Thermodynamics and Simple Harmonic Motion
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Thermodynamics
First Law of Thermodynamics (Law of Energy Conservation)
Energy cannot be created or destroyed; it can only change forms.
Formula: ΔU=Q−W
Where: ΔU is the change in internal energy of the system, Q is the heat added to the system, W is the work done by the system.
Explanation: The total energy in a closed system remains constant. Energy can be transformed from one form to another (e.g., heat energy to mechanical energy).
Second Law of Thermodynamics
Key Concept: The total entropy (disorder) of an isolated system always increases or remains constant.
Formulas:
Entropy Change for Reversible Process: ΔS=Qrev /T
Where: ΔS is the change in entropy. Qrev is the heat added in a reversible process. T is the absolute temperature. Qirr is the heat added in an irreversible process.
Entropy Change for Irreversible Process: ΔS>Qirr/ T Where: Qirr is the heat added in an irreversible process.
Clausius Statement:
ΔSuniverse≥0
Where: ΔSuniverse is the change in the entropy of the universe.
Carnot Efficiency: η=1−Tc/TH
Where: η is the efficiency of a Carnot engine. TC is the absolute temperature of the cold reservoir. TH is the absolute temperature of the hot reservoir.
Explanation: Heat flows naturally from hot to cold objects. Entropy, a measure of disorder, tends to increase in an isolated system. Many natural processes are irreversible and increase the overall entropy.
Simple Harmonic Motion (SHM)
Equilibrium Position: This is the position where the net force on the object is zero.
Amplitude (A): The maximum distance from the equilibrium position.
Period (T): The time it takes to complete one full cycle of motion.
Frequency (f): The number of cycles per unit of time. It is related to the period by f=1/T.
Phase (ϕ): Defines the initial state of the motion.
The position x(t) of an object in SHM as a function of time can be described by; x(t)=Asin(ωt+φ)
where: A is the amplitude. ω is the angular frequency, related to frequency and period by ω=2πf=2π/T.φ is the initial phase.
The velocity v(t) is the derivative of position with respect to time: v(t)=dx/dt=−Aωcos(ωt+φ)
The acceleration a(t) is the derivative of velocity with respect to time: a(t)=dv/dt=−Aω2sen(ωt+φ)a(t)
In SHM, the total mechanical energy remains constant and is exchanged between kinetic energy and potential energy: Potential Energy (U): U=1/2kx2, Kinetic Energy (K): K=1/2mv2, Total Mechanical Energy (E): E=K+U=1/2kA2. This is constant and depends only on the amplitude and the spring constant.
Simple Harmonic Motion is a type of oscillatory motion where a restoring force proportional to the displacement acts on the object. This motion is described by a sinusoidal equation involving parameters like amplitude, angular frequency, and phase. SHM is a fundamental model for understanding many types of periodic motion in physics.