Physics : Forces and Newtons Laws of Motion

Force

What is Force?

Often I have seen students struggling with this question. In my experience 99% of the students are not able to define this simple idea which we have already leant in class 9th.

Definition: Force is the "Pull" or "Push" on the body.

The definition is so simple yet it carries so much weight. It is because if you don't know what is pull and what is push. You will not be able to categorize forces as pulling or pushing or both. And you will end up drawing wrong Free body Diagram also called as FBD which we will introduce later in this chapter.

• Pull Forces always act away from the body
• Push Forces always acts towards the body

Type of Forces

Forces in general have been divided into 2 broad categories

1. Contact Forces : Contact Forces need Physical contact for their application. Examples are Tension, Normal Reaction , Friction, Spring Forces etc.
2. Non Contact Forces : They don't need any physical contact with the body to exert itself. Examples are : Gravitational Force, Electrostatic Force, Magnetic Force and Nuclear Force

Lets try to discuss few of these forces in this section which will help us to understand this chapter in a better way. Once we have understood the chapter using these limited number of forces then I will introduce other forces in this chapter and following chapters. And then we will again apply all the concepts learnt in this chapter using forces learnt later on.

Tension

Definition of Tension: Tension is a self-adjusting force that pulls along the length of a string or rope. It's electromagnetic in nature at the molecular level.

• In physics problems, we typically assume strings are massless unless otherwise specified.
• Strings only become taut when their ends are pulled apart. A slack string exerts no tension.
• Tension is always a pulling force that acts away from an object along the direction of the string.
• In an ideal massless string, the tension is uniform throughout its entire length.
• Pulleys can change the direction of the force exerted by a string without changing the magnitude of the tension.

Normal Reaction

Definition of Normal Reaction: It is the force which one surface exerts on the other at the contact point. It is a pushing force and acts perpendicular to the surface. And it is exerted by one surface on to another in order to prevent the surfaces from penetrating each other

Weight

Definition of Weight: It is the force with which earth attracts a body . It is given by

W=mg

Here m is the mass and g is the acceleration due to gravity

Inertial vs Non-Inertial Frame

Frame of Reference

Definition of Frame of Reference: A frame of reference is the context in which or from which an observer makes measurements and observations.

It is where coordinate system is fixed and observations are made from origin and a clock is used to measure the position and time of events in space.

Types of Frames of Reference:

1. Inertial Frames of Reference:
2. Definition of Inertial Frame: A frame that is either at rest or moving with constant velocity in a straight line. Newton's laws of motion are valid. Also called unaccelerated, Newtonian, or Galilean frames. c) Practical considerations: - Ideal inertial frames don't exist in the universe. - A frame can be considered inertial if its acceleration is negligible compared to the observed object's acceleration.
3. Examples: Earth as an inertial frame for observing a falling apple. Sun as an inertial frame for observing planetary motion.
4. Non-inertial Frames of Reference:
5. Definition of Non-Inertial Frame: Frames that are accelerating relative to an inertial frame. Newton's laws of motion do not directly apply. - Require additional considerations (like fictitious forces) to describe motion accurately.
6. Examples: Cars moving in circular motion. Accelerating elevators. Aircraft during takeoff.
   
   

Newton laws of Motion

Newton's First Law of Motion

A body continue to be in its state of rest or of uniform motion along a straight line, unless it is acted upon by some external force to change the state.

• If the net force acting on an object is zero, its velocity remains constant.
• This means an object at rest stays at rest, and an object in motion stays in motion with the same speed and direction, unless acted upon by an external force.
• Newton's first law defines the concept of inertia.

Inertia and Types of Inertia

Inertia is an object's resistance to changes in its state of motion.

There are three types of inertia:

i. Inertia of rest: An object's tendency to remain at rest.

Example:

• When we happen to shake the branch of a tree, we observe that the leaves or the fruits fall down. This is because the branches comes in motion, whereas the leaves or the fruits tend to remain at rest and hence fall down.

ii. Inertia of motion: An object's tendency to remain in uniform motion.

Example:

• When a moving car suddenly stops, we know that the person sitting in the car falls in the forward direction. This is because the lower portion of the person's body in contact with the car comes to rest, whereas the upper portion tends to remain in motion due to inertia of motion.
• A person runs a certain distance before taking a long jump. This is mainly because the velocity acquired by running prior to attempting a long jump is added at the time of jump, so that he or she can cover a long distance.

iii. Inertia of direction: An object's tendency to continue moving in the same direction.

Example:

• When a car moves around a curve, a person sitting inside it is thrown outward. This is to ensure his or her direction of motion.

Newton's Second Law of Motion

The rate of change of linear momentum of a body is directly proportional to the external force applied on the body and this change takes place always in the direction of the applied force.

Linear Momentum

Definition of Linear Momentum: Linear momentum is a measure of an object's translational motion. For a single particle, it's defined as the product of the particle's mass (m) and velocity (v).

Mathematically expressed as: p = mv

Linear momentum is a Vector Quantity

Direction: It aligns with the direction of the velocity vector.

Units: In SI units, momentum is measured in kg⋅m/s (kilogram-meters per second).

System of Particles:

• For a system of multiple particles, the net momentum is the vector sum of individual momenta.
• Example for two particles: Masses: m₁ and m₂ Velocities: v₁ and v₂ Total momentum: p_total = m₁v₁ + m₂v₂

Importance: Understanding linear momentum is crucial in analyzing collisions, propulsion systems, and conservation laws in physics.

If a body of mass , moves with velocity  then its linear momentum can be given by p  and if force  is applied on a body, then

Newton's Third Law of Motion

To every action, there is always an equal (in magnitude) and opposite (in direction) reaction.

• When a body exerts a force on any other body, the second body also exerts an equal and opposite force on the first.
• Forces in nature always occurs in pairs. A single isolated force is not possible.
• Any agent, applying a force also experiences a force of equal magnitude but in opposite direction. The force applied by the agent is called 'Action' and the counter force experienced by it is called 'Reaction.
• Action and reaction never act on the same body. If it were so, the total force on a body would have always been zero i.e. the body will always remain in equilibrium.

How to Solve Questions using Newton's Laws of Motion

In order to solve question you have to proceed in a sequential manner through 3 steps

Step 1:

Draw the Free Body Diagram. Also called as force Diagram or simply FBD.

What is Free Body Diagram?

A Free Body Diagram (FBD) is a crucial tool in mechanics that isolates an object of interest from its surroundings. In this diagram, the object is typically represented as a simplified shape, such as a point or a box, and all interactions between the object and its environment are depicted as force vectors.

These forces are shown as arrows pointing in the direction of the force and should be clearly labeled (e.g., F_gravity, F_normal, F_friction).

It's important to note that acceleration (a) should not be included as a force in the FBD. Instead, acceleration is the result of the net force acting on the object, calculated using Newton's Second Law (F_net = ma)

Also the forces which are acting on the body and not the forces that the body exert on other object should be shown. Also only external forces should be considered

Step 2:

Choose an appropriate co-ordinate axis. While setting the axis 2 things needs to be kept in mind

1. The axis should be such that maximum number of forces should lie along the axis. In this way you will drastically reduce your calculations
2. One of the axis should be along the direction of acceleration. Otherwise you have to take component of acceleration which will increase your calculations.

Step 3:

Resolve the forces along the selected axis and then apply Newtons Laws of motion.

Mathematically it is

• for accelerating direction

F=Ma

• for Non-accelerating direction

F=0

Question on Equilibrium

In-order to understand how to solve questions let me take some Example.

You can either Read about the questions in following Section or watch the Video

Question on Dynamics

In-order to understand how to solve questions let me take some Example.

You can either Read about the questions in following Section or watch the Video

Pseudo force

Motion in Accelerated Frames

1. Inertial vs. Non-inertial Reference Frames: a) Newton's laws of motion typically apply to observations made in inertial frames of reference. b) However, we can extend their application to non-inertial (accelerating) frames with some modifications.
2. Example: Block in an Accelerating Train a) Consider a block on a smooth surface inside an accelerating train compartment. b) As the train accelerates forward, an observer inside the train sees the block move backward.
3. Limitations of Newton's Second Law in Non-inertial Frames: a) An observer in the train might conclude that a force is acting on the block, causing its apparent backward acceleration. b) However, Newton's second law (F = ma) doesn't directly apply in this non-inertial frame. c) We cannot directly relate the observed acceleration to real forces acting on the block.
4. Introduction of Pseudo Force: a) To use Newton's second law in a non-inertial frame, we introduce a "pseudo force" or "fictitious force." b) This force acts in the direction opposite to the acceleration of the non-inertial frame. c) For our train example, the pseudo force explains the block's apparent motion towards the back of the train.
5. Characteristics of Pseudo Forces: a) Magnitude: F_pseudo = -ma', where m is the mass of the object and a' is the acceleration of the non-inertial frame. b) Pseudo forces appear to act on objects similarly to real forces. c) Key difference: Real forces always result from interactions between two objects, while pseudo forces don't have a physical source.
6. Purpose: The introduction of pseudo forces allows us to apply Newton's laws in non-inertial frames, making calculations and analysis more straightforward in these situations.
7. Thus, we may conclude that pseudo force is not a real force. When we draw the free body diagram of a mass, with respect to an inertial frame of reference we apply only the real forces (forces which are acting on the mass), but when the free body diagram is drown from a non-inertial frame of reference a pseudo force (in addition to all real forces) has to be applied to make the equation , valid in this frame also.

Constraint Motion

When the motion of two bodies are inter-related by some function Such that the displacement or velocity or acceleration of the two bodies or more than two bodies can be calculated if motion of any one body is known.

The relationship between the variables (x,v and a) is called as Constraint relationship. And the aim of this discussion is to know how to find out that relationship.

This discussion can be divided into different sections depending on the type of systems we encounter during problem solving.

1.String Constraint: (Block and Pulley)

When the two object are connected through a string and if the string have the following properties:

• The length of the string remains constant i.e., it is inextensible string
• Sting always remains taut i.e., does not slacks.
  Then the parameters of the motion of the objects along the length of the string have a definite relation between them.

Type 1: Fixed Pulley
If pulley is fixed then the velocity of all the particles of string is same along the string.

Type 2: Moving Pulleys
To understand this consider the following example in which pulley is moving with some velocity velocity  and both block have also have some velocity  respectively as shown in figure.

However, If we observe the motion of  and  with respect to pulley. Then the pulley is at rest so the blocks will have equal and opposite velocity. But that will be non inertial frame. We have to solve questions from Inertial frame.

2. Block and Wedge Constraint

Here if the following conditions are met

• Contact must not be lost between two bodies.
• Bodies are rigid.

Then, the relative velocity / acceleration perpendicular to the contact surface of the two rigid object is always zero. Wedge constraint is applicable for each contact. or you can say that components of velocity and acceleration perpendicular to the contact surface of the two objects is always equal if there is no deformation and they remain in contact.

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