## Sunday, November 1, 2015

### Free Body diagrams for Newton Laws of Motion Problems with Solutions

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 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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