Class 11 Physics Notes Chapter 5 (Chapter 5) – Examplar Problems (English) Book
Alright class, let's get straight into Chapter 5: Laws of Motion from your NCERT Exemplar. This chapter is fundamental not just for your Class 11 exams but forms the bedrock for much of classical mechanics, making it crucial for any government exam with a physics component. We'll focus on the core concepts and the kind of thinking needed to tackle Exemplar-level problems.
Chapter 5: Laws of Motion - Detailed Notes for Competitive Exams
1. Introduction & Pre-Newtonian Ideas:
- Aristotle's Fallacy: Believed an external force is required to keep a body in uniform motion. (Incorrect)
- Galileo's Contribution: Studied motion on inclined planes. Concluded that an object moving on a frictionless horizontal plane must neither have acceleration nor retardation, i.e., it should move with constant velocity. Introduced the concept of Inertia.
2. Inertia and Newton's First Law of Motion (Law of Inertia):
- Inertia: The inherent property of all bodies by virtue of which they cannot change their state of rest or of uniform motion along a straight line by themselves. It's a measure of resistance to change in the state of motion. Mass is a quantitative measure of inertia. More mass = More inertia.
- Newton's First Law: "Every body continues in its state of rest or of uniform motion in a straight line unless compelled to change that state by an external force applied upon it."
- This law defines force qualitatively (as an agent causing change in state of motion) and defines inertial frames of reference.
- Inertial Frame: A frame of reference in which Newton's first law is valid. Any frame moving with constant velocity relative to an inertial frame is also inertial. Earth is approximately an inertial frame for many practical purposes. Non-inertial frames are accelerated frames.
- Examples: Dust flying off a carpet when beaten, passenger leaning forward when a bus stops suddenly, passenger pushed outwards during a sharp turn (due to inertia of direction).
3. Newton's Second Law of Motion:
- Momentum (p): The quantity of motion possessed by a body. Defined as the product of mass (m) and velocity (v).
- p = mv
- Vector quantity; direction is the same as velocity.
- SI Unit: kg m s⁻¹
- Newton's Second Law: "The rate of change of linear momentum of a body is directly proportional to the net external force applied on the body, and this change takes place in the direction of the applied force."
- Mathematically: F_ext = dp/dt = d(mv)/dt
- For constant mass: F_ext = m (dv/dt) = ma
- This is the most commonly used form, but F = dp/dt is the more fundamental statement (applies even if mass changes, e.g., rocket propulsion).
- Force (F): An external effort (push or pull) that changes or tends to change the state of rest or uniform motion, or the direction of motion of a body.
- Vector quantity.
- SI Unit: newton (N). 1 N = 1 kg m s⁻².
- Dimensional Formula: [MLT⁻²]
- Key Points:
- F refers to the net external force (vector sum of all external forces).
- Internal forces within a system cannot cause acceleration of the system's center of mass.
- The law is valid only in inertial frames.
- Acceleration (a) is always in the direction of the net force (F).
4. Impulse (J):
- Defined as the product of the average force and the time interval for which it acts. It measures the total effect of a force over time.
- J = F_avg Δt
- Impulse-Momentum Theorem: From Newton's second law, F = dp/dt => F dt = dp. Integrating over a time interval Δt = t₂ - t₁, we get:
- ∫F dt = ∫dp => J = Δp = p_final - p_initial
- Impulse equals the change in linear momentum.
- Vector quantity; direction is the same as the change in momentum / force.
- SI Unit: N s or kg m s⁻¹ (same as momentum).
- Applications: Used for situations involving large forces acting for short durations (e.g., hitting a cricket ball, shock absorbers). To minimize the impact force (F = Δp/Δt), the time interval (Δt) is increased (e.g., cricketer pulling hands back while catching).
5. Newton's Third Law of Motion:
- "To every action, there is always an equal and opposite reaction."
- Mathematically: If body A exerts force F_AB on body B, then body B exerts force F_BA on body A, such that:
- F_BA = - F_AB
- Key Points:
- Action and reaction forces act on different bodies.
- They are equal in magnitude and opposite in direction.
- They occur simultaneously (no cause-effect relation in time).
- They are always of the same nature (e.g., if action is gravitational, reaction is also gravitational).
- Since they act on different bodies, they do not cancel each other out in terms of causing motion of individual bodies.
- Examples: Walking (pushing ground backward, ground pushes forward), recoil of a gun, rocket propulsion (expels gas downward, gas pushes rocket upward), swimming.
6. Conservation of Linear Momentum:
- From Newton's Second Law: F_ext = dp/dt
- If the net external force acting on a system is zero (F_ext = 0), then dp/dt = 0.
- This implies p = constant.
- Principle: "In the absence of a net external force, the total linear momentum of an isolated system remains constant (conserved)."
- p_initial = p_final
- m₁u₁ + m₂u₂ + ... = m₁v₁ + m₂v₂ + ...
- Applications: Collisions, explosions, recoil phenomena.
- Note: Momentum is conserved component-wise as well. If F_ext,x = 0, then p_x is conserved, even if F_ext,y ≠ 0.
7. Equilibrium of a Particle:
- A particle is in equilibrium if the net external force acting on it is zero.
- ΣF_ext = 0
- This implies the particle is either at rest (static equilibrium) or moving with constant velocity (dynamic equilibrium).
- For concurrent forces (forces acting at a single point), the vector sum must be zero.
- ΣF_x = 0, ΣF_y = 0, ΣF_z = 0
- Lami's Theorem: If three concurrent forces acting on a body keep it in equilibrium, then each force is proportional to the sine of the angle between the other two forces.
- F₁/sin α = F₂/sin β = F₃/sin γ (where α, β, γ are angles opposite to F₁, F₂, F₃ respectively).
8. Common Forces in Mechanics:
- Weight (W): Force of gravity acting on a body (W = mg). Acts vertically downwards towards the center of the Earth.
- Normal Reaction (N or R): Contact force perpendicular to the surface of contact. It's an electromagnetic force arising from the compression of surface molecules. It's a self-adjusting force (adjusts itself to prevent penetration).
- Tension (T): Force exerted by a string, rope, or cable when pulled taut. Acts along the string, away from the body it is attached to. Assumed massless and inextensible in ideal problems.
- Friction (f): Contact force parallel to the surface of contact, opposing relative motion or the tendency of relative motion. Electromagnetic in origin.
- Static Friction (f_s): Opposes the tendency of relative motion. Self-adjusting.
- 0 ≤ f_s ≤ f_s(max)
- Limiting Friction (f_s(max)): Maximum value of static friction, occurs just before motion starts. f_s(max) = μ_s N, where μ_s is the coefficient of static friction.
- Kinetic Friction (f_k): Opposes actual relative motion. Approximately constant value.
- f_k = μ_k N, where μ_k is the coefficient of kinetic friction.
- Generally, μ_k < μ_s.
- Rolling Friction (f_r): Friction when a body rolls over a surface. Much smaller than static or kinetic friction (μ_r << μ_k < μ_s).
- Laws of Limiting Friction:
- Depends on the nature of surfaces in contact and their roughness.
- Independent of the apparent area of contact.
- Directly proportional to the normal reaction (f_s(max) ∝ N).
- Angle of Friction (θ): Angle between the normal reaction (N) and the resultant of N and limiting friction (f_s(max)). tan θ = μ_s.
- Angle of Repose (α): Minimum angle of inclination of a plane with the horizontal such that a body placed on it just begins to slide down. tan α = μ_s. For smooth surfaces, α = 0. Numerically, Angle of Friction = Angle of Repose.
- Static Friction (f_s): Opposes the tendency of relative motion. Self-adjusting.
9. Dynamics of Uniform Circular Motion:
- An object moving in a circle with constant speed has changing velocity (due to changing direction).
- Requires a centripetal acceleration (a_c) directed towards the center of the circle.
- a_c = v²/r = ω²r (where v is speed, r is radius, ω is angular velocity).
- Centripetal Force (F_c): The net force directed towards the center, responsible for providing the centripetal acceleration.
- F_c = ma_c = mv²/r = mω²r
- Important: Centripetal force is not a new kind of force. It is provided by existing forces like tension, friction, gravity, normal reaction, or their components.
- Examples:
- Stone tied to string: Tension provides F_c.
- Car on a level circular road: Static friction provides F_c (v_max = √(μ_s rg)).
- Car on a banked road: Horizontal component of Normal reaction (and possibly friction) provides F_c.
- Banking angle for speed v without friction: tan θ = v²/rg
- Maximum safe speed on banked road with friction: v_max = √[rg (μ_s + tan θ) / (1 - μ_s tan θ)]
10. Solving Problems using Newton's Laws:
- Free Body Diagram (FBD): Essential first step.
- Isolate the body (or system) of interest.
- Represent the body as a point mass (if applicable).
- Draw all external forces acting on the body (Weight, Normal reaction, Tension, Friction, Applied forces). Do not show forces exerted by the body.
- Choose a suitable coordinate system (often aligning one axis with the direction of acceleration or potential motion).
- Resolve forces that are not along the axes into components.
- Apply Newton's Second Law (ΣF_x = ma_x, ΣF_y = ma_y) along each axis.
- Solve the resulting equations.
- Common Scenarios:
- Connected Bodies: Draw FBD for each body. Tension is internal to the system but external for individual bodies. Acceleration is usually the same for connected bodies (if string is inextensible).
- Pulley Systems: Assume massless, frictionless pulleys and inextensible strings unless stated otherwise. Tension is the same throughout a single continuous string.
- Lift (Elevator) Problems: Apparent weight = Normal reaction (N) exerted by the lift floor.
- Lift at rest or constant velocity (a=0): N = mg (Apparent weight = True weight)
- Lift accelerating up (a): N - mg = ma => N = m(g+a) (Apparent weight > True weight)
- Lift accelerating down (a): mg - N = ma => N = m(g-a) (Apparent weight < True weight)
- Lift in free fall (a=g): N = m(g-g) = 0 (Weightlessness)
Key Takeaways for Exams:
- Master FBDs.
- Clearly distinguish between static, limiting, and kinetic friction.
- Understand that centripetal force is a requirement for circular motion, provided by other forces.
- Know the conditions for conservation of linear momentum and how to apply it.
- Be precise about action-reaction pairs (act on different bodies).
- Understand the concept of inertial vs non-inertial frames.
Multiple Choice Questions (MCQs)
-
A passenger in a moving train tosses a coin. If the coin falls behind him, it implies that the train's motion is:
(a) Accelerated
(b) Retarded
(c) Uniform
(d) Along a circular track -
A block of mass M is placed on a flat surface. A force F is applied parallel to the surface to move the body. The frictional force f developed is proportional to the:
(a) Mass of the body
(b) Square of the mass of the body
(c) Normal reaction on the body
(d) Applied force F -
Consider the action-reaction pair in the case of a book resting on a table. Which of the following pairs is correct?
(a) Force exerted by book on table; Force exerted by table on book
(b) Weight of the book (force by Earth on book); Force exerted by table on book
(c) Force exerted by book on table; Weight of the book
(d) Weight of the book; Force exerted by book on Earth -
A body of mass 5 kg undergoes a change in speed from 20 m/s to 0.20 m/s. The momentum of the body would:
(a) Increase by 99 kg m/s
(b) Decrease by 99 kg m/s
(c) Increase by 101 kg m/s
(d) Decrease by 101 kg m/s -
A cyclist bends while taking a turn to:
(a) Reduce friction
(b) Generate required centripetal force
(c) Reduce apparent weight
(d) Reduce speed -
A bullet of mass 0.04 kg moving with a speed of 90 m/s enters a heavy wooden block and is stopped after a distance of 60 cm (0.6 m). The average resistive force exerted by the block on the bullet is:
(a) 270 N
(b) 300 N
(c) 200 N
(d) 180 N -
Which of the following statements about friction is true?
(a) Kinetic friction is always greater than limiting friction.
(b) Limiting friction is independent of the normal reaction.
(c) Static friction is a self-adjusting force up to its limiting value.
(d) Rolling friction is greater than sliding friction. -
A person stands in an elevator. In which situation does he find his weight less than actual weight?
(a) Elevator moving upwards with constant acceleration.
(b) Elevator moving downwards with constant acceleration.
(c) Elevator moving upwards with constant velocity.
(d) Elevator moving downwards with constant velocity. -
Conservation of momentum in a collision between particles can be understood from:
(a) Newton's first law only
(b) Newton's second law only
(c) Both Newton's second and third law
(d) Conservation of energy -
Inertia is the property of a body linked to tendency of a body:
(a) to change its position.
(b) to change its direction of motion.
(c) to change its momentum.
(d) to resist any change in its state of rest or uniform motion.
Answers to MCQs:
- (a) Accelerated (Coin has inertia of motion from initial speed; if train accelerates, train moves ahead faster, coin falls behind relative to passenger)
- (c) Normal reaction on the body (f_k = μ_k N and f_s(max) = μ_s N)
- (a) Force exerted by book on table; Force exerted by table on book (These act on different bodies, are equal and opposite). Pair (d) is also an action-reaction pair (gravitational force). However, (a) is the contact force pair related to the table interaction. In context of "book resting on table", (a) is the most direct pair.
- (b) Decrease by 99 kg m/s (Initial p = 5 * 20 = 100 kg m/s; Final p = 5 * 0.20 = 1 kg m/s. Change = Final - Initial = 1 - 100 = -99 kg m/s. Decrease is 99.)
- (b) Generate required centripetal force (The horizontal component of the normal reaction provides the necessary centripetal force).
- (a) 270 N (Use v² = u² + 2as => 0² = 90² + 2 * a * 0.6 => a = -8100 / 1.2 = -6750 m/s². Resistive Force F = ma = 0.04 * 6750 = 270 N)
- (c) Static friction is a self-adjusting force up to its limiting value.
- (b) Elevator moving downwards with constant acceleration (Apparent weight N = m(g-a)).
- (c) Both Newton's second and third law (F_ext = dp/dt; internal forces cancel in pairs due to 3rd law).
- (d) to resist any change in its state of rest or uniform motion.
Study these notes carefully, focusing on the underlying concepts and how to apply them using FBDs. Good luck with your preparation!