Free body diagrams
To solve problems using Newton
laws of motion, we need to learn the concept of free the body diagrams. This is
to identify all the forces acting on the body but will never consider the force
is applied by the body. It is simply because forces acting on the body will
cause the motion on the body but the force is applied by the body will cause
the motion on some other body.
We can consider number of the
bodies together as a system when all the bodies are having the same
acceleration. All parts of the system shall have the same acceleration and it
is the qualification to call the group of the bodies as a system. As forces a
vector quantity, we shall consider the direction of the force acting on the
body and the motion that is caused by that force.
We shall identify the direction
of the motion of the system or at least we can anticipate the direction of the
motion of the system. The forces acting along the direction of the motion shall
be treated as positive to another forces acting against the motion shall be
treated as negative.
We can explain that how we can
draw this kind of the free body diagram by taking some examples and by solving
some problems.
Problem
A monkey of mass 10 kg is
climbing up a rope with and acceleration two meter per second Squire. What is
the tension in the rope?
Solution
The motion of the Monkey is in
the upward direction which means the resultant forces in the upward direction.
Hence the force acting in the upward direction shall be treated as positive and
the force acting against the motion shall be treated as negative.
Tension in the string is a force
that causes the tightness in the body and it always acts towards the point of
suspension and away from the body. We can draw the free body diagrams as shown
below. The resultant force is the vector sum of the tension on the weight of
the monkey. Here tension being it is acting in the upward direction shall be
treated as positive. As the weight is acting in the downward direction which is
opposed to the motion of the monkey, it shall be treated as negative. The
equation can be further solved as shown below.
Problem
A fire man is sliding down along
a rope where the breaking stress of the rope is one by fourth of his weight.
What is the maximum acceleration with which he can slide down along the rope
without breaking the rope?
Solution
As the man is sliding down, the
resultant force is acting in the downward direction. Weight is the force that
is acting in the downward direction and the tension in the wire is the force that is acting in
the upward direction. Hence we can write the equation further resultant force
as the difference between the weight and the tension. In the place of the
tension we can substitute one by fourth of the weight as given in the problem and
further we can solve the problem as shown below.
Concept and application
Let us consider two bodies of
different masses are in contact with each other on a smooth horizontal floor.
Let a horizontal force F is applied on the first body from left to right and we
need to calculate the acceleration of the system and what is the contact force
basing on the Newton laws of motion.
Explanation
Let us consider two bodies are
having different masses lying on a smooth horizontal plane. Being the surface
is smooth, there is no question of friction and we can ignore the impact of
friction. As the force is applied from left to right the system is going to
move in the same direction. Hence the forces acting in the same direction shall
be treated as positive and the forces acting against the motion shall be
treated as negative.
When the first body is pushed by
a horizontal force, it will further push the second body with a certain force
and that force is called contact force. The contact force applied by the first
body on the second body acts like action on the second body. The second body
reacts by applying the same magnitude of the force on the first body and it
shall be treated as reaction. This action and reaction are equal in magnitude
opposite indirection but they are not acting on the same body. And hence they
are not going to cancel each other.
We can draw free body diagram for
each body as shown in the diagram below. We can notice that on the first body
two forces are acting. The first force is the horizontal force that we have
applied. The second force is the contact force applied by the second body on
the first body. We shall not consider the contact force applied by the first
body on the second body as it is not causing any motion in the first body.
We can draw the free body diagram
for a second body by considering the only force acting on it. It is nothing but
the contact force applied by the first body on the second body. It is acting in
the forward direction the second body also motion the forward direction.
By solving these two equations we
can get the value of the acceleration the contact force as shown below.
Application
Let us consider three different
bodies of different masses on a smooth horizontal plane. Let they are connected
with the strings between them and the first body horizontal force is applied as
shown. We want to calculate the tension in the each wire and the acceleration
of the system.
To find out these values we can
use the concept of the free body diagrams. We can consider all these bodies
together as a system because they’re all having the acceleration as a common
value. We can draw the free body diagram for each body as shown by considering
only the forces acting on that particular body. Will never consider the force
applied by the body during this process.
In the following diagram, three
free body diagrams are drawn for each body and that’s solved as shown below.
Atwood’s machine
This machine is a combination of
the two bodies connected over a smooth pulley with the help of a string. The
pulley is fixed to a rigid support. As the mass of the bodies are different one
body moves in the downward direction and automatically the other body motion
the upward direction. By considering the forces acting along the direction of
motion as positive and vice versa, we can draw the free body diagram and solve
the corresponding equations of motion as shown below.
Application
Let us consider two different
bodies connected with a string. Let one body is on the table and another body is
hanging over air and they connecting with a string and is passed over a pulley.
Assume that the surface as a smooth surface, we need to calculate the
acceleration of the system under the tension on the wire. We can also calculate
the tension acting in the pulley.
The body hanging in there will
move in the downward direction and a body on the table will move on the
horizontal direction. For a body hanging in the air weight is acting in the
downward direction and the tension in the string is acting in the upward
direction. Here weight will be automatically treated as positive and the
tension as negative while we are writing the equations of motion. The only
force acting on the body on the table is the tightness on the wire which is
nothing but the tension. By drawing the free body diagrams and by writing the
equations of motion with can solve this situation as shown below.
Application
Two bodies are connected over a
triangular wedge ith the help of a smooth pulley. We need to calculate the
tension on the wire as well as the acceleration of the system. We’ll be
following the same process to solve any of the bodies that are connected. The
process is simple as explained above. This could be explained in different
steps.
Steps to Solve Problem basing on Free body Diagrams
In the first step we shall
identify the direction of the motion of the bodies in the system.
In the second step we shall identify the forces acting on the bodies. In this step we are not going to consider the force is applied by the body and we are going to consider only the forces acting on the body which cause the motion on a body.
The third step we can draw the free body diagrams by clearly showing all the forces acting on the bodies.
In the fourth step we can draw the equations of motion. During this process we shall consider all the forces acting along the direction of the motion as positive and vice versa.
In the fifth step we can solve these equations and get the required answers.
In the second step we shall identify the forces acting on the bodies. In this step we are not going to consider the force is applied by the body and we are going to consider only the forces acting on the body which cause the motion on a body.
The third step we can draw the free body diagrams by clearly showing all the forces acting on the bodies.
In the fourth step we can draw the equations of motion. During this process we shall consider all the forces acting along the direction of the motion as positive and vice versa.
In the fifth step we can solve these equations and get the required answers.
In this problem the entire weight
is acting in the vertically downward direction and we can identify the
component of the weight is trying to pull the body in the downward direction.
This part of the weight shall be treated as positive and the tension on the
strings shall be treated as negative.
We can solve the problem as shown
below.
Problem
Calculate the acceleration in the
two different cases shown in the diagram. In the first case two different
bodies are connected over a pulley and in the second case the first body is
connected over a pulley under the second end a force is applied.
Solution
In the first case on the heavy
body weight is acting in the downward direction and other tension is acting in
the upward direction. As a heavy body is tend to move in the downward direction
its weight shall be treated as positive and attention shall be treated as
negative. By using the equation of motion and the free body diagram, we can
solve equation for acceleration in the particular case.
In the second case the tightness
is caused in the wide because of the applied force and there is no need to
consider that acceleration in this part.
A detailed solution is as shown
below.
Problem
Three bodies of equal masses are
connected over a pulley as shown. We need to calculate the tensions in the
string.
We can clearly draw free body
diagrams by considering all the forces acting on the body and solve the
equations of motion as shown below.
Frame of reference
Frame of reference is a system
which represents the position of a body. It is the way of explaining the
coordinates of a body with respect to something. If you’re in a bus, you are
the body and bus is your frame of reference. If you are on the earth, the earth
itself is treated like the frame of reference.
If the frame of reference is in
the state of rest or in the state of uniform motion, then it is called inertial
frame of reference. All Newton laws are very much widely read in the inertial
frame of reference.
If the frame of reference is
having a acceleration, then it is called non-inertial frame of reference.
Newton laws are not valid in this kind of frame of reference. But we don’t have
any other choice than applying the Newton laws of motion. So to nullify the
effect of non-inertial frame of reference, we do imagine new force on the body
who is equal to the force with which the frame of reference is moving. This
imagined forces called Virtual force or pseudo-force and it is imagined in the
opposite direction to the force that is acting on the frame of reference. This
imagined force do compensate the non-inertial frame for the body and it acts as
if like the bodies in the inertial frame.
Problem and Solution
Let us consider a problem and
solve it where the bodies in a non-inertial frame as shown below. In this case
the truck is having a horizontal acceleration and a simple pendulum is
suspended to the rigid support vertically. The truck is the frame of reference
and as it is having acceleration, it automatically behaves like a non-inertial
frame of reference. To nullify the impact of the non-inertial frame, we do
imagine a acceleration on the Bob of the simple pendulum who is equal to the
acceleration of the frame of reference in the opposite direction. It is further
solved as shown below.
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