System of Particles and Rotational Motion

• Physics forms the basis of various other sciences like engineering, chemistry, and astronomy.

System of Particles and Rotational Motion

• A system of particles is a collection of multiple particles interacting with each other and may also be subject to external forces.
• The center of mass of a system of particles is the point that moves as though all the mass of the system were concentrated there and all the external forces were applied there.
• The position vector of the center of mass of a system consisting of n particles is given by R = (Σ *m_ir_i) / M, where M is the total mass.
• The velocity of the center of mass is V = (Σ *m_iv_i) / M.
• The linear momentum of the system is P = M*V, which remains constant if no external force acts on the system.
• Torque is the rotational equivalent of force.
• Torque (τ) = r × F, where r is the position vector from the axis of rotation to the point where the force F is applied.
• The angular momentum L of a particle is defined as L = r × p = r × (m*v), where p is the linear momentum.
• The total angular momentum of a system of particles is the vector sum of the angular momenta of individual particles about a common point.
• The rate of change of the total angular momentum of a system of particles about a point is equal to the sum of the external torques acting on the system taken about the same point: dL/dt = τ_ext.
• If the total external torque on a system is zero, the total angular momentum of the system is conserved.
• For a rigid body rotating about a fixed axis, the component of angular momentum along the axis is L_z = Iω, where I is the moment of inertia and ω is the angular speed.
• The moment of inertia I of a rigid body about an axis is defined as I = Σ *m_ir_i², where r_i is the perpendicular distance of the ith point mass from the axis.
• The kinetic energy of rotation is K = (1/2) *Iω².
• Theorems of moment of inertia include the parallel axis theorem (I = I_cm + Ma²) and the perpendicular axis theorem (for 2D bodies, I_z = I_x + I_y).
• For rolling without slipping, the velocity of the point of contact with the ground is zero.
• The kinetic energy of a rolling body is the sum of translational and rotational kinetic energies: K = (1/2) *Mv_cm² + (1/2) *I_cmω².

Work, Energy, and Power

• Work is done when a force causes a displacement.
• Work done by a constant force is W = F ⋅ d = Fd cos θ, where θ is the angle between the force and the displacement.
• Work done by a variable force is calculated by integrating the force over the displacement: W = ∫ F ⋅ dr.
• Kinetic energy (K) is the energy possessed by an object due to its motion and is given by K = (1/2) *mv².
• The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy: W = ΔK = K_f - K_i.
• Potential energy (U) is the energy stored in a system due to its position or configuration.
• Gravitational potential energy is U = mgh, where h is the height above a reference point.
• Elastic potential energy for a spring is U = (1/2) kx², where x is the displacement from the equilibrium position and k is the spring constant.
• Conservative forces are those for which the work done is independent of the path taken and can be expressed as the difference in potential energy.
• Examples of conservative forces include gravity and spring force.
• Non-conservative forces are path-dependent, and the work done by them cannot be expressed as a change in potential energy. Friction is an example.
• The mechanical energy (E) of a system is the sum of its kinetic and potential energies: E = K + U.
• In the absence of non-conservative forces, the total mechanical energy of a system remains constant (conservation of mechanical energy).
• Power (P) is the rate at which work is done or energy is transferred: P = W/Δt = F ⋅ v.
• The SI unit of power is the watt (W).
• Average power is the total work done divided by the total time taken.
• Instantaneous power is the power at a specific instant in time.
• 1 horsepower (hp) = 746 watts.
• A collision is an event in which two or more objects exert forces on each other for a relatively short time.
• In an elastic collision, both kinetic energy and momentum are conserved.
• In an inelastic collision, momentum is conserved, but kinetic energy is not.
• A perfectly inelastic collision is one in which the objects stick together after the collision, resulting in the maximum loss of kinetic energy.

Motion in a Plane

• Scalar quantities have magnitude only (e.g., speed, mass).
• Vector quantities have both magnitude and direction (e.g., velocity, force).
• A vector A can be resolved into components along coordinate axes: A = *A_x**i + *A_y**j, where A_x and A_y are the scalar components and i and j are unit vectors.
• The magnitude of a vector A is |A| = √(A_x² + A_y²).
• Vector addition can be done using the triangle law or parallelogram law.
• Vector subtraction: A - B = A + (-B).
• Scalar (dot) product: A ⋅ B = AB cos θ = A_xB_x + A_yB_y, where θ is the angle between A and B.
• Vector (cross) product: A × B = AB sin θ n, where n is a unit vector perpendicular to the plane containing A and B, and θ is the angle between A and B. The direction is given by the right-hand rule.
• Displacement is the change in position: Δr = r₂ - r₁.
• Average velocity: v_avg = Δr/Δt.
• Instantaneous velocity: v = lim_Δt→0 (Δr/Δt) = dr/dt.
• Average acceleration: a_avg = Δv/Δt.
• Instantaneous acceleration: a = lim_Δt→0 (Δv/Δt) = dv/dt.
• For uniform acceleration in 2D: r(t) = r₀ + v₀t + (1/2) at² and v(t) = v₀ + at.
• Projectile motion is a special case of motion in a plane under constant gravitational acceleration.
• The trajectory of a projectile is a parabola.
• Time of flight (T) = (2 v₀ sin θ) / g, where v₀ is the initial velocity and θ is the angle of projection.
• Maximum height (H) = (v₀² sin² θ) / (2g).
• Horizontal range (R) = (v₀² sin 2θ) / g.
• Uniform circular motion involves an object moving at constant speed along a circular path.
• Centripetal acceleration (a_c) = v²/r = rω², directed towards the center of the circle.
• Centripetal force (F_c) = ma_c = mv²/r, required to maintain circular motion.
• Angular velocity (ω) = v/r = 2π/T, where T is the period.
• Relative velocity of A with respect to B: v_AB = v_A - v_B.

Motion in a Straight Line

• Displacement (Δx) is the change in position.
• Average velocity (v_avg) = Δx/Δt.
• Instantaneous velocity (v) = lim_Δt→0 (Δx/Δt) = dx/dt.
• Speed is the magnitude of velocity.
• Average acceleration (a_avg) = Δv/Δt.
• Instantaneous acceleration (a) = lim_Δt→0 (Δv/Δt) = dv/dt.
• For uniform acceleration (constant a):
• v = u + at (u is initial velocity)
• s = ut + (1/2) *at² (s is displacement)
• v² = u² + 2as
• s_n = u + (a/2) (2n - 1) (displacement in the nth second)
• Relative velocity of A with respect to B: v_AB = v_A - v_B.
• When two objects meet, their positions are equal.