometry of circles, we will bring in the linear velocity v and the radius of the circle r, in a circular motion.
=

Therefore,
=  =

or,
=

where v is the linear velocity of the object at any point.

Now, an Angular acceleration is defined as the rate of change of angular velocity with respect to time. It is denoted by another Greek letter . Now considering an infinitesimal study,

=

and hence,
=

or
=

[ = ].

= .

But, in a circular motion two types of accelerations exist. Let us explain this concept with a simple experiment.

Take a string and securely tie an object at one end, hold the other end and start rotating the string. You will notice that the string gets completely stretched. It means a force is exerted on the object from the center to the end and there by an acceleration ao in a radial direction is faced by the object. To counter this force a tension force developed in the string acting in the opposite direction. This tension force is called the centripetal force and the acceleration generated on the object is called the Centripetal acceleration and denoted as ac. Suppose the string snaps right at the end where the object is tied. The object, now, will invariable fly in a direction tangent to the circle at the point. This is because there exists a tangential force on the object and the acceleration due to this force is called the tangential acceleration and denoted as at.
The situation is described in the following diagram The string is turned in clockwise or anticlockwise direction, the magnitude of acceleration will be the same, which is given by,a = .

Constant Acceleration

When an object is moving then there is change in its speed and also its velocity. The change in speed is called velocity while the change in velocity is known as acceleration.  If an object is said to be in acceleration then its velocity is continuously changed. This is the state of acceleration. This is defined as the rate change of velocity with time.

This word acceleration is generally used to represent the increasing speed state.  It is not necessary to increase the speed, it also depends on velocity change and change in the direction of motion with time. This is the reason that it is a vector quantity which has both the direction and magnitude. Thus it can be occurred with increasing or decreasing or changing the direction of object motion.

Here we discuss about the different types of acceleration especially constant acceleration in which the velocity of moving object does not change in a particular period of time. Now we discuss its formula, motion graph, and some problem based on it.

Constant Acceleration Definition

Constant acceleration is the special case of acceleration. When a moving body changes its velocity by the equal amount per second, the body is said to be moving with the constant accelerationFor better understanding of the constant acceleration, consider the following tables:

Table - 1
 Time(s) Velocity (m) 1 2 2 4 3 6 4 8 5 10

Table -2
 Time(s) Velocity (m) 1 0 2 3 3 4 4 7 5 12

If we look at table-1 and table-2, we can see that in the table-1 the velocity is constantly varying with same amount per second so the acceleration in this case is constant. Now if we look at table-2, We can conclude that the velocity in this case is not varying constantly per second and hence the acceleration in this case is not constant.

From the above table it is concluded that the body moving with constant acceleration should have constant velocity per unit time.

Constant Acceleration Equations

Acceleration could be mathematically defined as,

a =

where,
v2 = velocity of the body at time t2
v1 = velocity of the body at time t1

For the body to move with constant acceleration the difference between its velocities between equal time intervals should be equal.

If the time interval is considered to be very small then the above equation could be rewritten as;

a =

The above equation is also known as constant acceleration formula.

Constant Acceleration Graph

Considering the above table 1, the graph of constant acceleration could be, Motion with Constant Acceleration

When a body is dropped from a height, it gets accelerated under the influence of gravity and its acceleration is equal to 9.8 m/sec2With the help of this statement, we can find the height from which the body is dropped. The acceleration in this case is constant. The relation between acceleration and displacement of the body is,
s = ut +  at2
Where,
s = displacement of the object
u = initial velocity of the object
a = acceleration (or deceleration) of the object
t = time taken by the object to get displaced by ‘s’ units

Non Constant Acceleration

Non constant acceleration is the most general description of motion. It is the rate of change in velocity. In other words, it means that acceleration changes during motion of the object. This variation can be expressed either in terms of position (x) or time (t).If the non constant acceleration is described in one dimension, we can easily extend the analysis to two or three dimensions using composition of motions in component directions.

For this reason, we shall confine ourselves to the consideration of non constant, i.e., variable acceleration in one dimension. For finding the 2-D and 3-D equations of the angular motion, we would find the equation of motion in each direction separately since the motion in one direction is independent of the motion in other direction.

Angular Acceleration

Angular Acceleration is the rate of change of the angular velocity of the object with respect to time.

Angular acceleration can also be defined, mathematically, as shown below,
=  =

=
Where,
= angular velocity of the object
= angular acceleration of the object
= linear tangential acceleration
= angular motion of the object
t = time

Angular Motion with Constant Acceleration

Consider the situation that an object is moving in a plain such that it is changing its direction frequently. The acceleration of the object is, say constant. Lets also consider that the initial velocity v0 at time t = 0 and velocity at time t is v, then the acceleration can be obtained as,

=

=

Solving it further we have,
v = v0 -  t

Considering that the body is moving in the angular direction, we can find the angular acceleration of the object also,

=

where,
= Angular Velocity of the object
= Angular Acceleration of the object
t = Time

Constant Acceleration Problems

Lets take up some problems to understand the uniform or constant acceleration better.

Solved Examples

Question 1: A bus starts from rest and accelerates uniformly over a time of 5 sec. In this time it covered a distance of 100 m. Find the acceleration of the bus?
Solution:

From question, we have;
s = 100m
t = 5 sec
Now using following equation;
s = ut + at
where, u is the initial velocity.

Putting the values of all the variables from the question we have;
a =
a =
Solving we have
a = 8 m/sec

So, the acceleration of the bus is 8 m/sec.

Question 2: For the better understanding of the constant acceleration, let us consider following table:

 Times(s) Velocity(m) 1 2 2 4 3 6 4 8 5 10

Find the acceleration of the object?
Solution:

For checking whether the object is traveling with constant velocity, find the acceleration between any two points and checks it with another two points. If both have equal results then it could be concluded that the object is moving with constant acceleration.
So, we will consider motion of the object from 1 sec to 3 sec and 2 sec to 5 sec.

a =
a =
a =  = 2m/sec
a =
a =
a =  = 2m/sec

As we see that the a and a are equal we can consider that the body is moving with constant acceleration.
This can also be obtained from the graph between the velocity and time.

Angular Acceleration

When an object is in motion then its velocity is changed continuously then it is called in acceleration state. We know that acceleration is the change in velocity of moving object with respect to time. When an object is moved on circular path then its velocity is the angular velocity. The angular velocity is in the perpendicular direction of the rotational plane and it is related to the change in the angular speed and measure with unit as radians per second or revolutions per second.

The angular acceleration shows the change in the angular velocity with time. It is not required to be this rotational acceleration in the same direction of angular velocity; for example, if a car is rolling on highway with increasing its speed then the direction of angular acceleration is on the left side of the axis of the wheel and it becomes disappears when the car stops and maintained its constant velocity. When it slows down then the acceleration is in the reverse direction. It also helps to give relation between circular motion and curve motion.

Here, we are discussing about the angular acceleration and its mathematical formula, and relation with torque.

Angular Acceleration Definitio

We know that the rate of change of displacement with respect to time is called velocity of the object. But if there is a change in the velocity while the body is going in uniform circular motion, what can we call it? how can we Calculate it?

Angular acceleration is the rate of change of angular Velocity with respect to time. It is a vector quantity.

It can be represented as:
=  .
or
=
where,
is the angular Velocity
at is the linear tangential acceleration
r is the radius of Circular path

Angular Acceleration Formula

If the angular velocity is constant, then algebraically it is defined as,
=
where,
is angular velocity
is the angle rotated
‘t’ is the time taken

It may be realized when the angular velocity is constant, the angular acceleration is 0. If the velocity is not constant, then the constant angular acceleration  is defined as

=  =

If the angular acceleration is not constant and varies from time to time, then we can only refer to average angular acceleration and instantaneous acceleration.

In such a case the angular acceleration formulas are,
av =  and i =  = .These equations help us in finding angular acceleration.

Angular Acceleration Units

As per the definition, an angular acceleration is the amount of angle covered per square of time. Hence dimensionally its unit must be the ratio of angle to the square of time. Then, in common terms, the angular acceleration unit is degrees/square of time like s2, min2, hr2. But in internationally accepted unit (SI unit), the unit of angle is radian (rad) and the unit of time is second (s). Hence the standard unit of Angular acceleration is rad/s2.
Hence it is more sensible to define the displacement in terms of the angle it has rotated. If the object completes one full rotation it comes back to the initial position on the circle, meaning the displacement is 0. But the angular displacement is 360o degrees or 2 radians. Therefore, in Circular motions, the rate of change of angle is defined as angular velocity and the rate of change of angular velocity is defined as ‘Angular acceleration’.

Average Angular Acceleration

Angular acceleration is the rate of change of angular velocity with respect to time. Suppose the body has moved some distance then if we consider initial and final point where the displacement gets an end, then its average angular acceleration is given by

Average angular acceleration,

where,
ω= final angular velocity
ωi = initial angular velocity
tf = final interval of time or the end of time
t= initial interval of time
It has the same unit as the angular acceleration, i.e., rad/s2.

Torque and Angular Acceleration

In case of linear motion, as per Newton’s second law a force is required to accelerate object. That force is defined as the product of the mass of the object and the acceleration created.

In case of circular motion the force that is required to impart angular acceleration is called ‘Torque’. In other words, torque is an angular force and it is denoted by the Greek letter  (pronounced as ‘tau’).
Also in rotational motion the moment of inertia I of the object plays the role of mass.

The torque in a circular motion is defined as,
= I

Constant Angular acceleration

If an object undergoes rotational motion about a fixed axis under a constant angular acceleration , its motion can be described with the following set of equations,

-  =  t.
-  =  t +   t2
=  + 2  ( - )

where  = angular speed of the rigid body at time t=0
= angular speed of the rigid body at time t
= angular accelerationAngular Acceleration to Linear Acceleration
As mentioned earlier, the definition of average angular velocity is the change in angle  with respect to time t.
=
where,
is the angle rotated in the time t

Now we will bring in the linear velocity v and the radius of the circle r, in a circular motion. Let  be the actual distance moved by the object along the circumference. As per the geometry of circles,
= ,
Therefore,  =
= .

Now if we say, v =
we can write
=  and hence,
=
where v is the linear velocity of the object at any point and its direction is along the tangent at that point.

Now considering an infinitesimal study,
=  =  =