Unit 2: Momentum, Energy and Rotational Motion

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.

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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

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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

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