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Physics

Newton's Laws of Motion

First Law of Motion

11
⚡ Quick Summary
If all the forces on an object cancel out (add up to zero), then the object either stays still or keeps moving at the same speed in the same direction. Basically, things don't change their motion unless a force makes them.
F = 0 implies a = 0, and a = 0 implies F = 0
  • If the (vector) sum of all the forces acting on a particle is zero then and only then the particle remains unaccelerated (i.e., remains at rest or moves with constant velocity).
  • a = 0 if and only if F = 0 (where 'a' is acceleration and 'F' is the net force).
  • Rest, motion, and acceleration are meaningful only with respect to a frame of reference.
  • Acceleration of a particle is different when measured from different frames of reference.

Newton's First Law and Frames of Reference

Class 11
⚡ Quick Summary
Newton's First Law (an object stays at rest or in motion unless acted upon by a force) only works in certain viewpoints called 'inertial frames'. Imagine you're in a car. If the car suddenly brakes, things inside seem to move even without being pushed. That's because the car is *not* an inertial frame during braking. An inertial frame is one that's not accelerating.
N/A
  • Newton's First Law is valid only in inertial frames of reference.
  • Inertial Frame: A frame of reference in which Newton's first law is valid. a = 0 if and only if F = 0.
  • Non-Inertial Frame: A frame of reference in which Newton's first law is NOT valid.
  • The earth is a good approximation of an inertial frame for routine affairs.
  • All frames moving uniformly (constant velocity) with respect to an inertial frame are themselves inertial.

Relationship between Inertial Frames

Class 11
⚡ Quick Summary
If you have one inertial frame (think of it as a steady, non-accelerating observer), then *any* other observer moving at a *constant* speed relative to that first observer is *also* in an inertial frame. They'll both see the same laws of physics working correctly.
a<sub>P,S</sub> = a<sub>P,S'</sub> + a<sub>S',S</sub>
  • If S is an inertial frame and S' is a frame moving uniformly with respect to S, then S' is also an inertial frame.
  • Mathematical relationship: aP,S = aP,S' + aS',S where:
    • aP,S is the acceleration of particle P with respect to frame S
    • aP,S' is the acceleration of particle P with respect to frame S'
    • aS',S is the acceleration of frame S' with respect to frame S
    Because S' moves uniformly with respect to S, aS',S = 0, therefore aP,S = aP,S'

Newton's Second Law of Motion

Class 11
⚡ Quick Summary
Newton's Second Law basically says F=ma. But to be accurate, the acceleration *must* be measured from an inertial frame. The total force on an object is equal to its mass times the acceleration measured from an inertial point of view.
F = ma (where 'a' is measured from an inertial frame)
  • The acceleration of a particle as measured from an inertial frame is given by the vector sum of all the forces acting on the particle divided by its mass.

Forces on a System

Class 11
⚡ Quick Summary
When analyzing a system, only consider forces acting on the system from objects outside of it. Don't worry about internal forces within the system itself.
N/A
  • When analyzing the forces acting on a system, focus only on external forces. These are forces exerted on the system *by objects other than the system itself*.
  • Internal forces, which are forces exerted between different parts of the system, do not need to be explicitly considered when applying Newton's laws to the entire system as a whole.

Relationship between Displacement of Block and Pulley

Class 11
⚡ Quick Summary
If a pulley connected to a block through a string is displaced by a distance 'x', the block will be displaced by twice that distance (2x). This is because the string length remains constant. The acceleration of the block is also twice the acceleration of the pulley.
Displacement of Block = 2 * Displacement of Pulley; Acceleration of Block = 2 * Acceleration of Pulley
  • When a pulley connected to a block via a string is displaced, the block's displacement is twice the pulley's displacement.
  • This relationship arises from the constant length of the string.
  • If the pulley is moved a distance x, the block moves 2x.
  • Acceleration of Block = 2 * Acceleration of Pulley

Tension in String and Acceleration (Massless Pulley)

Class 11
⚡ Quick Summary
When a massless pulley is pulled by a force F, the tension (T) in the string connected to it is F/2. This means the force is distributed equally along the two segments of the string connected to the pulley.
T = F/2
  • For a massless pulley, the net force acting on it is zero.
  • If a force F is applied to the pulley, and the string exerts tension T on both sides, then F - 2T = 0.
  • Therefore, the tension in the string is T = F/2.

Tension in a String with Mass

Class 11
⚡ Quick Summary
The tension in a string pulling a block that is accelerating depends on where you measure it. Tension is greater higher up the string because it must also support the string's weight.
T = (M + m)(g - a) T' = M(g - a)
  • When a block of mass M is pulled by a string of mass m and accelerates downwards with acceleration 'a', the tension T at a point A along the string can be calculated considering the block and the portion of string below point A as a system.
  • The tension T at point A is given by: T = (M + m)(g - a).
  • The tension T' at the lower end of the string (where m=0) is: T' = M(g - a).

Tension in a Massless String

Class 11
⚡ Quick Summary
If the string is super light (massless), the tension is the same everywhere in the string, as long as nothing heavy is hanging in the middle.
N/A
  • If the string is considered massless (its mass is negligible), the tension is the same throughout the string.
  • This holds true provided there are no massive objects or particles connected between the points where the tension is being considered.

Relationship between speed of ring and block

Class 11
⚡ Quick Summary
If a ring A slides with a speed v, the block descends with speed v cos θ.
v_block = v_ring * cos(θ)
  • If a ring A slides with a speed v, the block descends with speed v cos θ.

Tension in a String System with Multiple Blocks and Pulleys

Class 11
⚡ Quick Summary
In a system with a light string connecting blocks via pulleys, the tension is the same throughout the string segment between any two blocks or a block and a pulley. Tension can be different in different string segments.
N/A
  • If a light string connects multiple blocks via pulleys, the tension within a single continuous segment of the string is the same at all points.
  • However, the tension may be different in different segments of the string (e.g., between block A and pulley B versus between pulley B and block C).
  • If the pulley is also light (massless), the tension in the string segments on either side of the pulley are equal.

The Horse and the Cart

Class 11
⚡ Quick Summary
When a horse pulls a cart, it seems like the forces cancel out, but they don't because they act on different things! To understand the motion, you need to consider only the forces *on* the cart to find its acceleration, and only the forces *on* the horse to find its acceleration. The road's push on the horse is key for the horse's forward motion.
a<sub>cart</sub> = (F<sub>1</sub> - f')/M<sub>C</sub>; a<sub>horse</sub> = (f - F<sub>2</sub>)/M<sub>h</sub>
  • The cart is pulled forward by the horse with a force F1.
  • The horse is pulled backward by the cart with an equal force F2 (F1 = F2 = F). These forces don't cancel each other out in terms of causing acceleration because they act on *different* objects.
  • To analyze the cart's motion, consider *only* the forces acting *on* the cart. This includes F1 (the horse's pull) and a force from the road, f', which opposes the motion. The acceleration of the cart is (F1 - f')/MC, where MC is the mass of the cart.
  • To analyze the horse's motion, consider *only* the forces acting *on* the horse. This includes F2 (the cart's pull backward) and the force P from the road pushing the horse forward. The forward component of the road's force is 'f'. The acceleration of the horse is (f - F2)/Mh, where Mh is the mass of the horse.
  • The forces f and f' (road friction forces) are self-adjusting, ensuring that the horse and cart move together with the same acceleration.

Pseudo Forces

Class 11
⚡ Quick Summary
Describes the need to use 'Pseudo Forces' when analysing motion relative to an accelerating (non-inertial) frame of reference.
N/A
  • Introduces the concept of analyzing motion with respect to a non-inertial frame of reference.

Inertia

Class 11
⚡ Quick Summary
Inertia is like laziness! It's how much an object resists changes to its motion. A heavier object has more inertia, meaning it's harder to start moving or stop once it's moving.
F = ma (Implies that for a given F, larger m results in smaller a, hence larger inertia.)
  • Inertia is the unwillingness of a particle to change its state of rest or of uniform motion along a straight line.
  • If equal forces are applied to two particles, the particle with the smaller acceleration has greater inertia.
  • Larger mass implies larger inertia. A more massive object resists acceleration more than a less massive object when the same force is applied.

Tension in Strings and Acceleration of Connected Bodies

Class 11
⚡ Quick Summary
When masses are connected by strings and pulleys, the tension in the string and the acceleration of each mass are related. By applying Newton's Second Law to each mass and considering the constraints imposed by the strings (like equal tension or related accelerations), we can solve for these unknowns.
T = m₁a₀ (Tension equals mass times acceleration for a horizontally moving mass). m₂g - T/2 = m₂(a₀ - a) (Applying Newton's Second Law to a vertically moving mass, accounting for tension and acceleration). m₃g - T/2 = m₃(a₀ + a) (Applying Newton's Second Law to a vertically moving mass, accounting for tension and acceleration).
  • Tension in a string is the force exerted by the string. It acts in the direction of the string.
  • For a massless pulley, the net force on the pulley is zero.
  • Newton's Second Law (F = ma) is applied to each mass individually.
  • Constraints relating the accelerations of different masses are crucial for solving the problem. For example, if two masses are connected by a single string passing over a pulley, their accelerations are related.

Motion on an Inclined Plane in an Accelerating Frame

Class 11
⚡ Quick Summary
When an object slides down an inclined plane inside an accelerating elevator (or any accelerating frame), you need to consider a 'pseudo force' in addition to the usual forces like gravity and the normal force. This pseudo force accounts for the elevator's acceleration and affects the object's motion relative to the incline.
a = (g + a₀)sinθ (Acceleration of the particle down the incline with respect to the elevator frame, where θ is the angle of inclination). t = √(2L / ((g + a₀)sinθ cosθ)) (Time taken to reach the bottom of the incline, where L is the base length of the incline).
  • In an accelerating frame (like an elevator accelerating upwards with acceleration a₀), a pseudo force (ma₀) acts on the object in the opposite direction of the frame's acceleration.
  • Draw a free body diagram including all real forces (gravity, normal force) and the pseudo force.
  • Apply Newton's Second Law in the accelerating frame.
  • The acceleration calculated is with respect to the accelerating frame itself.

Inertial Frame of Reference

11
⚡ Quick Summary
Imagine you're in a car moving at a constant speed. If you drop a ball, it falls straight down. That's because you're in an inertial frame. It's a frame of reference where objects at rest stay at rest, and objects in motion stay in motion with a constant velocity, unless acted upon by a force. Newton's laws work perfectly in these frames.
Newton's First Law: ∑F = 0 implies a = 0 (in an inertial frame)
  • An inertial frame of reference is a frame in which Newton's laws of motion hold true.
  • In an inertial frame, an object at rest remains at rest, and an object in motion continues to move with constant velocity unless acted upon by a net external force.
  • A non-inertial frame of reference is one that is accelerating or rotating. In these frames, fictitious forces (like the Coriolis force) appear.

Forces in Accelerated Frames (Non-Inertial Frames)

11
⚡ Quick Summary
Think about being in a car that suddenly brakes. You feel thrown forward, even though no real force is pushing you. That feeling comes from being in a non-inertial frame (the accelerating car). We sometimes describe this with 'pseudo' forces.
F_pseudo = -m * a_frame
  • In non-inertial frames, we observe apparent forces called pseudo forces or fictitious forces. These forces are not due to actual interactions but arise from the acceleration of the frame itself.
  • The pseudo force is given by F_pseudo = -m*a_frame, where m is the mass of the object and a_frame is the acceleration of the non-inertial frame.

Tension in a String

Class 11
⚡ Quick Summary
Tension is the pulling force exerted by a string, cable, or similar object on another object. It acts along the direction of the string.
T = Force exerted by the string
  • Tension is a pulling force.
  • It acts along the length of the string.
  • In a light string (massless string), the tension is the same throughout the string.
  • When a string passes over a frictionless pulley, the tension remains the same on both sides of the pulley.

Frictionless Surfaces

Class 11
⚡ Quick Summary
Frictionless surfaces are ideal surfaces where there is no friction between objects. In reality, perfectly frictionless surfaces do not exist, but this assumption simplifies calculations.
Frictional Force (Ff) = 0
  • No frictional force acts between objects in contact with a frictionless surface.
  • This implies that the net force acting on an object is only due to applied forces or tension in strings.

Light String and Pulley

Class 11
⚡ Quick Summary
A light string or pulley means they have negligible mass, and any tension in the string is uniform throughout, and the pulley doesn't require extra force to rotate.
Mass of string (m) ≈ 0; Moment of Inertia of Pulley (I) ≈ 0
  • A light string is assumed to have zero mass.
  • A light pulley is assumed to have zero moment of inertia.
  • For a light string, the tension is the same at all points along the string.
  • For a light pulley, the tension on either side of the pulley is the same if the pulley is also frictionless.

Acceleration of Blocks Connected by Strings

Class 11
⚡ Quick Summary
When blocks are connected by strings and pulleys, they accelerate together. To find the acceleration, consider the entire system or individual blocks, applying Newton's second law (F=ma).
F = ma (Newton's Second Law)
  • Write down the forces acting on each block (tension, weight, normal reaction).
  • Apply Newton's second law (F = ma) to each block.
  • Relate the accelerations of the blocks if they are connected by strings (e.g., if one block moves a distance x, another block connected to it may move a different distance depending on the arrangement of pulleys).
  • Solve the resulting equations to find the acceleration(s).

Acceleration in Connected Systems

Class 11
⚡ Quick Summary
When objects are connected (like with a rope and pulley), they accelerate together. To find the acceleration, consider the net force on the *whole system* and the total mass of the system.
N/A - The solutions given don't provide formulas. The formulas need to be derived for each specific configuration.
Finding acceleration requires analysing forces acting on each mass and then considering the entire system to find the combined acceleration. This is applicable for multiple masses linked together via strings or pulleys.

Tension in Strings

Class 11
⚡ Quick Summary
The tension in a string is the force it exerts. It's important to consider the direction of the tension. It acts *away* from the object it's pulling on.
N/A - The solutions given don't provide formulas. The formulas need to be derived for each specific configuration.
Tension is a pulling force transmitted through a string, rope, cable or wire when it is pulled tight by forces acting from opposite ends. The tension force is directed along the length of the wire and pulls equally on the objects on the opposite ends of the wire.

Normal Reaction

Class 11
⚡ Quick Summary
The normal force is the force a surface exerts back on an object pressing on it. It's always perpendicular to the surface.
N/A - The solutions given don't provide formulas. The formulas need to be derived for each specific configuration.
When an object rests on a surface, the surface exerts a force on the object. This force is perpendicular to the surface and is called the normal force or normal reaction.

Pseudo Force

Class 11
⚡ Quick Summary
If you're in an accelerating frame (like a car speeding up or braking), you feel a 'fake' force called a pseudo force. It acts in the opposite direction of the acceleration.
F_pseudo = -m*a, where a is the acceleration of the non-inertial frame.
Pseudo forces are apparent forces that act on all objects in a non-inertial frame of reference (an accelerating frame). They arise because the frame of reference is accelerating relative to an inertial frame. The pseudo force is equal to the mass of the object multiplied by the acceleration of the non-inertial frame, and points in the opposite direction of the acceleration.

Angle of Banking

Class 11
⚡ Quick Summary
Banking of roads is done to provide necessary centripetal force for a vehicle to take a turn safely. The angle of banking depends on the speed of the vehicle and the radius of curvature of the road.
tan θ = v^2 / (rg)
The angle of banking (θ) is the angle at which a road is raised on the outer edge to help vehicles navigate a curve. It reduces reliance on friction and makes turning safer. The ideal angle depends on the vehicle's speed (v) and the radius of the curve (r), as well as the acceleration due to gravity (g).

Applying Newton's Laws with Friction

Class 11
⚡ Quick Summary
To solve problems with friction, draw a free-body diagram showing all forces. Then, apply Newton's second law (F = ma) in each direction (x and y) separately.
∑F = ma
  • Draw free-body diagrams to identify all forces acting on the object(s).
  • Apply Newton's second law (∑F = ma) in component form (∑Fx = max and ∑Fy = may).
  • Relate the friction force to the normal force using the appropriate coefficient of friction (static or kinetic).