7.1 Simple Harmonic Motion (SHM)

• The angular frequency ω of a mass-spring system with a spring constant of 50 N/m and a mass of 2 kg is 5 rad/s.
• The amplitude of a mass-spring system can be calculated using the formula A = √(x₀² + (v₀/ω)²)


  True
• What is the amplitude of a mass-spring system with an initial displacement of 0.1 m, initial velocity of 0.5 m/s, and angular frequency of 5 rad/s?

  0.1 m
• What is the displacement of a mass-spring system at t = 2 s if A = 0.1 m, ω = 5 rad/s, and φ = -0.1 rad?

  0.0995 m
• The velocity of a mass-spring system at t = 2 s if A = 0.1 m, ω = 5 rad/s, and φ = -0.1 rad is -0.4975 m/s.
• What is the angular frequency ω of a mass-spring system with a spring constant of 100 N/m and a mass of 2 kg?

  10 rad/s
• What is the total mechanical energy of a mass-spring system with a spring constant of 100 N/m and an amplitude of 0.2 m?

  2 J
• Match the energy type with its formula for a mass-spring system:

  Kinetic Energy ↔️ $(1 / 2)mv^{2}$

  Potential Energy ↔️ $(1 / 2)kx^{2}$

  Total Energy ↔️ KE + PE
• In SHM, acceleration is proportional to displacement and in the same direction.

  False
• Hooke's Law states that the restoring force is proportional to the displacement and the spring constant.


  True
• Match the condition with its description:

  Restoring Force Proportional to Displacement ↔️ The force acting on the object must be proportional to its displacement from equilibrium

  Hooke's Law Obedience ↔️ The restoring force follows Hooke's Law
• The spring constant in Hooke's Law represents the stiffness of the spring.


  True
• What is the role of a restoring force in motion?

  Pulls object to equilibrium
• The equation for the restoring force in SHM is F = -kx
• In SHM, the amplitude represents the maximum displacement from equilibrium.


  True
• The acceleration in SHM is proportional to the displacement and is opposite in direction.
• What are the key characteristics of SHM?

  Periodic motion, proportional acceleration
• SHM is a special case of periodic motion where the restoring force obeys Hooke's Law.


  True
• What is the mathematical expression for Hooke's Law?

  F = -kx
• Match the quantity in SHM with its description:

  Displacement ↔️ Distance from equilibrium

  Velocity ↔️ Rate of change of displacement

  Acceleration ↔️ Rate of change of velocity
• Acceleration in SHM is proportional to displacement.


  True
• In SHM equations, ω represents the angular frequency.
• What type of functions describe displacement, velocity, and acceleration in SHM?

  Sinusoidal
• Match the quantity with its equation in SHM:

  Displacement ↔️ x = A cos(ωt)

  Velocity ↔️ v = -Aω sin(ωt)

  Acceleration ↔️ a = -Aω² cos(ωt)
• The total mechanical energy in SHM remains constant.


  True
• Potential energy in SHM is maximum at the maximum displacement from equilibrium.
• Steps to derive SHM equations using Newton's Second Law

  1️⃣ Apply Newton's Second Law: \( F = ma \)

  2️⃣ Substitute restoring force: \( F = -kx \)

  3️⃣ Define angular frequency: \( \omega^2 = \frac{k}{m} \)

  4️⃣ Write the differential equation: \( \frac{d^2x}{dt^2} + \omega^2 x = 0 \)

  5️⃣ General solution: \( x(t) = A \cos(\omega t + \phi) \)
• What are the two essential conditions for SHM?

  Restoring force ∝ displacement
• For SHM, the mathematical expression for the restoring force is F ∝ -x.
• What are the two essential conditions for an object to undergo Simple Harmonic Motion (SHM)?

  Restoring force proportional to displacement and Hooke's Law obedience
• The restoring force in SHM is proportional to the object's displacement from equilibrium.
• The restoring force in SHM follows Hooke's Law.


  True
• Match the feature with its description:

  Restoring Force ↔️ Pulls object back to equilibrium

  Other Forces ↔️ Can cause or change motion
• The velocity in SHM is the first derivative of displacement.
• What is the principle underlying energy conservation in SHM?

  Total mechanical energy remains constant
• What is the equation for potential energy in SHM?

  PE = \frac{1}{2}kx^2
• Steps in deriving SHM equations using Newton's Second Law:

  1️⃣ Apply Newton's Second Law: F = ma

  2️⃣ Substitute restoring force: -kx = ma

  3️⃣ Define angular frequency: ω^2 = k/m

  4️⃣ Write differential equation: d^2x/dt^2 + ω^2 x = 0

  5️⃣ General solution: x(t) = A cos(ωt + φ)
• Angular frequency in SHM is defined as ω^2 = k/m.


  True
• What is the general solution for displacement in SHM?

  x(t) = A cos(ωt + φ)
• What is the second-order differential equation for SHM?

  $\frac{d^{2}x}{dt^{2}} +$$\omega^{2} x =$$0$