Understanding Key Physics Principles and Theorems

Posted on Feb 15, 2025 in Physics

Steiner’s Theorem

Determining the moment of inertia about an axis of symmetry can be quite complicated by integration. Steiner’s Theorem simplifies this determination. It provides a relationship between the moment of inertia about an axis through a random point of the plane and the moment of inertia about a parallel axis through the center of mass.

I_e = ∫ (x² + y²) dm = ∫ [x² + (d + y)²] dm = ∫ (x² + y²) – 2dy dm + d² ∫ dm = I_z + Md²

Graphical Determination of Instantaneous Center

a) When velocities are not parallel: Speeds must be parallel to a direction that passes through the Center of Instantaneous Rotation (CIR). Perpendicular lines are drawn to their points of application, and the intersection point is the CIR.

b) When velocities are parallel: Two lines are drawn, one passing through the point of application and another at their ends. The intersection point of these lines is the CIR.

Principle of Conservation of Mechanical Energy

If only conservative forces are acting on a particle, the work done is equal to the decrease in potential energy and also equals the increase in the particle’s kinetic energy:

-ΔU = W = ΔKE

Therefore:

ΔKE + ΔU = 0

This yields:

E_m = K + U = constant

The mechanical energy remains constant if only conservative forces do work on the particle.

Theorem of Active Forces

W_total = W_conservative + W_{non-conservative} = ΔKE

W_conservative = -ΔU = U_a – U_b

Replacing the expression of W_conservative:

E_p(A) – E_p(B) + W_{non-conservative} = E_c(B) – E_c(A)

Grouping terms:

E_p(A) – E_c(A) + W_{non-conservative} = E_p(B) + E_c(B)

Since E_m = E_p + E_c, we have:

E_m(A) + W_{non-conservative} = E_m(B)

If W_{non-conservative} is frictional work, it is negative. If it is engine work, it will be positive.

Specific Heat at Constant Pressure/Volume

• C_p: The heat that must be exchanged by 1 kg of a gas at constant pressure to change its temperature by one degree.
• C_v: The same, but at constant volume.

Temperature Property

A common feature of thermodynamic systems in equilibrium. It is measured with a thermometer at thermal equilibrium with the system.

Joule’s Law

The passage of charge from a higher to a lower potential is possible because of potential energy (E_p). This energy is converted into kinetic energy, increasing the speed of the particles. Due to multiple interactions between particles, this energy is converted into heat. This heat increases the temperature of the conductor and dissipates into the air.

dQ = I dt v₁-v₂ = IR

W = I dt IR (replacing W = dq (v₁ – v₂))

W = I²Rt

Usually expressed as power, i.e., energy per unit time:

P = I²R

Hall Effect

The emergence of an electric field on a conductor within a magnetic field. If a conductor or semiconductor material has a current flowing through it and is placed within a magnetic field, a magnetic force will redistribute the charge carriers, creating a voltage and thus an electric field perpendicular to both the magnetic field and the original electric field. This new field is called the Hall electric field.

Quantitative explanation:

F_e = F_m -> eE = evB

E = vB

v₁ – v₂ = Ed -> v₁ – v₂ = V_Bd

The electric field, once balanced with the magnetic field, satisfies the above expression.