This chapter explores the concepts of systems of particles and the principles of rotational motion, which are crucial for understanding the mechanics of real-life extended bodies.
SYSTEM OF PARTICLES AND ROTATIONAL MOTION – Formula & Equation Sheet
Essential formulas and equations from Physics Part - I, tailored for Class 11 in Physics.
This one-pager compiles key formulas and equations from the SYSTEM OF PARTICLES AND ROTATIONAL MOTION chapter of Physics Part - I. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
X = (m₁x₁ + m₂x₂) / (m₁ + m₂)
X is the center of mass of two particles, with m₁ and m₂ as their respective masses, and x₁ and x₂ as their positions from an origin. This formula shows how to calculate the weighted average position of two masses.
V = P / M
V is the velocity of the center of mass, P is the total linear momentum, and M is the total mass. This formula emphasizes conservation of momentum in systems.
τ = r × F
τ is the torque, r is the position vector from the pivot point to the point of force application, and F is the force vector. Torque quantifies the rotational effect of a force applied on a body.
l = r × p
l is the angular momentum, r is the position vector, and p is the linear momentum of a particle. This defines how angular momentum is related to position and motion.
I = Σ(mᵢrᵢ²)
I is the moment of inertia about an axis, mᵢ is the mass of particle i, and rᵢ is the distance from the axis. This indicates how mass distribution affects rotation.
K = 1/2 Iω²
K is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity. This parallels the formula for kinetic energy in linear motion.
L = Iω
L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. This captures the relationship in rotational motion analogous to p = mv.
α = τ / I
α is the angular acceleration, τ is the net torque, and I is the moment of inertia. This shows the relationship between rotational force and resulting acceleration.
θ = θ₀ + ω₀t + 1/2αt²
This is the equation of angular displacement, where θ₀ is the initial angular displacement, ω₀ is the initial angular velocity, and α is the angular acceleration.
ω = ω₀ + αt
ω is the final angular velocity with initial velocity ω₀ and angular acceleration α. This relates initial conditions to motion over time.
Equations
F = ma
F is the total force acting on a system, m is the total mass, and a is the acceleration. This is the foundation of Newton's second law for linear motion.
τ_total = Στ_external
The total torque on a rigid body is equal to the sum of external torques acting on it, reflecting how external forces affect motion.
dL/dt = τ
This states that the rate of change of angular momentum (L) of a system is equal to the net torque (τ) applied. It is a key principle in dynamics.
L_total = L_initial
In the absence of external torques, the total angular momentum of a system remains constant, highlighting the conservation of angular momentum.
X = Σ(m_ix_i) / Σm_i
The position of the center of mass is obtained by summing the products of masses and their positions and dividing by the total mass.
Y = Σ(m_iy_i) / Σm_i
Similar to the X formula, this gives the Y-coordinate of the center of mass for a system of particles.
Z = Σ(m_iz_i) / Σm_i
This provides the Z-coordinate of the center of mass for a three-dimensional system.
v = rω
In a circular path, the linear velocity (v) of a particle is related to its angular velocity (ω) and the radius (r) of its motion.
p = mv
Momentum (p) of a particle is the product of its mass (m) and its linear velocity (v), establishing the relationship in linear dynamics.
α = dω/dt
Angular acceleration (α) is defined as the rate of change of angular velocity over time, akin to linear acceleration.
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