Unit 2: Momentum, Energy and Rotational Motion

Momentum and Energy
Rotational Motion
Moment of Inertia

Momentum and Energy

Conservation of momentum

Linear Momentum (p) of a particle is the product of its mass and velocity: p = mv.

Newton's Second Law states that the net external force on a system is equal to the rate of change of its total linear momentum:

F_net = dP_total / dt

The Principle of Conservation of Momentum follows directly from this:

If the net external force on a system is zero (F_net = 0), then dP_total / dt = 0, which means P_total = constant.

In words: In the absence of a net external force, the total linear momentum of a system remains constant.

This principle is fundamental in all of physics.
Application: Collisions (elastic and inelastic) and explosions. In a collision, the internal forces are huge, but the net *external* force on the two-particle system is usually negligible.
Initial Momentum = Final Momentum
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Conservation of energy

Energy is the capacity to do work. The total mechanical energy (E) of a system is the sum of its Kinetic Energy (K) and Potential Energy (U).

E = K + U

Kinetic Energy (K): Energy of motion. K = (1/2)mv²
Potential Energy (U): Stored energy due to position or configuration. It is only defined for conservative forces.
- Gravitational P.E.: U = mgh
- Elastic (Spring) P.E.: U = (1/2)kx²

A force is conservative if the work it does is path-independent (e.g., gravity). A force is non-conservative if the work it does depends on the path (e.g., friction).

The Principle of Conservation of Mechanical Energy states:

If only conservative forces are doing work within a system, the total mechanical energy (K + U) of the system remains constant.

K_initial + U_initial = K_final + U_final

If non-conservative forces (like friction) are present, they do work (W_nc) that dissipates energy: W_nc = ΔE = E_final - E_initial.

Work Energy theorem

This theorem provides a direct link between the net work done on an object and its kinetic energy.

The net work (W_net) done by the *net force* (F_net) on a particle as it moves from point 1 to point 2 is:

W_net = ∫ F_net · dr

By derivation (using F=ma, a=dv/dt, dr=v dt):

W_net = (1/2)mv₂² - (1/2)mv₁²

W_net = K_final - K_initial = ΔK

In words: The net work done on an object equals the change in its kinetic energy. This is true for *all* forces, conservative and non-conservative.

Rotational Motion

Angular velocity (ω) and Angular momentum (L)

Angular Velocity (ω): The rate of change of angular position (θ). ω = dθ/dt. It is a vector pointing along the axis of rotation (by the right-hand rule).

Angular Momentum (L): The rotational analog of linear momentum. For a single particle, it is defined as:

L = r × p

Where r is the position vector from the origin and p is the linear momentum.

For a rigid body (a system of particles) rotating with angular velocity ω about a fixed axis, the total angular momentum simplifies to:

L = Iω

Where I is the Moment of Inertia of the body about that axis. This is the direct rotational analog of p = mv.

Torque (τ)

Torque (τ) is the rotational analog of force. It is the "turning effect" of a force.

τ = r × F

Where r is the position vector from the axis of rotation to the point where the force F is applied.

Torque as the rate of change of Angular Momentum

We can derive the rotational version of Newton's Second Law by differentiating L = r × p with respect to time:

dL/dt = d/dt (r × p)

Using the product rule: dL/dt = (dr/dt × p) + (r × dp/dt)

The first term is (v × p) = (v × mv) = m(v × v). The cross product of a vector with itself is zero.
The second term uses F_net = dp/dt. So, (r × dp/dt) = (r × F_net).

By definition, (r × F_net) is the net torque τ_net. Therefore:

τ_net = dL/dt

This is the fundamental law of rotational dynamics. If the Moment of Inertia `I` is constant, this simplifies to:

τ_net = d(Iω)/dt = I (dω/dt) = Iα (where α is angular acceleration). This is the analog of F=ma.

Conservation of angular momentum

From the law τ_net = dL/dt, we get the conservation principle:

If the net external torque on a system is zero (τ_net = 0), then dL/dt = 0, which means L = constant.

In words: In the absence of a net external torque, the total angular momentum of a system remains constant.

L_initial = L_final or I_initial ω_initial = I_final ω_final

Key Applications:

Ice Skater: When a skater pulls her arms in, her mass is closer to the axis of rotation. This decreases her moment of inertia (I). To keep L = Iω constant, her angular velocity (ω) must increase, and she spins faster.
Kepler's Second Law: A planet orbiting the Sun feels a central gravitational force. This force creates no torque (τ = r × F = 0). Therefore, the planet's angular momentum (L) is conserved. This conservation leads to the "equal areas in equal times" law.

Moment of Inertia

The Moment of Inertia (I) is the rotational equivalent of mass. It measures an object's resistance to angular acceleration.

A large `I` means it is hard to start or stop the object's rotation.
It depends on the mass of the object and, more importantly, the distribution of that mass relative to the axis of rotation.

Definition:

For a system of discrete particles: I = Σ mᵢrᵢ²
(where rᵢ is the perpendicular distance of mass mᵢ from the axis)
For a continuous body: I = ∫ r² dm

Radius of Gyration (K)

The radius of gyration is a conceptual distance. It is the distance from the axis of rotation where all the object's mass (M) could be concentrated into a single point, such that this point-mass has the same moment of inertia as the actual object.

I = MK² or K = √(I / M)

Two objects with the same mass M can have very different K values (e.g., a solid sphere vs. a hollow shell).

Calculation of Moment of Inertia

Calculating MOI often involves integration. For this syllabus, knowing the standard results is key. We also use two important theorems.

Theorem 1: Perpendicular Axis Theorem

Applies to: 2D (planar) objects only.
Statement: The MOI about an axis perpendicular to the plane (z-axis) is the sum of the MOIs about any two perpendicular axes lying *in* the plane.
I_z = I_x + I_y

Theorem 2: Parallel Axis Theorem

Applies to: Any 3D object.
Statement: The MOI (`I`) about *any* axis is the MOI about a *parallel* axis through the center of mass (`I_cm`), plus the total mass (M) times the squared distance (d²) between the axes.
I = I_cm + Md²

Standard Formulas for MOI

Object	Axis of Rotation	Moment of Inertia (I)
Rectangular Bar (or Rod) (Mass M, Length L)	Through center (CM), perpendicular to length	I = (1/12) ML²
Rectangular Bar (or Rod) (Mass M, Length L)	Through one end, perpendicular to length	I = I_cm + M(L/2)² = (1/3) ML²
Solid Cylinder (Mass M, Radius R)	Along the central axis of the cylinder	I = (1/2) MR²
Solid Cylinder (Mass M, Radius R, Length L)	Through CM, perpendicular to central axis	I = (1/4)MR² + (1/12)ML²
Cylindrical Shell (or Hoop) (Mass M, Radius R)	Along the central axis	I = MR²
Solid Sphere (Mass M, Radius R)	Through center (any diameter)	I = (2/5) MR²
Spherical Shell (Hollow) (Mass M, Radius R)	Through center (any diameter)	I = (2/3) MR²

(The syllabus specifically lists "Rectangular bar, cylinder and shell". The bolded items are the most common interpretations required.)