Force and Laws of Motion

Notes by Mohit Sir

Force

It is defined as push or pull which tries to change or changes its state.

Effects of Force

Balanced Forces

Balanced forces are two or more forces acting on an object where the magnitudes are equal, but the directions are opposite, resulting in a net force of zero.

\[ \text{50N} \leftarrow \text{Rope} \rightarrow \text{50N} \]

Rope does not move

Unbalanced Forces

When forces are not equal. It always causes motion.

\[ \text{70 N} \rightarrow 🚗 \leftarrow \text{60 N} \]

Car moves in direction of greater force

Newton's Laws of Motion

Galileo Galilei's Contribution

Galileo first of all said that objects move with a constant speed when no forces act on them. This means if an object is moving on a frictionless path and no other force is acting upon it, the object would be moving forever. That is, there is no unbalanced force working on the object.

But practically it is not possible for any object. Because to attain the condition of zero, unbalanced force is impossible. Force of friction, force of air and many other forces are always acting upon an object.

Newton's Laws of Motion

First Law (Law of Inertia)

Any object remains in the state of rest or in uniform motion along a straight line, until it is compelled to change the state by applying external force.

If Fnet = 0, then v = constant

Inertia

Each and every body has a tendency to resist its change in state of motion. This property of body is called "Inertia".

  • Inertia of a body is directly proportional to its mass.

Types of Inertia

  1. Inertia of Rest: Tendency of a body to remain in its state of rest (e.g., when a bus suddenly starts, passengers fall backward).
  2. Inertia of Motion: Tendency of a body to remain in its state of uniform motion (e.g., when a bus suddenly stops, passengers fall forward).
  3. Inertia of Direction: Tendency of a body to maintain its direction of motion (e.g., mud flying off a rotating tire).

Linear Momentum

  • Momentum is the power of motion of an object.
  • The product of velocity and mass is called the momentum. Momentum is denoted by 'p'.

    • Momentum of the object = Mass × Velocity

    Or, \( p = m \times v \)

    Where, \( p \) = momentum, \( m \) = mass of the object and \( v \) = velocity of the object.

    Example

    If we drop two bodies, one lighter and one heavy, from the top of a tower simultaneously, both will have same velocity at bottom, but the heavier one will have greater momentum.

    Newton's Second Law of Motion

    The rate of change of momentum of a body is directly proportional to the applied unbalanced force.

    \[ F \propto \frac{\Delta p}{\Delta t} \]

    \[ F = k \times \frac{m(v - u)}{t} \] (where k = 1)

    \[ F = m \times a \]

    Definition of 1 Newton: When an acceleration of 1 m/s² is seen in a body of mass 1 kg, then the force applied on the body is said to be 1 Newton.

    Newton's Third Law of Motion

    To every action there is an equal and opposite reaction.

    \[ F_{AB} = -F_{BA} \]

    Applications:

    1. Walking is enabled by IIIrd law.
    2. A boat moves back when we deboard it.
    3. A gun recoils.
    4. Rowing of a boat.

    Law of Conservation of Momentum

    When two (or more) bodies act upon one another, their total momentum remains constant (or conserved) provided no external forces are acting.

    Initial momentum = Final momentum

    \[ m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \]

    Suppose, two objects A and B each of mass m₁ and mass m₂ are moving initially with velocities u₁ and u₂, strike each other after time t and start moving with velocities v₁ and v₂ respectively.

    Now,
    Initial momentum of object A = m₁u₁
    Initial momentum of object B = m₂u₂
    Final momentum of object A = m₁v₁
    Final momentum of object B = m₂v₂

    So,
    Rate of change of momentum in A, \( F_1 = \frac{m_1 v_1 - m_1 u_1}{t} \)
    \[ = \frac{m_1 (v_1 - u_1)}{t} \quad \text{(i)} \]

    And
    Rate of change of momentum in B, \( F_2 = \frac{m_2 v_2 - m_2 u_2}{t} \)
    \[ = \frac{m_2 (v_2 - u_2)}{t} \quad \text{(ii)} \]

    We know from IIIrd law of motion,
    F₁ = -F₂

    So,
    \[ \frac{m_1 (v_1 - u_1)}{t} = \frac{m_2 (v_2 - u_2)}{t} \quad \text{[From equations (i) & (ii)]} \]

    Or
    \[ m_1 v_1 - m_1 u_1 = -m_2 v_2 + m_2 u_2 \]

    So
    \[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]

    Thus,
    Initial momentum = Final momentum

    Example 1: Recoil Velocity of Pistol

    A bullet of mass 20 g is fired horizontally with a velocity of 150 m/s from a pistol of mass 2 kg. Find the recoil velocity of the pistol.

    Solution:

    Given:

    Using conservation of momentum:

    Initial momentum = Final momentum

    \[ 0 = m_1v_1 + m_2v_2 \]

    \[ 0 = 0.02 \times 150 + 2 \times v_2 \]

    \[ 3 + 2v_2 = 0 \]

    \[ v_2 = -1.5 \, \text{m/s} \]

    The negative sign indicates the gun recoils in the opposite direction to the bullet.

    Example 2: Collision of Hockey Players

    Two hockey players (A: 50 kg moving at 4 m/s and B: 60 kg moving at 3 m/s) get entangled while chasing and fall down. Find their common velocity after collision.

    Solution:

    Given:

    Total initial momentum:

    \[ p_{\text{initial}} = m_Au_A + m_Bu_B \]

    \[ = 50 \times 4 + 60 \times 3 = 200 + 180 = 380 \, \text{kg m/s} \]

    After collision, combined mass moves with velocity v:

    \[ p_{\text{final}} = (m_A + m_B)v = 110v \]

    \[ 380 = 110v \]

    \[ v = \frac{380}{110} = 3.454 \, \text{m/s} \]

    Questions

    Very Short Answer Type Questions

    1. Can force be negative? When?
    2. What is the tendency of a body to resist its change of state called?
    3. Inertia is also measured by ______ of an object.
    4. Higher the mass of an object, higher is its ______.
    5. Acceleration is determined by ______ which is also mass of the object.
    6. Why does the load from the cage above the seats in a bus fall down when suddenly brakes are applied?
    7. When a tree is shaken, its fruits and leaves fall down. Why?
    8. Define Momentum of a body.
    9. On what factors does the momentum of a body depend?
    10. Why it is difficult to walk on a slippery road?

    Short Answer Type Questions

    1. Quantity of motion contained in a body is ______.
    2. Unit of momentum is ______.
    3. Define 1 Newton.
    4. Although we know that a moving body keeps moving indefinitely until an external force is applied on it, then why does a ball stop when we slide it on ground?
    5. Why is it difficult to stop a truck suddenly than a motorbike?
    6. When a metro suddenly stops all the passengers fell forward on its floor. Why does this happen?
    7. We have a huge atmosphere above us that exerts a huge pressure on our shoulders, head and whole body. Why don't we get crushed under it?
    8. A coin of mass 1 kg and a stone of mass 5 kg are thrown down the Eiffel Tower with an acceleration of 10 m/s². Which one would reach the ground early and why?
    9. Give applications of 1st law of motion (inertia).
    10. (a) Friction is measured in ______. (b) Distinguish between balanced and unbalance forces.

    Long Answer Type Questions

    1. (a) Derive first law of Newton from second law. (b) Find the force required to stop a car of mass 100 kg with two passengers each of 50 kg sitting inside, if it is moving at 60 km/hr speed and takes 5 s to stop.
    2. Two balls A and B of masses 40 g and 50 g are moving at speeds of 40 m/s and 30 m/s respectively. If after colliding, B starts moving with a velocity of 25 m/s, what is the velocity of A?
    3. A girl of mass 30 kg jumps on a cart of mass 5 kg with a velocity of 10 m/s. Find the velocity with which she and cart start moving after she jumps on it.
    4. (a) Why does a gunman get a jerk on firing a bullet? (b) Calculate the momentum of a toy car of mass 200 gm moving with a speed of 5 m/s. (c) State the law of conservation of momentum.
    5. For how long should a force of 100 N act on a body of 20 kg so that it acquires a velocity of 100 m/s?
    6. (a) Find the acceleration produced by a force of 5 N acting on a mass of 10 kg. (b) Which would require a greater force: (a) accelerating a 10 gm mass at 5 m/s² or (b) a 20 gm mass at 2 m/s²?
    7. The velocity of a body of mass 10 kg increases from 4 m/s to 8 m/s when a force acts on it for 2s. (a) What is the momentum before the force acts? (b) What is the momentum after the force acts? (c) What is the gain in momentum per second? (d) What is the value of force?

    Objective Type Questions

    Multiple Choice Questions

    Question Options
    1. A truck and a car are moving with equal velocity, on applying brakes, both will stop after certain distance and then: (a) Truck will cover less distance before stopping.
    (b) Car will cover less distance before stopping.
    (c) Both will cover equal distance.
    (d) None of the above.
    2. In which of the following cases, the net force is not zero? (a) An object floating in air
    (b) A ball freely falling from a certain height.
    (c) A cork floating on the surface of water
    (d) All the cases.
    3. A force acts on a body of mass 3kg such that its velocity changes from 4 m/s to 10 m/s. The change in momentum of the body is: (a) 42 kg m/s
    (b) 2 kg m/s
    (c) 18 kg m/s
    (d) 14 kg m/s
    4. While opening a tap with two fingers, the force applied are: (a) equal in magnitude
    (b) Parallel to each other
    (c) opposite in direction
    (d) All of the above
    5. The engine of a car produces an acceleration of 4 m/s² in a car, if this car pulls another car of same mass, what is the acceleration produced? (a) 8 m/s²
    (b) 2 m/s²
    (c) 4 m/s²
    (d) 0.5 m/s²
    6. A force 100N acts in a body mass 2kg for 10 sec. The change in the velocity of the body is. (a) 100 m/s
    (b) 250 m/s
    (c) 500 m/s
    (d) 1000 m/s

    Assertion and Reason Type Questions

    Instructions

    Choose the appropriate answer:

    1. If both assertion and reason are CORRECT and reason is the CORRECT explanation of the assertion.
    2. If both assertion and reason are CORRECT but reason is NOT THE CORRECT explanation of the assertion.
    3. If assertion is CORRECT but reason is INCORRECT
    4. If assertion is INCORRECT but reason is CORRECT
    5. If both assertion and reason are INCORRECT
    1. Assertion: If the net external force on the body is zero, then its acceleration is zero.
      Reason: Acceleration does not depend on force.
    2. Assertion: If two objects of different masses have same momentum, the lighter body possesses greater velocity.
      Reason: For all bodies momentum always remains same.
    3. Assertion: Newton's third law of motion is applicable only when bodies are in motion.
      Reason: Newton's third law applies to all types of forces (gravitational, electric or magnetic force etc.)

    Concept Mapping

    Force And Laws Of Motion

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