Linear Momentum (p) of a particle is the product of its mass (m) and velocity (v).
Formula: p = mv
Newton's Second Law states that the net external force (Fnet) on a system is equal to the rate of change of its total linear momentum (Ptotal).
Newton's Second Law: Fnet = dPtotal / dt
The Principle of Conservation of Linear Momentum follows directly from this:
Principle: If the net external force acting on a system is zero (Fnet = 0), then the total linear momentum (Ptotal) of the system remains constant.If Fnet = 0, then dPtotal / dt = 0 => Ptotal = constant
This is extremely useful for analyzing collisions, explosions, etc., where internal forces are large but external forces are negligible.
Example: A 2kg object (A) at 10 m/s hits a 3kg stationary object (B). They stick together.
Pinitial = pA + pB = (2 kg)(10 m/s) + (3 kg)(0 m/s) = 20 kg·m/s
Pfinal = (mA + mB)vf = (2+3)vf = 5vf
By conservation: Pinitial = Pfinal => 20 = 5vf => vf = 4 m/s.
Energy is the capacity to do work. The total energy of an isolated system is always conserved (Law of Conservation of Energy). In mechanics, we often focus on Total Mechanical Energy (E), which is the sum of Kinetic Energy (K) and Potential Energy (U).
Formula: E = K + U
Kinetic Energy (K): Energy of motion. K = (1/2)mv²
Potential Energy (U): Stored energy due to position or configuration (e.g., gravitational Ug = mgh, elastic Ue = (1/2)kx²).
The Principle of Conservation of Mechanical Energy states:
Principle: If only conservative forces (like gravity, spring force) do work within a system, the total mechanical energy (E = K + U) of the system remains constant.Kinitial + Uinitial = Kfinal + Ufinal
If non-conservative forces (like friction, air resistance) are present, they do work (Wnc) and the mechanical energy is not conserved. The change in energy equals the work done by these forces.
ΔE = ΔK + ΔU = Wnc
The Work-Energy Theorem provides a direct link between the net work (Wnet) done on an object and the change in its kinetic energy (ΔK).
Derivation:
Net Work, Wnet = ∫ Fnet ⋅ dx
Using Fnet = ma = m(dv/dt): Wnet = ∫ m(dv/dt) ⋅ dx
Using the chain rule, (dv/dt)dx = (dx/dt)dv = v dv.
Wnet = ∫v1v2 m v dv = m [ (1/2)v² ]v1v2
Wnet = (1/2)mv₂² - (1/2)mv₁²
Work-Energy Theorem: Wnet = Kfinal - Kinitial = ΔK
Significance: This theorem is universal. The *net* work (done by *all* forces, conservative and non-conservative) equals the change in *kinetic* energy. This is different from the conservation of mechanical energy, which deals with K and U.
The Centre of Mass is a specific point in a system or object that moves as if all the system's mass were concentrated at that point and all external forces were applied there.
For a system of discrete particles (m₁, m₂, ...) at positions (r₁, r₂, ...):
Formula (Position): RCM = (m₁r₁ + m₂r₂ + ...) / (m₁ + m₂ + ...) = (Σmiri) / Mtotal
For a continuous body with density ρ:
Formula (Continuous): RCM = (1/Mtotal) ∫ r dm = (1/Mtotal) ∫ r ρ(r) dV
The motion of the CM is governed by Newton's Second Law for the whole system:
Formula (Motion): Fnet, external = Mtotal aCM
This means internal forces (like in an explosion) do not change the motion of the CM.
The Centre of Gravity is the point where the net gravitational torque on the object is zero. It's the "balance point" of the object.
Key Difference:
When are they the same? In a uniform gravitational field (like near the Earth's surface, for small objects), the force of gravity (g) is the same on all parts of the object. In this case, the CM and CG are in the same location.
For a very large object (like a skyscraper), 'g' is slightly weaker at the top than the bottom, so the CG would be very slightly lower than the CM. For this syllabus, you can assume they are identical.
Angular Velocity (ω): The rate of change of angular position (θ). It is a vector pointing along the axis of rotation (Right-Hand Rule).
Formula: ω = dθ / dt
Angular Momentum (L): The rotational equivalent of linear momentum.
For a single particle, L is defined relative to an origin:
Formula (Particle): L = r × p = r × (mv)
For a rigid body rotating about an axis, L is:
Formula (Rigid Body): L = Iω
Where I is the Moment of Inertia.
Torque is the rotational equivalent of force. It is the "twist" that causes a change in rotational motion (an angular acceleration).
For a force F applied at a position r from the axis:
Formula: τ = r × F
Torque is also the rate of change of angular momentum (Rotational Newton's Second Law):
Formula: τnet = dL / dt
From τnet = dL / dt, we get the conservation principle:
Principle: If the net external torque (τnet) acting on a system is zero, then the total angular momentum (Ltotal) of the system remains constant.If τnet = 0, then dL / dt = 0 => L = constant
This means: Linitial = Lfinal => Iinitialωinitial = Ifinalωfinal
Example: An ice skater spinning with arms out (high I, low ω) pulls their arms in (low I). To conserve L, their angular velocity (ω) must increase dramatically.
Moment of Inertia (I) is the rotational equivalent of mass. It measures an object's resistance to angular acceleration.
For a system of discrete masses (mi) at distances (ri) from the axis:
Formula (Discrete): I = Σmiri²
For a continuous body:
Formula (Continuous): I = ∫ r² dm
Radius of Gyration (K): The distance from the axis where all the object's mass (M) could be concentrated to give the same moment of inertia.
Formula: I = MK² => K = √(I / M)
You need to know the standard formulas for these shapes:
For a thin rectangular bar of mass M and length L, axis through the center and perpendicular to the length:
Formula (Bar, center): I = (1/12)ML²
For a solid cylinder of mass M and radius R, axis along the cylinder's main axis of symmetry:
Formula (Solid Cylinder, main axis): I = (1/2)MR²
This is the same as a flat disc.
For a thin cylindrical shell (or hoop) of mass M and radius R, axis along the cylinder's main axis:
Formula (Hollow Cylinder/Hoop): I = MR²
All the mass is at the same distance R from the axis.